

LIBRARY OF CONGRESS. 

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UNITED STATES OF 



MAY 28 1884 





































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MECHANICS’ AND ENGINEERS’ 


POCKET-BOOK 

OF 

TABLES, KULES, AND FORMULAS 

PERTAINING TO 

MECHANICS, MATHEMATICS, AND PHYSICS: 

INCLUDING 

AREAS, SQUARES, CUBES, AND ROOTS, ETC.; 

LOGARITHMS, HYDRAULICS, HYDRODYNAMICS, STEAM AND 
THE STEAM-ENGINE, NAVAL ARCHITECTURE, 
MASONRY, STEAM VESSELS, 

MILLS, ETC.; 

LIMES, MORTARS, CEMENTS, ETC.; 

ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS, ETC., ETC. 


Korty-fiftli Edition.. 


—— v - 

BY CHAS. H. HAS WELL, 

* \ 

CIVIL, MARINE, AND MECHANICAL ENGINEER, MEMBER OF THE AM. SOC. OF CIVIL ENGINEERS, OF 
ENGINEERS’ CLUB, PHILADELPHIA, OF THE N. Y. ACADEMY OF SCIENCES, OF THE 
INSTITUTION OF NAVAL ARCHITECTS, ENGLAND, CORRESPONDING MEMBER OF 
THE AMERICAN INSTITUTE OF ARCHITECTS, AND OF THE BOSTON 
SOCIETY OF CIVIL ENGINEERS, AND ASSOCIATE MEMBER OF 
THE N. Y. MICROSCOPICAL SOCIETY, ETC. 

ft f •* 

jr . t H *• v u 


An examination of facts is the foundation of science. 

-:-— i — ; , ; t | o i >* f- 





NEW YOItJ^: ; 

HARPER & BROTHERS, PTTBLL 

FRANK: LIN SQUARE. 

1 884 . 













i 


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l ‘ 







By CHAS. H. HASWELL, 

Civil, Marine, and Mechanical Engineer, M. Am. Soc. C. E. 


MENSURATION. 

For Tuition and Reference, containing Tables of Weights and Measures; Mensura¬ 
tion of Surfaces, Lines, and Solids, and Conic Sections, Centres of Gravity, &c. 
To which is added, Tables of the Areas of Circular Segments, Sines of a Circle, 
Circular and Semi-elliptical Arcs, &c. By Chas. H. Haswell, Civil and Marine 
Engineer, &c. Fifth Edition. 12mo, Sheep, 90 cents. 

MECHANICS’ TABLES. 

Containing Areas and Circumferences of Circles, Sides of Equal Square ; Circum¬ 
ferences of Angled Hoops, angled Outside and Inside ; Cutting of Boiler-Plates, 
Covering of Solids, &c., and Weights of various Metals, &c. Miscellaneous Notes, 
comprising Dimensions of Materials, Alloys, Paints, Lacquers, &c.; U. S. Tonnage 
Act, with Diagrams, &c. By Chas. H. Haswell, Civil and Marine Engineer, &c. 
Second Edition. 12mo, Cloth, 75 cents. 

Published by HARPER & BROTHERS, New York. 


Either of the above Works, or the Pocket-Book, will be sent by Mail, postage 
prepaid, on receipt of the price. 







Entered according to Act of Congress, in the year one thousand eight hundred and 

eighty-four, by 

HARPER & BROTHERS, 

In the Office of the Librarian of Congress, at Washington. 




INSCRIBED 


TO 


CAPTAIN JOHN ERICSSON, LL.D., 

AS A SLIGHT TRIBUTE TO HIS GENIUS AND ATTAINMENTS, 
AND IN TESTIMONY OF THE SINCERE REGARD 
AND ESTEEM OF HIS FRIEND, 


THE AUTHOR. 





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PREFACE 

To the Forty-fifth. Edition. 


The First Edition of this work, consisting of 284 pages, 
was submitted to the Mechanics and Engineers of the United 
States by one of their number in 1843, who designed it for 
a convenient reference to Rules, Results, and Tables con¬ 
nected with the discharge of their various duties. 

The Twenty-first Edition was published in 1867, consisted 
of 664 pages, and, in addition to the original design of the 
work, it was essayed to embrace some general information 
upon Mechanical and Physical subjects. 

The Tables of Areas and Circumferences of Circles have 
been extended, and together with those of Weights of Metals, 
Balls, Tubes, Pipes, etc., of this and some preceding editions 
were computed and verified by the author. 

This edition is a revision and an entire reconstruction of 
all preceding, embracing amended and much new matter, as 
Masonry, Strength of Girders, Floor Beams, Logarithms, etc., 
etc. 

To the young Mechanic and Engineer it is recommended 
to cultivate a knowledge of Physical Laws and to note re¬ 
sults of observations and of practice, without which eminence 
in his profession can never be attained; and if this work 
shall assist him in the attainment of these objects, one great 
purpose of the author will be well accomplished. 





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INDEX. 


A. Page 

Abutments and Arches. 604 

Acids. 188 

Adulteration in Metals, Proportion 

of, in a Compound . 216 

Aerodynamics. 614 

Aerometry, Course of Wind .675 

“ Distance of Audible Sounds 674 

“ Pneumatics .673-676 

“ Resistance of a Ptane Sur- 

“ face . 675 

“ Resistance to a SteamVessel 

in Air or Water. .911 

“ To Compute Height of a Col¬ 
umn of Mercury to Induce 

an Efflux of Air . 675 

u Velocity ,and Pressure of 

“ Wind .674, 911 

“ Volume of Air discharged 
“ Through an Opening, 

etc.674-676 

“ Weight of Air .675 

Aerostatics. 427-431 

11 Elevation by a Barometer 428 
“ “ by a Thermometer 429 

“ Velocity of and Sound... 428 
“ Velocity of Air flowing 

into a Vacuum. .428 

Ages of Animals. 192 

Air, and Steam . 737 

“ Atmospheric .431-432 

“ Consumption of. . 432 

“ Decrease of Temperature by Al¬ 
titudes . 522 

“ Expansion of . 520 

“ Flow of in Pipes .745,746,909 

“ Pressure and Resistance of . _648 

‘‘ Resistance of different Figures in 646 
“ Velocity Lost by a Projectile .... 648 
“ Volume and Weight of Vapor 

in .68, 69 

“ “ of and Gas in a Furnace. 760 

“ “ Pressure, and Density of.. 521 

“ “ Pressure, Temperature, 

and Density of . 522 

“ “ Required per Hour, etc.. 525 

Alcohol. 194 

“ Elastic Force of Vapor of.. 707 
“ Proportion of, in Liquors.. 204 

Ale and Beer Measures. 45 

Algebra, Symbols and Formulas..22, 23 

Alimentary Principles. 200 

Alligation ... 106 

Alloys and Compositions.634-637 

Almanac, Epacls and Dominical Let¬ 
ters, 1800 to 1901. 73 

Altitudes, Decrease of Temperature 

&]/. 522 

American Gauge.118, 120 

Analysis of Organic Substances.. 190 

“ of Foods and Fruits . 201 

“ of Meat, Fish, and Vegetables 200 


Page 

Anchors and Kedges. 174 

“ Cables, Chains, etc.173,174 

“ Diameter of a Chain Cable. 175 

“ Experiments on . 175 

“ Length of Chain Cables .... 175 

“ Number and Weight of.,.. 174 

“ Resistance to Dragging .. ,. 175 

Ancient and Scripture Lineal Meas¬ 
ures. 53 

il Weights. 53 

Angle and T Iron, Weight of. . 130 

Angles and Distances Corresponding 

to a Two-foot Rule . 160 

Angles, To Describe, etc. 222 

Angles, To Plot and Compute Chord of 359 

Animal and Human Sustenance. 203 

Animal Food.200-207 

“ Power....432-440 

“ “ Birds and Insects 438 

“ “ Camel . 438 

“ “ Crocodile . 438 

“ “ Day's Work. . 434 

“ “ Dog . 438 

“ “ Horse. 435,436,437,439,440 

“ ■“ Llama .438 

“ “ Men 433,434,438,439 

“ “ Mule and Ass . 437 

“ “ on Street Rails or 

Tramways . 

“ “ Ox . 438 

Animals, Proportion of Food for _205 

Annuities .no-m 

“ Amount of . m 

“ “ at Compound 

Interest. ... 111 

“ Present Worth . no 

Yearly Amount that will 

Liquidate a Debt . no 

Anti-attrition Metal.636 

Apartments, Buildings, Ventilation of 524 

Appendix.913 

Aqueducts, Roads, and Railroads ... 178 

Arc, To Describe .. .225, 227, 228 

Arches and Abutments. 604 

Arches and Walls.602 

Areas of Circles.231-236 

“ by Logarithms.. .236, 252 

‘ ‘ When Composed of an 

Integer and a Frac¬ 
tion . 236 

Area of a Circle, When Greater than 

any Contained in Tables .235, 252 

Areas of Segments of a Circle. .267-269 
“ of Circles, by Birmingham 

W.G .236 

“ “ Greater than in 

Table . 235 

“ of Zones of a Circle_269-271 

“ of Zones, To Compute . 271 

“ and Circumferences of Cir¬ 
cles by loths and i2ths. .243-257 









































































11 


INDEX, 


Page 

Apothecaries’ or Fluid Measure. 46 

44 Weight. 32 

Arithmetical Progression. 101 

Artesian Well.179, 198 

Ash. 482 

Asphalt.481, 689 

Asphalt Composition. 593 

“ Pavement . 690 

Ass. 437 

Atmosphere, To Compute Volume of 

Vapor in . 68 

Atmospheric Air. 431 

“ Proportion of Oxygen 

and Carbonic Acid at 
Various Locations .... 432 

“ Carbonic Acid Exhaled 

by Man .432 

Avoirdupois Weight. 32 

Axle, Compound. 627 

B. 

Babbitt’s Anti-attrition Metal. 636 

Baking of Meats, Loss by . 206 

Balances, Fraudulent. 65 

Balloon, Capacity and Diameter of.. 218 

Balls, Cast Iron and Lead. 153 

Balls, Lead, Weight and Dimension of 501 
Barometer, Elevations by Readings. 429 

“ Height of. . 429 

“ Indications ..429 

“ Weather Glasses . 430 

“ Weather Indications ... 431 

Barrel, Dimensions of . 30 

Beam or Girder Trusses. 823 

“ General Deduction 824 

Beams, Deflection of..770-777 

11 Dimensions of which a Struct¬ 
ure can Bear . 644 

44 Elements of Wrought Iron , 

Rolled . 807 

“ Elliptical Sided . 826 

“ Floor , Headers, Trimmers , 

etc.835-838 

“ Formula of Transverse Stress 801 
44 of Unsymmetrical Section , 

Neutral Axis and Strength 

of. ... 820 

44 or Girders , Moments of.. .621, 622 

44 Shearing Stress . 622 

Bearings for Propeller Shafts. 473 

Beet Root and Beet Root Sugar. 207 

Beeves and Beef, Comparative Weights 

of. . 35 

Beils, Weight of. . 180 

Belt, Equivalent, and Wire Rope.... 167 

Belting. 907 

Belts and Belting.441-443 

“ Width of . 441 

Bench Marks. 85 

Beton or Concrete. 593 

Birds. 440 

“ and Insects .196, 438 

Bissextile or Leap Year. 70 

Black and Galvanized Sheet Iron.... 129 
Black and Galvanized Sheet Iron, 

Weight of . 129 

Blast Furnace. 529 


Page 

Blast Furnace, Pipe of a Locomotive. 907 

Blasting. 443 

“ Boring Holes in Granite .. 444 
“ Charge of Gunpowder for.. 444 

“ Effects . 444 

“ Weight of Explosive Mate¬ 
rials in Holes . 444 

Blasts and Draughts, Effects of. . 746 

Blower and Exhausting Force. 898 

Blowers, Fan. 447 

Blowing Engines.445, 898 

“ Memoranda . 448 

44 Power of | etc. 446 

“ Pressure of Blast . 447 

“ Root's Rotary . 449 

“ To Compute Dimensions of 

a Driving Engine... 446 

“ “ Elements of. . 447 

“ “ 44 of a Fan-Blower 448 

“ “ Power of a Centrifugal 

Fan . 448 

“ 44 Volume* of Air trans¬ 
mitted . 447 

Blowing Oef. 726 

Board and Timber Measure. 61 

Boiler, Steam . 739 

“ and Ship Plates . 828 

“ Areas and Ratio of Grate and 

Heating Surface , etc. 741 

“ Draught .739,744,745 

“ “ and Blasts , Compara¬ 
tive Effect of. . 746 

“ 44 Velocity of . 746 

“ EvaporativeCapacityof Tubes 742 

“ Evaporation , Effects of for 

Different Rates of Combus¬ 
tion . 743 

“ Evaporation, Power of._ _757 

“ Fuel that may be Consumed.. 742 

“ Heating Surfaces .740 

“ Loss of Pressure by Flciv of 

Air in Pipes . 745 

‘ 4 Minimum Fuel Consumed per 

Square Foot of Grate . 740 

44 Power . 760 

44 Rate of Combustion . 760 

“ Relation of Grate , Heating 

Surface , and Fuel . 741 

44 Result of Experiments with a 

Steam Jet . 746 

‘ 4 Results of Operation of . 743 

44 “ of Operation of Vari¬ 
ous Designs of Boiler . 744 

44 Riveting . 907 

“ Safety Valves . 746 

‘ 4 Steam .739-745, 829 

44 Steam Heating. .. 526 

44 Steam Room . 748 

44 Volume of Furnace Gas per 

Lb. of Coal . 760 

44 Weights of . 759 

Boiling of Meats, Loss by . 206 

Boiling-Points . 5I y 

Bolts and Nuts, Dimensions and 

Weights of -156. 157 

English Standard... 158 
44 French Standard... 158 



























































































INDEX. 


Ill 


Page 


Bolts and Nuts, Square Heads . 159 

“ Tenacity of . 198 

“ Wrought-iron, Experiments on. 783 

“ and Plates . 749-757 

Boring and Turning Metal. 197 

“ Instruments, Tempering of... 197 

“ Wells. 197 

Brain, Weights of. .. 192 

Brass, Sheet, Weight of. . 142 

“ Plates, Weight of. .118,119, 146 

“ Wire, Weight of. .120, 121 

Brass. 636 

“ Castings, Weight of.. .,. 155 

“ Tubes, Weight of.... . 142 

“ Weight of. .136, 149 

Braziers’ and Sheathing Sheets. 155 

Bread. 207 

Breakwaters. 181 

Breast-wheel. 568 

Brick Walls. 603 

Brickwork.597, 801 

Brick or Compressed Fuel.907 

Bricks.598, 599 

“ Crushing Resistance of. .908 

“ Volume of and Number in a 

Cube Foot of Masonry .599 

Bridge, Britannia Tubular.178 

“ Highest . 907 

“ Iron . 178 

“ New York, Erie, and West¬ 
ern Railroad . 178 

“ New York and Brooklyn ... 178 

“ Suspension . 842 

Bridge Plates and Rivets. 830 

Bridges. 178 

“ Lengths and Spans of . . 181 

“ Resistance of. . 645 

“ Suspension, Length of Span 

of . 199 

“ Suspension .178,842 

Bridles or Stirrups, for Beams . 838 

British and Metric Measures, Com¬ 
mercial Equivalents of . 906 

Broccoli. 207 

Bronze. 637 

“ Malleable . 907 

Browning or Bronzing Liquid. 874 

Builders’ Measure. 46 

Building Department, T&jq'iaraneihsq/! 907 
Building Stones, Expansion and Con¬ 
traction of. . 184 

Buildings, Walls of. . 189 

‘ ‘ Protection of. .907 

Buoyancy of Casks. 192 

Burns and Stings, Application for... 196 

Buttress. 696 

C. 

Cabbage. 207 

Cables, Chain, Weight and Strength of 168 

“ Chain, Breaking Strain and 

Proof of. . 169 

“ Circumference of. . 171 

“ Galvanized Steel . 163 

“ Strength of . .168,170 

Cables, Ropes, Hawsers, Anchors, 

. and Chains.163-175 


Page 

Calculus. 24 

Calendar, Ecclesiastical.... 70 

“ Gregorian or N. S.70, 71 

Caloric .504, 614 

Caloric Engine, Ericsson’s. 903 

Canal, Suez, Wia . 912 

Canals. 181 

“ Flow of Water in . 550 

“ Locks .183,553-555 

“ Power of a Horse . 848 

“ Traction on . 848 

“ Transportation of. . 193 

Caudles, Gas, Light of etc .583, 584 

Cannon Ball, Flight of. . 495 

Capillary Tube. . . 358 

Cargoes, To Ascertain Weight of. 176, 177 
“ Units for Measurement of .. 176 

Carrot. 207 

Cascades and Waterfalls. 184 

Case Hardening.644, 786 

Cask Gauging. 377 

Casks, Buoyancy of. . 192 

“ Ullage . 378 

Cast Iron.637, 765, 783, 784 

u and Lead Balls, Weight of. 153 
“ Balls, Weight and Diam¬ 
eter of . . 153 

“ Bars, Experiments on.... 780 
“ Crushing Weight of Col¬ 
umns . 768 

“ Columns, Weight Borne 

Safely . 768 

“ Pipes, Weight of. .132, 133 

“ Plates, Weight of. . 146 

“ To Compute Weight of a 

Bar or Rod . 131 

“ Weight of. , .136, 155 

Castings, Shrinkage of. . 218 

“ Weight of, by Pattern . 217 

Catenary, To Describe . 230 

Cathedral, St. Peter’s.. 179 

Cattle and Horses, Transportation of. 192 

“ Weight of ... 35 

Cauliflower. 207 

Cement. 907 

Cements.589, 871-873 

Cements, Limes, Mortars, and Con¬ 
cretes .588-597 

Central Forces.449-454 

Centre of Gravity.605-608 

“ Of a Vessel and Dis¬ 
placement .653, 658 

“ To Ascertain Centre 

“ of. . 605 

Centre of Gyration .609-6x1 

“ “ To Compute Ele¬ 

ments and Cen¬ 
tre of. .610, 611 

Centres of ■ Oscillation and Per¬ 
cussion .612-614 

“ Centre of. in Bodies of Va¬ 
rious Figures.. .613 

“ Centre of, To Compute. .612-614 
Centrifugal Fan, Elements and Power 

of a Fan Blower , etc.448 

Centrifugal Pump. 579 

Chain, To Set out a Right Angle with. 69 








































































































IV 


INDEX. 


Page 

Chain Cables, Breaking Strain and 

Proof of . . 169 

Chaining over an Elevation. 69 

Chains and Ropes. ... 457 

“ Anchors, etc., Diameter and 
Length for a Giv&h Weight 

of Anchor . 175 

Characters and Symbols. 21 

Charcoal..194, 480 

“ Produce of. . . .481 

Cheese, Composition of. . 205 

Chemical Composition of some Com¬ 
pound Substances .461 

Chemical Formula, To Convert . 190 

Chimney Draught.. 907 

Chimneys.179, 180, 904 

“ and Smoke Pipes .748, 749 

Chinese Wall. 179 

Chinese Windlass.627 

“ or Indian Ink.903 

Chronological Eras and Cycles. .. 26 

Chronology.70-74 

“ To Ascertain Years of 

Coincidence . 74 

Churches and Opera-Houses. 180 

Circular Arcs, Length of. .260-262 

Circular Measure...113,114 

“ Motion..... 618 

Circulating Pumps. 749 

Circles, Areas of.231-236 

“ “ by Logarithms.... 236 

“ “ by Wire Gauge... 236 

“ Sides of a Square of... .258-259 

Circumferences of Circles.237-242 

“ “ by Logarithms. 242 

“ “ by Wire Gauge 242 

Cisterns and Wells, Excavation of... 63 

“ “ Capacity of ..... 63 

Civil Day.37, 70 

Civil Year. 70 

Cloth Measure. 27 

Clouds. 430 

Coal, Anthracite. 480 

“ Average Composition of Heat 

of Combustion and Evapora- 

rative Power of. .486 

“ Bituminous . 479 

“ “ Caking Splint or Hard 

Cherry or Soft . 479 

“ “ Cannel . 479 

“ “ Chemical Composition , 

Varieties of. .479 

“ Consumption of to Heat 100 

Feet of Pipe . 527 

“ Effective Value of . 908 

“ Elements of Various .480 

“ Fields, Areas of. . 191 

' 1 Japan . 9°9 

“ Measure .1. 46 

Coast and Bay Service.908 

Cocks, Composition, and Copper 

Pipes. 150 

Cohesion... 614 

Coins, British Standards . 38 

il To Convert U. S- to British 

Currency . 39 

“ U. S., Weight and Fineness of. 38 


Page 

Coins, Values of. . 39 

Weight , Fineness , and Mint 

Value of Foreign . 39 

Coke. 480 

Cold, Extremes of in Various Coun¬ 
tries . 191 

“ Greatest..... .908 

College, Oxford,. 179 

Collision or Impact.580-582 

Color Blindness. 195 

Colors for Drawings. 196 

Colors, Proportion of for Paints _ 66 

Columns. 180 

Combination. 112 

Combustion....458-466 

‘ ‘ Composition and Equi va- 

lents of Gases _ ..... 460 

“ Chemical Composition of 

some Combustibles .461 

“ Evaporative Power of 1 

Lb. of a given Combus¬ 
tible .462 

“ Heat of .463 

“ Heating Powers of Com¬ 
bustibles .461, 462 

“ Of Fuel .463-465 

“ Rate of. . 760 

“ Relative Evaporation of 

Combustibles... 465 

“ “ Volumes of Gases 

or Products of per Lb. 

of Fuel . 465 

“ Temperature of. . 462 

“ To Compute Consumption 

of Fuel .446 

“ “ Volume of Air 

Chemically Consumed 
in Complete Combustion 

of 1 Lb. of Coal . 459 

“ Volume of Air Required. 465 

1 ‘ Weight and Specific Heats 

of Products of Combus¬ 
tion , etc. 462 

Compass, Degrees, etc., of Each Point 54 

Composition and Alloys.634-637 

Compound Axle or Chinese Windlass 627 

Compound Interest. 108 

Compound Proportion. 95 

Concretes, Limes, Mortars, and 

Cements.588-597 

Concrete, Cements, and Mortar.595 

“ or Belon . 593 

Cones. 353 

Conic Sections.380-384 

“ Conoid . 380 

“ Ellipse or Hyperbola , To De¬ 
termine Parameter of.. .380, 381 
“ Ellipse, To Compute Area of 

Segment of . 382 

“ Hyperbola .383 

‘ ‘ To Compute Abscissae.. 383 

“ “ Area of .384 

“ “ Diameters.... 383,384 

“ “ Lenqth of any Arc 384 

“ Parabola . . 382 

“ “ To Compute Area of ... 383 

“ “ Ordinate or Abscissa 382 























































































INDEX. 


V 


«( 

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u 


Page 

Contractility. 614 

Copper. 750 

“ Braziers’ and Sheathing.... 131 
“ Plates, Weight of... 118, 119, 146 

‘‘ Pipes. . 150 

“ Rods or Bolts, and Pipes, 

Weight of. . 148 

Sheet, Weight of. . 135 

Tubes, Weight of .140, 144 

Weight of.....'. .136, 155 

“ Wire, Weight of . 120, 121 

“ Wire Cord . 123 

Copying. 29 

Cord, Copper Wire . 123 

Cordage, Friction and Rigidity of.. 472 

Corn Measure. 198 

Corrosion of Iron Steel..908 

Corrugated Iron Roof Plates, Weightof 131 

Co-secants and Secants....403-414 

“ “ To Compute, etc. 414 

Cosines and Sines..390-402 

“ “ To Compute, etc. 401, 402 

Cost, of Family of Mechanics in 

France . 908 

Co-tangents and Tangents.416-426 

“ “ To Compute, etc. 426 

Cotton Factories. 899 

Couple.. 614 

Couplings of Shafts .'.. 796 

Coursing and Leaping. 440 

Crane, Steam Dredging . 899 

‘ ‘ Wood . 900 

Cranes.179, 433, 455-457 

“ Machinery of. . 457 

“ To Compute Dimensions of 

Post . 456 

“ “ Stress on .455 

“ “ Stress upon Strut .... 451 

Crank. 433 

“ Turning . 433 

Cream, Percentage of, in Milk . 205 

Crocodile. 438 

Crops, Mineral Constituents Absorbed 

by . 189 

Croton Aqueduct. 178 

Crushing Strength.764-769 

Cube Measures. 30 

Cube Root, To Extract . 97 

Cubes, Squares, and Roots.272-302 

“ “ To Compute and to As¬ 
certain, etc.300-302 

Cucumber. 207 

Currency, To Convert U. S. to British 39 

Current Wheel. 570 

Curvature and Refraction. 55 

Cut Nails, Tacks, Spikes, etc. 154 

Cutters (Yachts) . 895 

Cycle, Dominical or Sunday Letter.. 70 

“ Lunar or Golden Number - 71 

“• Of Sun . 70 

Cycloid, To Describe . 228 

Cyclones. 675 

Cylindrical Flues and Tubes, Hollow 827 

D. 

Dams, Embankments, and Walls . 700-703 
Day, Astronomical . 7 ° 


u 

u 

u 

u 

u 

u 

u 

u 


87 

88 


86 


Page 

Day, Marine or Sea . 37 

“ Sidereal . 37 

Solar and Civil . 37 , 7 ° 

Day's Work . 434 

Decimals. 92-94 

Deer Park, Copenhagen . 179 

Deflection.770-781 

Delta Metal . 384 

Departures, Table of . 54 

Desiccation. 513 

Detrusive or Shearing Strength, 

782, 783 

Dew Point, and To Ascertain . 68 

Diamond Weight . 32 

Diamonds, Weight of . 193 

Diet, Daily , of a Man .202, 207 

Digestion of Food. 206 

Dip of Horizon . 60 

Discount or Rebate... 109 

Displacement of a Vessel. 653 

Distances, Steaming. 86 

Between Cities of U. S. ..... 184 

East and West . 187 

Principal Ports of 
World. 

“ of U.S... 
Various Ports of Eng¬ 
land, Canada, and 

U.S. . 

Distances and Angles Corresponding 

to a Two foot Rule . 160 

Distances and Geographic Levelling.. 56 

“ . “ Measures.. 54 

Distemper . 593 

Distillation. 514 

Fog .438, 440 

Domes and Towers.179, 180 

Dominical Letters, and Epacts . 73 

“ or Sunday Letter . 70 

Drainage of Lands. 691 

Drains, Diameter and Grade of, to 

Discharge Rainfall . 906 

Draught.739, 744, 746 

Drawing and Tracing Paper . 29 

“ or Pushing . 433 

Drawings, Colors for . 196 

‘‘ Dimensions of, for U. S. 

Patents . 198 

Dredger and Hopper Barge .899, 900 

Dredging, and Cost of. . 197 

‘ ‘ Machines . 899 

Drilling.445, 477 

Drowning Persons, Treatment of... 187 

Dry Measure.30, 31 

Dualin. 503 

Duodecimals. 94 

Dynamite. 443 

“ Cellulose . 444 

Dynamics.616-620 

“ Circular Motion . 618 

‘ 1 Decomposition of Forces... 620 

“ Motion on an Inclined 

Plane . 619 

“ Uniform Motion .617, 618 

“ Work Accumulated in Mov¬ 
ing Bodies . 619 

“ “ By Percussive Force . 620 






































































































VI 


INDEX, 


E. Page 

Earth.188, 198 

“ Area and Population of ..... 188 

“ Elements of Figure of . . 61 

“ Motion of . 70 

‘ 1 Pressure of. .695 

“ Weight of per Cube Yard .... 468 

“ Weights of . 33 

Earthwork.467, 468 

“ Bullc of Rock . 468 

‘ ‘ jS'umber of Loads and Vol¬ 
ume of per Day . 908 

“ Shovelling . 908 

“ Volume of Transported 

per Day . 908 

Easter Day . 71 

Ecclesiastical Year . 70 

Egyptian and Hebrew Measures . 53 

Elastic Fluids, Specific Gravity of.. 215 

Elasticity. 195, 614 

“ Coefficients of . . 761 

“ Modulus of. . 762 

‘ ‘ Relative, of Metals . 780 

Electric and Gas Light. 198 

“ Light, Candle-Power of.... 908 

Electrical Weights . 34 

Elementary Bodies. 190 

Elevation by a Barometer ... 428 

Elevations and Heights of Various 

Places . 183 

Elliptic Arcs, Length of. .263-266 

“ “ To Ascertain Length 

of. . 266 

Ellipse, To Describe , etc.226, 380 

Embankment and Excavation.466 

“ Walls and Dams. .700-703 

“ Weight of a Cube Foot 

of Materials . 694 

Endless Ropes. 167 

Engines and Machines. 898 

“ Elements of . 898 

“ and Sugar-mills, Weights of 908 
Ensigns, Pennants, and Flags, U. S.. 199 
Epacts, and Dominical Letters. ... 73 

Equation of Payments. 109 

Equilibrium, Angles of. .694 

“ Of Forces . 616 

Ericsson’s Caloric. 903 

Establishment of the Port for Several 

Locations . 85 

Ether, Elastic Force of Vapor . 707 

Evaporation of Water . 514 

Evaporative Power of Tubes . 513 

Evolution. 96 

Excavation and Embankment of 

Earth and Rock.. . 192 

Exhausting Fan and Blower. 898 

Expansion. 614 

F. 

Fan Blowers. 447 

Fascines . 690 

Fellowship. 99 

Fence Wire . 164 

Fig .' 207 

Filter Beds . 851 

Filtering Stone .909 

Filters for Waterworks . 184 


Page 

Fire Bricks . 600 

“ Clay . 597 

Fire-Engine, Steam .904, 909 

Fish, Meat , and Vegetables, Analysis 

of. . 200 

Flags, Ensigns, and Pennants, U. S.. 199 

Flax Mill. 476 

Floating Bodies, Velocity of. .909 

Flood Wave. 912 

Floors and Loads, Factor of Safety. 841 

“ “ Weight of. . 841 

Flour.. 207 

Mills . 900 

Flues and Tubes.747, 827 

“ Corrugated .909 

“ or Furnaces . 756 

Fluid Measure.. .30, 46 

Fluids, Impact and Resistance of.. 577 

“ Lamp and Gas . 584 

“ Percussion of . 579 

Flutter Wheel. 571 

Fluxes for Soldering or Welding.... 636 

Fly Wheel. 451 

Flying of Birds . 440 

Food, Animal .200-207 

“ Digestion of .... 206 

“ Nutritive Constituents and Val¬ 
ues of... . 202 

“ Nutritive Equivalents of. . 205 

“ Proportion Expended by Ani¬ 
mals . 205 

Foods, Analysis of ...201, 203 

“ Sutritious Properties of ..... 204 
“ Relative Values of.. .202, 203, 204 
“ Thermometrical Powers and 

Mechanical Energy of..... 205 
Forces, Composition and Resolu¬ 
tion of... 615 

“ Division of.. . 614 

u Equilibrium of. . 616 

“ Percussive . 620 

“ Decomposition of. .620 

Foreign Measures and Weights. .48-52 

Fortress Monroe . 179 

Foundation Piles.198, 909 

Fractions.89-91 

Fraudulent Balances. 65 

Freeboard . 666 

Friction.469-478, 571, 662 

“ and Rigidity of Cordage... 472 

“ Application of Results .474 

“ Bearings of Propeller Shafts 473 

“ Coefficients of Axle .471 

“ “ of Motion . 470 

“ “ of, of Journals. 470 

“ “ To Determine.. 471 

“ Frictional Resistances .475 

“ Grain Conveyers . 478 

“ Launching Vessels . 478 

“ Mechanical Effect of To 

Compute . 47! 

i; of Bottom of Vessels .gog 

“ of Pivots .472 

“ of Planed Surfaces .gog 

“ of Steam-Engines and Ma¬ 
chinery .475 

“ of Walls and Earth .698 









































































































INDEX. 


Vll 


Page 

Friction, Relative Value of Angles.. 472 

“ Results of Experiments. 474, 475 

“ Rolling .. 473 

“ Steam-engine . 478 

“ Steamers . 478 

Tools . 476 

“ Value of Unguents . 471 

“ Wood-sawing . 477 

Frictional Resistances... 475 

“ of Machinery, Results of 

Experiments upon.. .475-478 

Frigorific Mixtures.193, 516 

Fruits, Analysis of . 201 

“ Proportion of Acid and Sugar 203 

Fuel ...479-487, 513 

Area of Grate and Consump¬ 
tion of. . 5x3 

Ash . . 482 

Average Composition of... 485, 486 

Brick or Compressed . 90 

Elements of. . 486 

Lignite . 481 

Liquid . 484 

Miscellaneous .487 

“ Peat . 482 

“ Produce of Charcoal . 481 

“ Relative Values of . 483 

“ Tan . 482 

“ Values , Weights , and Evapora¬ 
tive Power of. . 483 

“ Wood., ... 481 

Furnaces.528, 756 

Fusible Compounds. 634 

G. 


u 

u 

ii 

it 

it 

it 

it 


Galvanized Sheet Iron, Thickness 


and Weight of. . 124 

Charcoal Iron . 163 

“ Iron Wire Rope . 162 

“ Steel Cables . 163 

Gas.585-587 

“ and Electric Light . 198 

“ Atmospheric Engine . 587 

“ Coal . 484 

“ Candles, etc..». 584 

“ Engines . 587 

‘ ‘ Flow of ... 586 

“ Mains, Dimensions of . 587 

“ Pipes. .138,160,586 

“ Pipes, Thickness of. . 123 

“ Steam and Hot-air Engines .909 

“ Temperature of. . 587 

“ Tubing, and Number of Burners, 

Regulation of . 586 

“ Volume of. .586, 587 

“ Volume of Furnace per Lb. of 

Coal . 760 

“ Weight of a Cube Foot of. . 215 

Gases, Expansion of . 520 

Gauges , Wire.. .118-123 

Gauging Cask... . 377 

Geographic Levelling by Boiling-Point 55 
Geographic Measures and Dis¬ 
tances. . .... .54, 56 

Geographical and Nautical Meas¬ 
ures. 26 

Geometrical Progression.103-105 


Page 

Geometry. 219^230 

“ Angles . 222 

“ Arcs .227 

“ Catenary ... 230 

‘ ‘ Circles . 224 

“ Cycloid and Epicycloid .. 228 

“ Ellipse . 226 

“ Hexagon. . . 223 

“ Hyperbola . 230 

“ Involute . 229 

“ Length of Elements . 221 

u Lines .221, 222 

“ Octagon . 223 

“ Parabola . 229 

“ Polygon . 223 

“ Rectilineal Figures....... 222 

“ Spiral . 230 

Geostatics and Geodynamics...._614 

Gestation, Periods of. . 192 

Girder, Beam, etc., General Deduction, 824 

Girders, Beams, Lintels, etc. 822 

Girders and Beams ...805, 806 

“ “ Centre of and Ver¬ 

tical Distance of 
Centres of Crush¬ 
ing and Tensile 

Stress . 819 

££ “ Deflection of. . 840 

“ “ Dimensions and 

Load of. _839, 840 

“ “ Factors of Safety. 

821, 841 

“ “ Moment of Stress 

of. .621, 623 

11 “ Trussed . 823 

“ “ Tubular . 775 

Glass Globes and Cylinders. 831 

Glass, Window . 124 

Glazing. 197 

Glues... 874 

Gold Sheet, Thickness of . 119 

Golden Number or Lunar Cycle... 71 

Gooseberry .. 207 

Governors. 452 

Grade, Reduction of, to Degrees.... 359 

Grain, and Roots, Weights of. . 34 

“ Conveyers . 478 

‘ ‘ Weights per Bushel . 32 

Graphic Delineation op Stress, with 

a Uniform Load , etc. 623 

“ Operation, Solution of Ques¬ 
tions by . 905 

Gravel.. 690 

“ or Earth Roads . 688 

Gravitation....487-496 

“ Accelerated and Retard¬ 
ed Motion . 494 

££ Average Velocity of a 

Moving Body . 495 

“ Formulas to Determine 

“ the Various Ele¬ 
ments .490 

££ ££ ££ of Retarded 

Motion . 492 

“ General Formulas for 

Accelerating and Re¬ 
tarding Forces .495 




































































































Vlll 


INDEX, 


Pftge 

Gravitation, Gravity and Motion on 

an Inclined Plane. 492,493 

‘ ‘ Inclined Plane .493, 494 

“ Miscell. Illustrations ... 496 

“ Promiscuous Examples. 489 

‘ ‘ Relat ion of Time, Space , 

and Velocities ... .488-491 
“ Retarded Motion —492,496 

“ /Space.489 

“ To Compute Action of.. 488 

“ “ Velocity of a Fall¬ 

ing Stream of 

Water . 496 

“ “ Action of by a 

Body Projected Up¬ 
ward or Downward.. 490 

“ Velocity . 489 

“ “ due to a Given 

Height of Fall and 
Height due to Given 

Velocity . 488 

Gravity of Bodies. 208 

‘ ‘ at Various Locations at Level 

of Sea . 487 

“ Centre of . 605 

“ Various Formulas for .488 

Grecian Measures and Weights . 53 

Gregorian Calendar ...70, 71 

Grindstones . 478 

Grouting .593, 594, 598 

Gudgeons and Shafts.790, 797 

Gun Barrels, Length of. . 198 

Gun Cotton .443 

Gun Metal, Weight of .136, 146, 149 

Gunnery.497-503 

• ‘ Charge, Range, and Veloci¬ 
ty, To Compute .497, 499 

“ Comparison of Forces of a 

Charge in Various Arms. 502 
‘ ‘ Experiments with Ordnance. 

498, 500 

“ Lead Balls, Weight and Di¬ 
mensions of. . 501 

‘ 1 Number of Percussion Caps 

corresponding to B Gauge. 502 
“ Penetration of Lead Balls 

in Small Arms . 500 

“ Penetration of Shot and 

Shell, etc.498, 500 

“ Proportion,Powder to Shot. 502 

“ Report of Board of Engi¬ 
neers, U.S. A., for Fortifi¬ 
cations, etc. 499 

‘ ‘ Velocity and Ranges of Sh ot 

and Shells . 498 

“ Windage and Waddings .. 501 

Gunpowder.443, 502 

“ Charge of .444 

“ Heat and Explosive Pow¬ 
er of etc. 503 

“ Proof of . 402 

“ Properties and Results of 503 

“ Relative Strength of for 

Use under Water -'503 

Gunter’s Chain. 26 

Gyration, Centre of. .609-611 

‘ ‘ Centre of of a Water-wheel 611 


H. Page 

Hammers Steam. 179 

Hancock Inspirator. 901 

Hawsers, Wire, and Ropes and Ca¬ 
bles, Comparison of . 169 

Hawsers, Ropes, and Cables—169-172 
“ “ Strength and Cir- 

cumf irence of .. 171 

“ “ Weight of. . 172 

Hay and Straw. 198 

Heat. 504-529 

“ Available Expended .909 

“ Capacity for .505,507 

“ Communication . 515 

“ “ and Transmission of... 510 

“ Condensation . 515 

“ Conduction or Convection of... 514 
“ Congelation and Liquefaction.. 516 
“ Decrease of Temperature by Al¬ 
titudes . 522 

“ Degrees of Fahrenheit to Reau¬ 
mur and Centigrade, and Con¬ 
trariwise . 523 

“ Desiccation . 513 

“ Distillation . 514 

“ Effect upon Various Bodies . 518 

“ Evaporation .512, 513, 5x4 

“ Expansion and Dilatation of a 

Bar or Prism.. .. 519 

‘ ‘ Expansion of Water ..519, 520 

“ “ of Liquids, Gases , and 

Air . 520 

“ Extremes of in Various Coun¬ 
tries ... 191 

“ Frigorific Mixtures .193,516 

‘ 1 Heating and Evaporating Water 

by Steam . 511 

“ Latent .508.509 

“ Latent, of Fusion . 509 

“ Latent, of Steam, To Compute.. 707 
“ Mean Temperatures of Various 

Localities .... 192 

“ Mechanical Equivalent of, in 

Steam . 705 

Melting and Boiling Points .... 517 

“ OfSun... . 193 

u Perpetual Congelation or Snow 

Line .. 192 

“ Proper Temperature of Enclosed 

Spaces .. 526 

“ Quantities of, Transmitted f rom 
Water to Water through Plates, 

etc. 51 x 

“ Radiation of . . 50^ 

‘ 1 Radiating and Absorbing Power 

of Various Bodies .... . 510 

“ Reduction of by Surf aces . 525 

“ Reflection . 5 IO 

“ Refrigerator, To Compute Sur¬ 
face of .512 

“ Relative Power of Various Sub¬ 
stances .. 5x0 

“ Sensible . ^07 

Specific .505,506, 507 

“ Temperature by Agitation .524 

“ Temperature of a Mixture of 
Like and Unlike Substances, 

To Compute . 506 









































































INDEX. 


IX 


Page 

Heat, To Compute Volume or Pressure Hydraulics 
of a Constant Weight of Air 
or other Gas , etc., for a Given 

Temperature . 522 

“ To Reduce Degrees of Different 

Scales . 523 

“ Transmission of through Glass . 511 
“ Underground Temperature .... 519 
“ Volume , Pressure , and Density 
of Air at Various Tempera¬ 
tures .. 521 

Heating, Air, Length of Pipe Required 525 

by Hot Water . 524 

Hebrew and Egyptian Measures and 

Weights. 53 

Height and Elevation of Various 

Places . 183 

Heights, Corresponding to Boiling 

Points of Water . 519 

Hemp and Wire Rope, Circumference 
of for Rig¬ 
ging . 172 

“ “ General Notes 167 

“ “ Relative Di¬ 

mensions of 172 
“ “ Weight and 

Strength of 172 
“ “ Weight of... 166 

Hemp Rope, Iron and Steel. 164 

“ and Iron Wire . 168 

“ Circumference of . 169 

“ Destructive Strength of. 171 

Iron and Steel , Relative 

Dimensions of . 168 

High Water, Time of... .74, 75 

Hills or Plants in an Acre . 193 

Hoggin. 690 

Hoisting Engines.901 

Honey.. 207 

Hoop Iron, Weight of. .129, 131 

Hopper Barge and Dredger.899, 900 

Horizon, Dip of. . 60 

Horizontal Wheels. 572 

Horse. 436 

Horse-Power.441, 733, 758 

“ Tractive Power of.... 436 

“ Transmission of. . 188 

Horses, Age of. . 186 

“ Labor of etc. 435-437 

“ Performance of. .439, 440 

u Transportation of... . 192 

“ Weight of. . 35 

Horseshoe Nails, Length of. . 153 

Horseshoes and Spikes. 152 

Hulls of Vessels, Diameter of Rivets. 830 

Human and Animal Sustenance. 203 

Hydraulic Ram... 561 

“ Cement. 591 

Hydrostatic Press..561, 901 

Hydraulics.529-557 

“ Canal Locks . 553~555 

“ Circular Sluices, etc.... 537 

li Coefficients of Circular 

Openings or 

Sluices . 536 

“ u In Clack or Trap 

Valve or Cock.. 546 

A* 


Page 

, Coefffits of Friction. 544-546 
u of Resistance in 
Bent or Angular 
Circular Pipes, 
Valve Gates , or 
Slide Valves ... 545 
Computation of Volume 

of Discharge . 533 

Curvatures , Radii of... 544 

Curves and Bends . 545 

“• Coefficients of Re¬ 
sistance . 545 

Cylindrical Ajutage .... 549 
Discharge from a Notch. 541 
l .‘ from Conduits or 

Pipe* . 530 

“ from Vessels not 
Receiving any 

Supply . 538 

“ from Vessels of 

Communication 541 
“ from Irregular- 

shaped Vessels. 542 
11 of Water in Pipes 
for any Length 
and Head , etc., 

547 , 548 

“ or Effluxfrom Va¬ 
rious Openings 
and Apertures. 532 
“ under Variable 

Pressures . 540 

Experiments on Dis¬ 
charge of Fluids, from 
Reservoirs , Conduits, 

or Pipes ..529, 531 

Flow and Velocity in 
Rivers , Canals, and 

Streams .550, 552 

“ in Lined Channels. 551 
“ of Water in Beds .. 542 
Forms of Sections of 

Canals . 543 

Friction in Pipes and 

Sewers .543, 544 

“ of Liquids . 531 

Height of a Jet , To Com¬ 
pute . 913 

Jets d'Eau . 550 

Miner's and Water Inch 557 
Miscel. Illustrations. 556, 557 

Prismatic Vessels . 539 

Rectangular Weir.. .532-535 
Reservoirs or Cisterns .. 541 
Sluice Weirs or Sluices.. 535 
Submerged or Drowned 
Orifices and Weirs.. 553 
To Compute Depth of Flow 

over a Sill , etc. ... 534 
“ Fall of a Canal or 
Conduit to Conduct 
and Discharge a 
Given Quantity of 
Water per Second. 914 
“ Head and Discharge 
of Water in Pipes 
of Great Length... 914 























































X 


INDEX, 


Page 

Hydraulics, To Compute Head or 
Height of Water 
from Surface of 
Supply to Centre 

of Discharge . 544 

“ “ Vertical Height of a 

Stream Projected 

from a Pipe . 549 

“ Volume of Water 
flowing in a Div¬ 
er . 543 

“ Short Tubes, Mouth¬ 

pieces, and Cylindri¬ 
cal Prolongations or 

Ajutages . 536-537 

‘ ‘ Triangular, Trapezoid¬ 

al, Prismatic Wedges, 
Sluices, Slits, etc.... 538 

u Variable Motion . 543 

“ Velocity of Water or of 

Fluids . 531 

“ Vena Contractu . 529 

“ Weirs or Notches . 539 

Hydrodynamics.. ..558-580 


U 

a 

a 

u 

u 

u 

u 

u 


u 
<( 
<< 

a 

u 

u 

u 

u 

a 

u 

u 

u 

u 

a 
a 
11 
u 


u 

a 

a 

u 

u 


u 

u 


Barker’’s Mill . 577 

Boyden Turbine _ 574 

Breast Wheel - 568-570 

Centrifugal Pump . . 579 

Current Wheel . 570 

Flutter Wheel . 571 

Fontaine Turbine.. . 574 
Friction of Journals 

or Gudgeons .. 571 

Horizontal Wheels.. 572 
Hydrostatic Press... 561 
Hydraulic Ram , 

561, 562 

Impact and Reaction 

Wheels . 576 

Impulse and Resist¬ 
ance of Iluids.577, 578 

Jonval Turbine . 575 

Memoranda . 571 


Overshot Wheel .. 563-566 
Percussion of Fluids. 579 
Pipes , Elements of .. 561 
u To Compute 
Thickness of etc... 560 
PonceleVs Wheel .... 567 
Pressure and Centre 


Of .558-560 

Reaction Wheel .576 


Tangential Wheel ... 576 

Tremont Turbine _ 576 

Turbine and Water 
Wheels, Compari¬ 
son Between . 579 

Turbines.. .572-576 

Undershot Wheel _ 566 

Victor Turbine ..... 576 

Water Power. . 562 

Water-Pressure En¬ 
gine . 579 

Water Wheels . $63 

“ Diameter and 
Journal of a 
Shaft, etc.... 571 


Page 


Hydrodynamics, Water-wheels, Di¬ 
mensions of 

Arms . 571 

“ Whitelaw's Wheel... 576 

Hydrometers... 67 

“ To Compute Strength of 

a Spirit . 67 

Hygrometer. 68 

“ To Ascertain Dew-point. 68 

“ To Compute Volume of 

Vapor in Atmosphere. 68 

Hyperbola, To Describe . 230 

Hyperbolic Logarithms. .. .331-334, 712 


I.. 

Ice. 912 

“ and Snoiv . 849 

‘ ‘ Boats and Speed of. .896, 909 

“ Strength of. . 195 

Impact and Reaction Wheels. 576 

“ or Collision. 580 

Impenetrability.,. 195 

Inclined Plane .6x9, 628 

Incubation of Birds, Periods of . 192 

Indicator. 724 

Inertia, Moment of a Revolving Body , 

To Compute . 609 

“ Moment of, to Ascertain Ap¬ 
proximately . 659 

of a Revolving Body , To 

Compute .616 

“ of a Solid Beam . 819 

Ink, Chinese or India .. 907 

Inks. 875 

Insects and Birds. 196 

Interest, Simple and Compound. 107, 109 

Involute, To Describe .229 

Involution. 96 

Iron.. 637-640 

“ Bolts in Wood, Tenacity of ..... 198 

“ Bridge, and Iron Pipe Bridge.. 178 

“ Cast Iron . 637 

“ Pipes, Weight of . 148 

“ “ Wrought Iron . 639 

“ Rope, Hemp, Iron, and Steel ... 164 
“ “ Hemp and Steel, Ultimate 

Strength and Safe Load 

°f- . 165 

“ u Variable Motion . 543 

“ Steel and Hemp Rope, Relative 

Dimensions of. .168, 172 

“ Wire and Hemp Rope . 168 

“ “ Gauge, Weight and Length. 163 

“ “ Weight of. .120,124 

Irregular Bodies, Volume of . 870 

. J. 

Jets d’Eau. 550 

Jewish Measures. 53 

Julian Calendar. 70 

Jumping, Leaping, etc. 439 


K. 

Hedges and Anchors, Weight and 

Number of. . 

Khorassan, or Turkish Mortar. 

Knot.. 


i74 

592 

27 





































































INDEX. 


XT 


L. Page 

Labor .433, 434, 436, 468 

Lacquers . 875 

Laitance. 593 

Lakes, Areas, Depth , and Height 

of. .181, 182 

Lamps, Fluids, and Gas. 584 

Land Measure. 29 

Larrying. 598 

Laths..i... 603 

Latitude . 198 

Latitude and Longitude .76-80 

Launching Vessels, Friction of. .478 

Lead . 640 


“ Encased Pipe , Weight of. . 151 

“ Measure . 32 

“ Pipe . 831 

“ Pipe, Weight of .139, 150 

“ Plates, Weight of . . 146 

“ Weight of .136,151 

Lead and Cast - iron Balls, Weight 

“ “ and Volume of. . 153 

“ “ To Compute Weight of 155 

Leap or Bissextile Year. 70 

Leaping, Jumping, etc.439, 440 

Leaves. 207 

Lee-way or Drift of a Vessel .910 

Legal Tenders. 38 

Lenses and Mirrors. 670 

Levelling, Geographic.55, 56 

Lever.624, 626 

Lifting. 439 

Light.195, 583-587 

“ Decomposition of . 583 

• “ Gas, Volumes and Temperature 

of-..- .585, 587 

“ Gas and Electric ... 198 

“ Intensity of . 585 

“ Loss of by Use of Globes . 584 

“ Refraction . 584 

Relative Intensity , Consump¬ 
tion, and Cost of Various 

Modes of Illumination . 584 

“ Standard of. .910 

Lightning. 430 

Lignite.479, 481 

Limes, Cements, Mortars, and Con¬ 
cretes.588-597 

“ Cements, and Mortars, Ex¬ 
periments of Gen. Gillmore. 596 


“ and Cements . 594 

“ Asphalt Composition .593 

“ Concrete or Beton . 593 

“ Conclusions from Experiments 590 

“ General Deductions . 596 

“ Pozzuolana. . 589 

“ Slaking . 594 

“ . Stucco . 591 

“ Transverse Strength . 596 


“ Turkish Plaster or Hydraulic 


Cement ... 591 

Lines, To Draw, Bisect, etc . 221 

Liquid Measure .30, 31, 46 

Liquids, Expansion of.. . 520 

“ Volume of at Boiling-Points 518 
Liquors, Proportion of Alcohol in... 204 
‘ ‘ Proof of Spir ituous....... 218 

Lithro-fracteur . 443 


Page 

Locomotive ■“ Experiment ”.902 

“ Axles . 910 

Locomotives, Operation of. .681-685 

“ Adhesion .681,685 

“ Tractive Power .681 

“ Train Resistances . 682 

Log Lines. 27 

Logarithm of a Number . 23 

Logarithms.305-310 

“ Hyperbolic .331-334 

“ of Numbers .311-330 

Longitude, To Reduce to Time . 54 

“ and Latitude . 76 

“ Lengths of a Degree . 60 

“ of Observatories in Time. 80 

Luminous Point. 195 

Lunar Cycle or Golden Number. 71 

Lunar Month. 70 

M. 

Macadamized Roads.687, 690 

Machines and Engines, Elements of 898 

Magnetic Variation.57-59 

“ Bearing of N. Y. . 184 

Magnetism..... 614 

Malleable Castings. 639 

“ Cast Iron . 785 

Manganese Bronze. 832 

Manures. 188 

Marine Day.. 37 

Marine Steamers and Engines. .886, 887 

“ Auxilliary Freight . 887 

“ Fire boat . 887 

“ Freight and Passenger _886, 887 

“ Iron Cruiser . 886 

“ “ Freight and. Passenger 

Propellers . 886 

“ Steel Launch. .. 887 

Marine Steam Vessels and En¬ 
gines .-.887-891 

“ Composite Yachts . 888 

“ Cutter . 889 

“ Ferry Boat . 889 

“ Iron Yachts . 888 

“ Light Draught . 889 

“ Side Wheels .889, 890 

“ Steel Launch . 887 

“ “ Yachts .888,889 

“ Torpedo Boats, Iron , Steel, and 

Composite . 889 

“ Wood Side Wheels, Passengers 

and Deck Cargo.. .890, 891 

“ “ Propellers . 891 

“ f ‘ Towing . 891 


“ Arch, To Compute Depth of.. 605 
“ Brick, Stone, and Granite, 595-600 

“ Designation of .602-603 

“ Estimate of Materials and 

Labor , etc.604 

“ Rubble . 601 

“ Stone .600, 603 

Technical Terms .597, 598 

Mason’s and Dixon’s Line. 188 

Mastic. 593 

Materials, Strength of .761-841 

Matter... .. 194 
















































































































Xll 


INDEX. 


Page 

Mean Proportion.... 94 

Measures, Ale and Beer . 45 

“ Apothecaries’ . 47 

“ Avoirdupois . 32 

“ Board and Timber . 61 

“ British and Metric, Com¬ 
mercial Equivalents of, 906 

“ Builder'’s . 46 

“ Circular .113, 114 

“ Cloth. . 27 

“ Coal . 33 

* ‘ Copying . 29 

“ Corn .. 198 

‘ ‘ Cube . 30 

££ Dry . 30 

“ Foreign of Value .39-43 

“ Foreign Memoranda . 43 

‘ 1 Geographic, and Distances 54 

“ Geographical and Nauti¬ 
cal . 26 

“ Grain . 32 

“ Grecian . 53 

“ Gunter’s Chain. .. 26 

“ Hebrew and Egyptian ... 53 

£i Jewish .................. 53 

“ Land . 29 

“ Liquid . 31 

“ Men and Women . 35 

Metric -27-33, 3 6 , 44, 4 6 , 47 

“ “ Equivalent Value, TJ. S., 

and Old and New 

C.S .28, 30, 33 

“ “ Power and Work. 36 

“ “ Temperatures.... 37 

■ “ “ Velocities . 37 

“ 11 Volumes .36,46 

“ “ Weights and Press¬ 
ures .• 36 

“ Miscellaneous ... .27, 29, 31, 46 

“ Nautical . 30 

“ Old and New TJ. S. Ap¬ 
proximate Equivalents. 33 

“ of Length .26,44 

“ of Offal in a Beef and Sheep. 35 

“ of Paper . 29 

“ of Surface .29,44 

“ of Timber, Local Stand’ds 62 

££ of Time. .. 37 

“ of Value . 38 

“ of Volume ... .30, 45 

“ of Weight. .32, 47 

“ Pendulum . 27 

“ Roman Long . ;.... 53 

Ropes and Cables... . 26 

“ Scripture and Ancient... 53 

££ Shoemaker’s . 27 

“ Timber, English . 62 

“ Troy .32,47 

“ Vernier Scale . 27 

“ Wine and Spirit . 45 

“ Wood.......... .33,47 

Measures and Weights .26-35 

“ English and French, 44, 45 

“ Foreign .48-53 

Meat, Analysis of, and of Fish and 

Vegetables .200 

“ To Preserve .196 


Page 

Meats, Roasting of. ..206 

Mechanical Centres .605-614 

“ Centre of Gravity .. .605-608 

“ “ of Gyration of a 

Water-wheel.... 611 
“ “ of Gyration... 609-611 

“ Elements of Gyration... 610 

“ Ratio of Gyration .609 

Mechanical Powers. .624-634 

££ Compound Axle or Chi¬ 
nese Windl ass .627 

“ Inclined Plane . 628 

‘ ‘ Lever .624-626 

Pulley . 632 

“ Rack and Pinion .628 

“ Screw . 630 

“ “ Differential .632 

“ Wedge .630 

“ Wheel and Axle .626 

Mechanics .614-623 

“ Accumulated Work . 619 

“ Decomposition of Forces.. 620 

‘ ‘ Dynamics .616-620 

“ Moment . 614 

“ Moments of Stress on Gir¬ 
ders .621-623 

“ Motion on an Inclined 

Plane . 619 

“ Solid, Fluid, and Aeri¬ 
form Bodies . 614 

“ Uniform Motion .617 

‘ ‘ Work by Percussive Force 620 

Melting-Points. 517 

Memoranda .907, 912 

“ Cast and Wrought Iron 

and Steel . 832 

Men. 433-435 

Mensuration of Areas, Lines, Sur¬ 
faces, and Volumes, 335-378 
“ Any Figure of Revolution, 

358 , 376 

“ “ Plane Figure . 359 

“ Arc and Chord, etc., of a 

Circle .. 343-345 

“ Area Bounded by a Curve. 342 

“ Capillary Tube . 358 

“ Cask Gauging . 377 

“ Chord of an Angle . 359 

‘ ‘ Circle .342 

“ “ Section of . 346 

“ “ Segment of .346 

. “ Circular Zone . 349 

“ Cones .353,363 

“ Cube and Parallelopipedon 360 

Cycloids . 352 

“ Cylinder .350,363 

11 “ Sections . 357 

“ Ellipsoid, Paraboloid, and 
Hyperboloid of Revolu¬ 
tion .357 375 

£[ Helix (Screw) . 354 

“ Irregular Bodies . 377 

“ “ Figures .341 

‘‘ L inks . 353,370 

u Lune . 252 

.“ of Areas, Lines, and Sur- 

f aces . 335-359 


































































































INDEX. 


Xlll 


Page 

Mensuration of Areas, Lines, Sur¬ 
faces, and Volumes.360-378 

“ Parallelograms . 335 

“ Polygons .338, 343 

“ Polyhedrons . 362 

“ Prism .350,360 

“ Prismoid .351, 361 

Pyramids .354,365 

“ Reduction of Ascending or 
Descending Line to Hori¬ 
zontal Measurement . 359 

“ Regular Bodies, 340,341,362,364 

“ Rin( P . 353,368 

“ Sphere .347, 367 

11 Spheroids or Ellipsoids . 348, 368 

“ Spindles .355, 370-374 

“ Spirals . 355 

“ To Ascertain Area of any 

Plane Figure . 359 

“ To Compute Chord of an 

Angle .359 

“ To Plot Angles . 359 

“ Trapezoid . 338 

“ Triangles .335-338 

■ u Ungulas . 351 

“ Useful Factors . 343 

“ Wedge .350, 361 

Mercurial Gauge . 9x0 

Meta-Centre of Hull of a Vessel , To 

Compute .... . .659, 913 

Metal Products ofU.S. .910 

Metals, Alloys and Compositions .. 634 
“ Comparative Quality of Va¬ 
rious . 821 

“ Lustre of. . 194 

“ To Compute Weight of. . 131 

“ To Compute Weight of by 

Pattern . 217 

Milk, Nutritive Values and Constitu¬ 
ents of . 202 

“ Percentage of Cream . 205 

“ Relat ive Richness of of Several 

Animals . 207 

“ To Detect Starch in . 196 

Mineral Constituents from an Acre 

of Soil . 189 

Mineral Waters, Analysis of etc..850, 851 

Minerals, Hardness of. . 193 

Miner’s Inch... 537 

Mining. 445 

Mining Engines and Boilers.901 

Mining Ropes. 165 

Mirage.195, 669 

Mirrors and Lenses. 670 

Miscellaneous Elements.188-198, 906 

“ Mixtures.871-879 

“ Operations and Illus¬ 
trations .879-885 

Mississippi River, Silt in . 910 

Models, Strength of . 644 

Molasses. 207 

Molding and Planing. 476 

Molecules, Velocity, Weight, and Vol¬ 
ume of . 194 

Moment. 614 

Momentum.,... 195 

Monoliths. 179 


Page 

Month, Lunar... . 70 

Months, Numbers of . 74 

Moon’s Age, To Compute . 74 

Mortar ....590, 592, 595 

Mortars, Limes, Cements, and Con¬ 
cretes.588-597 

Motion, Accelerated and Retard¬ 
ed • .. 494 , 495 

Motion of Bodies in Fluids.645 

“ Pressure, Velocity, Time, etc., 

To Compute . 648 

‘ 1 Resistances of Areas and Dif¬ 

ferent Fig's in Water or Air. 646 
Mountains and Passes, Heights of... 182 

Mowing. 433 

“ Machine .910 

Mule. 437 

Mural Efflorescence. 593 

N. 

Nails, Length and Number of... 153, 154 
“ and Spikes, Retentiveness of... 159 

National Road. 178 

Natural Powers. 198 

Nautical Measure. 30 

Naval Architecture.649-667 

“ Angles of Courses and Sails.. 665 

“ “ Heel and of Steady 

Heel .664, 665 

“ Area of Sails, etc.663, 664 

“ Centre of Effort, To Compute 

Location of. .659 

“ “ of Gravity of Hull, En¬ 
gines, etc.658 

“ Centres of Lateral Resistance 

and Effort. .658 

“ Course and Apparent Course 

of Wind . 666 

“ Curve and Coefficients of Dis¬ 
placement . 657 

“ Displacement, and its Centre 

of Gravity .653, 655 

“ Elements of a Vessel .653, 660 

“ “ of Capacity and Speed 

of Several Types of 

Stcdmers . 660 

“ Experiments upon Forms of 

Vessels . 649 

£ ‘ Freeboard .666 

11 Lee-way . 666 

“ Masts, Location of. . 664 

“ Memoranda . 667 

‘ ‘ Meta- Centre of Hull of a Vessel, 

To Compute .659, 913 

“ Moment of Inertia and Meta- 

Centre . 659 

11 Pitch of Propeller and Slip of 

Side Wheels . 662 

“ Plating Hulls . 667 

“ Power Utilized in a Steam- 

vessel . 662 

“ Relative Positions of Lateral 

Resistance and Centre of 

Effort .659 

u Resistance of Bottoms of Hulls, 663 

“ “ to Wet Surface of Hull, 

To Compute . 653 





























































































XIV 


INDEX. 


Page 

Naval Architecture, Rudder Head. 667 
“ Sailing , Ratio of Effective 

Area of Sails, etc., 

to Wind . 663 

“ “ Power of a Vessel .... 665 

“ Sails, Area and Trimming of, 

664, 665 

* ‘ Surface, Bottom, and Immersed 

Hull , To Compute . 653 

“ Stability .649,650 

44 44 Results of Experiments 

upon .649, 650 

“ “ To Compute Statical 

and Dynamical .651 

44 44 To Determine Mean of, 

of Hall of a Vessel.. 650 

44 To Compute Displacement , Ap¬ 

proximately, and Co¬ 


efficients of. . 655 

44 “ Elements of Power Re¬ 

quired to Careen a 

Vessel . 652 

44 44 Power Required in a 

Steam - vessel , Speed 
and IP, and Coeffi¬ 
cient of. . 661 

“ Wind, Effective Impulse of'... 665 

Needle, Magnetic Variation of. . 57 

“ Decennial Variation of . 58 

44 Variation of it in U. S', and 

Canada . 59 

Neutral Axis of a Beam . 820 

New and Old Style.37, 70 

Niagara, Falls of. . 198 

Nitro-Glycerine. 443 

Non-conductibility of Materials.911 

Notation. 25 

Number of Direction. 71 


Numbers, Properties and Powers of.. 98 

44 ffh and ffh Powers of. .303, 304 
44 To Compute 4th, 5 th, and 
6th Power , and 4th and 

5 th Root of .. 304 

Nutritive Equivalents ,Human Milk, 1 205 


0. 

Obelisks. 179 

Observatories, Latitude and Longi¬ 
tude . 80 

Oceans and Seas, Depths and Areas.. 182 
Offal, Weight of, in a Beef and Sheep. 35 

Oil, Yield of, from Seeds . 189 

“ Cake and Vegetables , Nutritious 

Properties of. .204 

44 Proportions of in Air-dry Seeds. 203 
Oils, Petroleum, Schist, and Pine- 

Wood . 484 

Old and New Style.37, 70 

Onion. 207 

Opera-Glasses. 671 

Opera-Houses. 180 

Optics .668-671 

4 4 Elements of Mirrors and Lenses 670 

44 Refraction .. . .668, 669 

44 To Compute Dimensions or Vol¬ 
ume of an Image .668 

Ordnance, Energy of. .910 


Page 

Organic Substances, Analysis of, by 

Weight . 190 

Orthography of Technical Words 

and Terms .915-922 

Oscillation and Percussion, Centres 

of. .612-614 

“ Centre of, in Bodies of 

Various Figures .613 

44 To Compute Centreof 612,614 

Overshot Wheel .*._563 


P. 

Painting.... 66 

Parabola, To Describe . 229 

Park, Deer. 179 

Parsnips. 207 

Passages of Steamboats... 896 

44 Steamer and Sailing Vessels. 897 
Passes, Mountains, and Volcanoes... 182 

Pavement, Asphalt . 690 

“ Block Stone .689, 690 

44 Granite .690 

“ Rubble Stone . 689 

44 - Telford .688 

44 Wood .689, 690 

Pavements, Roads, and Streets... 686-690 

Payments, Equation of . 109 

Peat.482 

Pendulum Measure. 27 

Pendulums. 452 

“ Centre of Gravity of.... 453 

44 Lengths and Number of 

Vibrations of. 453, 454 

“ Time of Vibration . 454 

Pennants, Ensigns, and Flags, U.S... 199 
Percussion and Oscillation, Centres 

of. .612-614 

Performances of Men, Horses, etc. 438 

Perimeter of a Figure . 912 

Permutation. 100 

Perpetuities. 112 

Petroleum, Elastic Force of Vapor.. 707 


44 Coefficient of Resistance of 

Earth , To Compute .672 

“ Pneumatic . 672 

“ Ringing Engine . 672 

“ Sheet Piling . 672 

44 Sinking . 673 

“ To Compute Weight of Ram.... 672 


44 Extreme Load a Pile will Bear 913 

44 Retaining Walls of Iron . 196 

Piling of Shot and Shells. 65 

Pillar at Delphi . 179 

Pipes. 747 

“ Lead and Tin, Weight of. . 139 

14 Riveted Iron and Copper, Weight 

of . 148 

44 Steam , Gas, and Water . 138 

44 To Compute Thickness of. . 560 

44 “ Weight of Metal... 147 

“ Tin, Weight of . i 5I 

44 Lead Encased. .. 151 

Pise. 593 

Pivots, Friction of .472 


















































































INDEX. 


XV 


Page 

Planing Cast Iron and Molding.476 

Plank Roads. 688 

Plants or Hills, in an Acre. 193 

Plaster, Turkish. 591 

Plastering.197, 604 

Plate Bending. 476 

Plates, Test of\ and Bolts .749, 753 

“ Thickness of by Wire Gauges 121 
“ To Compute Thickness of.... 751 

“ Wrought-Iron Shell .750 

Plating Iron Hulls-. 667 

Ploughing. 433 

Pneumatics.—Aerometry.673-676 

Pointing. 598 

Poisons, Antidotes and Treatment of. 185 

Poles and Spars. 62 

Poncelet’s Wheel. 567 

Population, Comparative Density of 
and Number of Persons 
in a House in Different 

Cities . 910 

“ of Earth . 188 

“ of Principal Cities . 187 

Position.98, 99 

Potato. 207 

Power and Work, Metric. 36 

“ and Mechanical Energy of 10 
Grains of Various Substances 
when Oxidized in Human 

Body . 205 

“ Motive .. 910 

t; Movers and Transmitters of... 797 

“ of a Quantity, To Ascertain 

Value of. . 359 

“ Required to Draw a Vessel up 

an Inclined Plane . 910 

“ To Sustain a Vehicle on an In¬ 
clined Road . 845 

“ Transmission of. . 167 

Powers, Natural . 198 

“ of first 9 Numbers . 98 

“ of ffh and 5 ih Numbers .303 

“ of 6th Number, etc . 304 

Probability.114-117 

Progression.101-105 

Proof of Spirituous Liquors. 218 

Propellers..730, 886-891 

Propeller Steamers, Ordinary Distri¬ 
bution of Power in . 911 

Properties op Numbers. 98 

Proportion.94-96 

Pulley.433, 632 

Pumping. 433 

Pumping Engines.902-903 

Pumps , Centrifugal .. 911 

“ Circulating . 749 

Pushing or Drawing. 433 

Pyramids, Statues, etc. 178 

Q. 

Quartermasters, Service Train of.... 198 

R. 

Race-Courses, English, Length of.'... 52 

Rack and Pinion. 628 

Railroads. 178 

Rails, Iron and Steel . 812 


Page 

Rails, Tangential Angles for Chords 

and Curves .677, 678 

“ To Define a Curve . . 677 

‘ ‘ To Determine Elevation of Out¬ 
er Rail .679 

Railways.677-685 

“ Curves by Offsets . 678 

“ Elevation of Rail 678, 679 

“ Operation of Locomotives. 

681-685 

“ Points and Crossings .678 

“ Radii of Curves .... 679 

‘ 1 Rise per Mile and Resist¬ 
ance to Gravity . 679 

“ Sidings . 677 

“ To Compute Weight of Rail 679 

“ “ Load a Locomotive 

will Draw up an 

Inclination .680 

“ “ Maximum Load 

Drawn up the 

Maximum Grade 

it can Attain... 680 

“ “ Resistance of Grav¬ 
ity .679 

“ “ Traction,Adhesion, 

etc. 680 

“ Velocity of Trains . 680 

Railway Trains . 911 

Rainfall, Volume of. . 850 

Reaction and Impact Wheel. 576 

Reaping. 433 

Rebate or Discount. 109 

Reciprocals. 304 

Rectilineal Figures, To Describe .222 

Refraction. 668 

“ of Light . 584 

Refrigerator, Surface of . 512 

Rendering. 598 

Resilience of Woods.. 763 

Retaining Walls. 695 

“ of Iron Piles . 196 

Revetment Walls.694-700 

‘ ‘ Surcharged .699 

“ To Compute Elements of 696 

Ringing Engine.672 

River Steamboat.913 

River Steamboats and Engines. .892, 914 

“ Traction on . 848 

Rivers, Current of. . 193 

“ Descent of Western . 188 

“ Flow of Water in .550, 551 

“ Lengths of . 183 

“ Obstruction in . 551 

Riveted Joints, Comparative Strength 

of etc.829, 829 

“ Experiments on . 783 

Riveting. 753-757 

Rivets. 830 

“ Diameter of, etc.756, 829 

‘ ‘ Memoranda .830 

Roads. 178 

“ Streets, and Pavements, 686-690 

“ Bituminous .690 

“ Concrete . 689 

“ Construction . 686 

“ Corduroy .688 




































































































XVI 


INDEX, 


Page 

Roads, Getters, Fillers, and Wheelers, 
Proportion of,\ in Different 

Soils . 688 

“ Grades . 686 

“ Macadamized .687, 690 

“ Materials , etc. 689 

“ Metalling .690 

‘ ‘ Miscellaneous Notes . 690 

“ Plank . 688 

“ Rolling , Sprinkling, etc.690 

“ Ruts . 687 

‘ ‘ Sweeping , Watering, and Wash¬ 
ing . 690 

“ Telford . 688 

Roadway, Construction of. . 687 

Rock. 467 

Rock and Earth, Excavation and 

Embankment of . 192 

Rocks, Bulk of . 468 

“ Weight of per Cube Yard .... 468 

Roman Calendar . 71 

“ Indiction. To Compute . 71 

“ Long Measures . 53 

Roof Plates, Corrugated, Weight of... 131 

Roofs of Buildings . 179 

“ Wooden .. 189 

Root, To Compute , of an Even Power. 98 

1 ‘ To Extract any . 97 

Roots and Grains, Weights of . 34 

Roots, Square and Cure.272-302 

“ To Compute 4 th and 6th . 301 

Ropes. 782 

“ and Cables. 26 

“ and Chains of Equal. Strength. 165 

“ Cables, Chains, etc.163-175 

“ Endless . 167 

“ Equivalent, and Belts .. 167 

“ Hawsers and Cables .169-172 

u u Pq Compute Strain of 170, 1 ji 
“ “ “ Circumference of. 171 

“ “ “ Weight of . 172 

“ Hemp and Wire. . 167 

“ “ Iron, and Steel .164, 168 

“ “ Weight and Strength of. 172 

“ Mining . 165 

‘ ‘ To Compute Stress, Tension, and 

Deflection . 166 

“ “ Circumference of 

etc. 169 

“ Transmission of Power . 167 

“ Wire .161-172 

“ “ Experiments on . 169 

Rowing. 433 

Rubble Stone Pavement.689 

Rule of Three. 95 

Running.438, 440 

S. 

Safety Valves... 746 

Sago. 207 

Sailing. 663 

“ Vessels, Iron .894, 895 

Sails, Propulsion and Area of. .663 

“ Trimming of. .665 

Saline Saturation.726 

“ Matter in Sea Water . 727 

Sand. 599 


Page 


Sandstones. 193 

Saw Mill...904 

Sawing Stone and Wood. 19$ 

Saws, Circular .197,477,911 

“ Vertical and Band . 477 

Scale or Sediment in Boilers . 726 

Scales, To Divide a Line, etc. 221 

“ Weighing without . 66 

Scarfs, Resistance of. . 841 

Screw. 630 

“ Cutting .477 

“ Differential . 632 

“ Propeller, Pitch and Speed of. 662 
Screw Propeller, Friction of Engines 663 
Scripture and Ancient Measures.. 53 

Sea Depth. 184 

Seas, Depth and Area . 182 

Secants and Cosecants.403-414 

“ “ To Compute Degrees, 

Minutes, etc. 414 

Seeds, Number of, in a Bushel, and 

per Sq. Foot per Acre . 193 

“ Proportion of Oil in Air-dry. 203 
Segments of a Circle, Area of.. 267, 268 
“ To Compute Area 

of . 268 

Sewers.691,692 

“ Drains, Diameter and Grade 

of, to Discharge Rainfall.. 906 

“ Pipes and Sewage .692 

Shaft, Bearings for Propeller .473 

Shafts.778, 793, 794, 796, 797 

‘ ‘ and Gudgeons . 790 

“ Deflection of. .778 

1 ‘ Supports of. .796 

Shearing or Detrusive Strength, 

782,783 

Sheathing and Braziers’ Sheets. 155 

“ and Braziers' 1 Copper _131 

“ Nails, Weight of. . 135 

Sheet Iron, Galvanized .124, 129 

Sheet Piling.672 

Shingles. 63 

Ship and Boiler Plates. 828 

Shoemaker’s Measure. 27 

Shot and Shell, Piling of. . 65 

Shot, Chilled and Drop . 906 

“ No., Diameter , and Numbers 

of . 906 

Shrinkage of Castings. 218 

Shrouds, Hemp and Wire . 173 

Side Wheels, Area of Blades .662 

“ Friction of Engines.. 662 

11 Slip of. .662 

Sides of Equal Squares.258, 259 

Silt, in Mississippi River . 910 

Silver Sheet, Thickness of. . 119 

Simple Interest. ... 107 

Simpson’s Rule, To Compute, Area... 344 

“ Volume of an Irreg¬ 
ular Body . 870 


“ To Compute . 

“ “ Number of Degrees, 

etc. 

Sixth Power of a Number, To Com¬ 
pute . 


304 







































































































INDEX. 


XVII 


Page 

Skating. 439 

Slackwater, Canal, etc., Traction on. 848 

Slaking. 594 

Slate, To Compute Surface of. . 64 

Slates and Slating. 64 

Slates, English . 64 

“ Weight per 1000 and Number 

Required to Cover a Square 64 

Slide Valves.731, 733 

Smelting of Ore. 445 

Slotting. 477 

Smoke Pipes and Chimneys. 748 

Snow-flakes. 195 

“ Line of Perpetual Congelation. 192 

“ Melted . 195 

Snow and Ice. 849 

Solar Day and Year. 70 

Solders.634, 636 

Soldering. 875 

Sound. 195 

“ To Compute Velocity of. . 428 

Soundings, To Reduce to Low Water. 60 

Spars and Poles. 62 

Specific Gravity, To Ascertain , of a 
Body Heavier or Lighter 

than Water . 209 

“ of a Body Soluble in Water. 209 

“ of a Fluid .-209 

Specific Gravity and Weight . .208-215 

“ To Compute Weight of a Body 215 
“ “ Proportions of Two In¬ 

gredients in a Com¬ 
pound , etc.216 

“ Weights and Volumes of Va¬ 
rious Substances in Ordi¬ 
nary Use . 216 

Spikes and Nails, Retentiveness of ... 159 
Spikes, Ship, Boat, and Railroad 

152, 154 

‘ 1 and Horseshoes . 152 

“ and Nails . 159 

“ Wrought - Iron Nails and 

Tacks . 154 

Spiral, To Describe . 230 

Spires. 180 

Spirituous Liquors, Dilution Per Cent. 191 

“ Proof of . 218 

“ Proportion of Alcohol. 191 

Springs, Deflection of. . 779 

Square and Cube Root, To Compute, 
of a Higher Number than 

contained in Table . 300 

“ To Ascertain, of a Number 
consisting of Inte¬ 
gers and Decimals 301 

“ < c of Decimals . 302 

Square Root, To Extract . 97 

Square, To Ascertain One that has 

Same Area as a given Circle . 259 

Squares, Cubes, and Square and 

Cube Roots.272-302 

u To Compute Square and 

Cube Roots of Roots of 
Whole Numbers and of 
Integers and Decimals ... 97 

Squares, Sides of Equal, in Area to a 
Circle.258, 259 


Page 

Stability.693-703 

“ Dynamical . 651 

“ Dynamical Surface . 651 

“ of Earth .695 

“ of Models . 649 

“ 'To Ascertain, of a Body .. 693 

“ To Careen a Vessel .652 

To Compute Statical .651 

“ “ Equilibrium of Walls 701 

“ Stability . 701 

“ To Determine Measure of 

Hull of a Vessel, etc.... 650 

Staging, Coach . 44 o 

Staining. 876 

Starch, Proportion of, in Vegetables.. 205 

Stars, Velocity of. . 198 

Statics.615-617 

“ Composition and Resolution 

of Forces.. . 615 

“ Equilibrium of Force .616 

Statues, Pyramids, etc. 178 

Stay Bolts, Diameter, etc. 752 

Steam.704-727 

“ and Air, Mixture of. . 737 

“ Average Pressure, etc.711, 712 

“ Blowing off of Saturated Water 726 

! ‘ Boilers . 829 

“ “ Zinc Foil in . 912 

“ Clearance, Effect of. . 715 

‘ ‘ Compound Expansion .720-724 

“ Conclusions on Actual Effi¬ 
ciency of. . 724 

“ Condensing Surface, Experi¬ 
ments on .9x1 

“ Density or Specific Gravity of 706 

“ Expansion . 710 

“ “ Effects of . .713, 7i 8, 719 

“ 11 Points of .712 

“ Feed Water, Gain in, at High 

Temperatures, To Compute.. 719 
“ Gain in Fuel, To Compute.... 725 
“ Gaseous and Total Heat of... 710 

“ Hammers . 179 

“ Heat of Saturated . 705 

“ Heating Co. of N. Y. .. 904 

“ Indicator . 724 

t! Injector . 736 

“ Latent Heat of. . 707 

“ Mean Pressure, To Compute.. 713 
“ Mechanical Equivalent of .... 705 

“ Pipes, Gas, etc. 138 

‘ * Point of Cutting off, and Press¬ 
ure at .7x0 

“ Pressure of .705, 710, 711 

“ “ Weight, and Temper¬ 
ature . 705 

“ Properties of, of Maximum 

Density . 717 

“ Ratio of Expansion . 7x0 

‘ ‘ Relative Effect of Equal Vol¬ 
ume of. . 714 

“ Saline Saturation in Boilers.. 726 

“ Saturated .704, 708, 716 

“ Scale or Sediment, Removal of 726 

“ Specific Gravity of. .704, 706 

“ Superheated . 717 

Temperature of. . 705 


B 



















































































XVU 1 


INDEX. 


Page 

Steam, Temperature of Water in a 

Condenser . 707 

“ To Compute Volume of Water 

Evaporated per Lb. of Coal 725 

“ “ Consumption of Fuel .... 725 

“ Total Effect of 1 Lb. of. . 714 

“ “ Heat of Saturated . 707 

“ Velocity of to Compute . 710 

“ Volume of a Cube Foot of Wa¬ 
ter in . 706 

“ “ of Cylinder, to Compute. 715 

“ “ of to Raise a Given Vol¬ 
ume of Water . 706 

“ “ of Water Contained in.. 706 

“ “ of Water to Raise or Re¬ 

duce to any Required 

Temperature . 707 

“ Weight of. . 705 

“ Wire Drawing . 718 

“ Woolf Engine.; . 722 

Steamboats, River, and Engines.892,893 

“ Passages of . 896 

“ Wood Side Wheels . 892 

“ “ Ferry Boat . 893 

“ “■ Passenger and Light 

Freight .893, 894 

“ 11 Stern Wheels .. .893, 894 

•Steam-Engine...727-760 

“ and Boilers, Cost of per Day. 904 

“ Area of Feed Pump . 736 

" Injection Pipe . 735 

• 73?~745 


“ Boiler. 


“ “ Evaporative Power oj. 757 

“ Weights of . 759 

“ Circulating Pumps . 749 

“ Condensing . 727 

“ Elements and Capacity of 

Steam Pumps . 738 

“ Evaporation . 747 


Fire. 


9°4 


“ Flues and Tubes, . 747 

“ Friction of Side Lever . 478 

“ General Rules .728, 730 

“ EP, to Compute, etc .733,734 

“ Injection Pipe , to Compute 

Volume of Flow . 735 

“ Non-condensing . 728 

“ Plates and Belts .749-753 

“ Practical Efficiency of . 737 

“ Propeller, to Compute Thrust 

°f . 731 

“ Propellers . 7 o X 

“ Relative Cost of .’ 737 

“ Results of Operation of. . 737 

“ Riveting .753-757 

“ “ General Formulas.. 7^7 

“ Safety Valves . 746 

“ Smoke Pipes and Chimneys. 
t . m r, 748, 749 

“ Stay Bolts . 752 

“ Steam Room . 748 

“ Slide Valves . 731 

“ “ To Compute and As¬ 

certain Lap, Breadth 
of Ports, Portion of 1 
Stroke, Lead, etc.731-733 
“ Volume of Circulating Water 735 


Page 

Steam-Engine, Volume of Injection 

and Feed Water .735, 736 

‘ 1 Volume of Water Required to 

be Evaporated . 734 

“ Water Surface . 748 

“ Water- Wheels ..730 

“ Weights of .758,759,911 

Steam Fire-Engine. 904 

Steam Pumps, Elements and Capaci¬ 
ties of . ... 738 

Steam-Vessel, Power Utilized in _662 

Steam-Vessels, Resistance to, in Air 

and Water . 911 

Steamer “Great Eastern”. 179 

Steamers, Friction of Screws .478 

Steaming Distances. 86 

Steel.640-643, 750, 783, 787, 827 

Cables, Galvanized . 163 

Columns, Crushing Weight of. 768 
Hemp and Iron Ropes, Relative 

Dimensions of. . 168 

Hemp, and Iron Wire Rope... 164 

Locomotive Tubes . 138 

Manufacture of. .642 

Plate . 830 

Plates, Weight of .118, 119, 146 

Relative Dimensions of. . 172 

Rolled and Bar, Weight 0/1.134, 135 
Rope, Hemp and Iron, Round 

and Flat . 164 

Weight of. .136, 149 

Wire, Weight of .120, 121 

Sterling, Pound. 38 

Stings and Burns, Application for... 196 

Stirling’s Mixed Iron. 785 

Stirrups or Bridles, for Beams .838 

Stone and Ore Breakers. 903 

Stone, Expansion and Contraction 

of. . 184 

Hauling . 468 

Masonry . 595 

Resistance of, to Freezing .... 184 

•• Saiving .196, 904 

“ Voids in a Cube Yard . 690 

Stones, Cements, etc. 766 

Streams, Flow of Water in . 550 

Street Rails or Tramways.435, 902 

Streets, Roads, and Pavements .. 

686, 690 

Strength of Materials.761-841 

Crushing Strength. ..764-769 

“ Comparative Value of Long 

Solid Columns . 769 

“ of Cements, Stones, etc.... 596 
“ of Columns , to Compute 

Weight of. . 768 

“ of 2-inch Cubes .767 

“ of Various Materials. .765, 766 

“ Resistance of Rivets .908 

“ Riveted Joints .828,829 

“ Safe Load of Columns, 
Arches, Chords, etc., of 

Cast Iron . 766 

“ Weight borne with Safety 

by Cast-Iron Column... 768 
“ Woods , Destructive Weight 

of Column of. . 769 

























































































INDEX. 


XIX 


Page 

Strength of Materials, Crushing 
Strength of Woods, Rel¬ 
ative Value of Various . 769 
‘ ‘ Wrought-Iron Cylinder and 

Rectangular Tubes . 767 

Deflection .770-781 

“ and Distributed Weight for 

Limits of to Compute... 778 
1 ‘ Cast-Iron Bars and Beams, 

777 , 778 

“ Continuous Girders or 

Beams of Wood .772 

“ Formulas for Beams .. .771, 772 
“ General Deductions....... 779 

“ Mill and Factory Shafts... 779 
“ of a Shaft from its Weight 

alone . 778 

“ of Bars, Beams, Girders, 

etc.770-782 

“ of Rectangular Bars or 

Beams of Cast Iron.... 777 

“ of Wrought-Iron Bars . 773 

“ of Wrought-Iron Rolled 

Beams . 774 

“ Rails .... 776 

“ Relative Elasticity of Vari¬ 
ous Materials . 780 

“ Results of Experiments.... 780 
“ Shafts of Wrought Iron — 778 
“ To Compute, and Compar¬ 
ative Strength of Cast- 
Iron Flanged Beams.. 778 
“ “ and Weight that may be 

Borne by a Rectan¬ 
gular Bar or Beam 

of Cast Iron . 777 

“ “ Maximum Load that 

may be Borne by a 
Rectangular Beam. 773 
“ u of and Weight Borne by 
a Rectangular Bar 

or Beam . 773,774 

“ 11 of Cast-Iron Flanged 

Beams . 777 

“ Woods . 772 

“ Wrought Iron and Woods.. 772 
“ Wrought - Iron Bars or 

Beams .773, 774 

“ Wrought - Iron Riveted 

Beams . 774 , 775 

“ Wrought - Iron Tubular 

Girders . 775 

Detrusive or Shearing Strength. 782,783 
“ Comparison between it and 

Transverse . 782 

“ of Woods . 782 

Results of Experiments. 782, 783 
“ Riveted Joints, Cast Iron, 

Treenails, and Woods... 783 

“ Shearing . 783 

“ To Compute Length of Sur¬ 
face of Resistance of 
'Wood to Horizontal 

Thrust . 782 

“ Wrought and Cast Iron 
Riv'd Joints , Steel, Tree¬ 
nails, and Woods . 783 


Page 

Strength of Materials, Elasticity 

and Strength .761-764 

“ Coefficients of. . 761 

“ Comparative Resilience of 

Woods... 763 

‘ ‘ Extension of Cast-Iron Bar 762 

“ Modulus of Cohesion .763 

“ “ of Elasticity 762,763 

“ “ of Elasticity and 

Weight . 763 

“ “ of Elasticity, Height 

of, to Compute... 763 

li “ of, to Compute 762 

“ Weight a Material will 
Bear without Permanent 

Alteration . 763 

Tensile Strength ...784-789 

“ Elements Connected with 
Resistance of Various 

Bodies .. 786 

“ Malleable Iron . 785 

“ Manganese Bronze . 832 

“ of Cast and Wrought Iron, 

784, 785 

“ of Tie-rods . 787 

“ of Wrought Iron . 785 

“ Rat io of Ductility and Mal¬ 
leability of Metals .787 

“ Steel, Bars and Plates. .787, 788 

“ Stirling's Mixed Iron . 785 

1 ‘ Various Materials .788-790 

Torsional Strength . 79°~797 

“ Couplings . 796 

‘ ‘ Hollow Shafts .792, 794 

“ Journals of Shafts, etc.... 796 

“ Metals and Woods . 793 

“ Mill and Factory Shafts.. 797 
“ Minimum and Maximum 
Diameter of Shafts, For¬ 
mulas for . 796 

“ of Various Materials . 793 

“ Shafts and Gudgeons... 790-795 

“ To Compute, of Shafts . 794 

li “ Diameter of a Shaft to 
Resist Lateral 

Stress . 791 

“ “ 11 of Shafts of Oak 

or Pine . 793 

“ “ “ of a Centre Shaft. 794 

“ “ “ of Solid and Hollow 

Shafts. .791, 792, 794 

Transverse Strength .799-841 

“ Bars, Beams, Cylinders, 

etc.801-805 

“ “ Girders, or Tubes, 

Comparative Value 

of . 824 

“ Bow-string Girder . 812 

“ Brick-work . 801 

“ Bridge Plates and Rivets.. 830 

“ Cast-Iron .814-817 

“ “ and Woods . 798 

“ “ Girders and Beams. 813 

“ Channel and Deck Beams, 

and Strut Bars .. 808 

“ Comparative Qualities of 

Various Metals. .821 





























































XX 


INDEX, 


Strength of Materials, Transverse , 
Comparative Strength and 
Deflection of Cast-Iron 

Beams . 809 

‘ ‘ Cylinders, Flues, and Tubes, 

827, 828 

“ Cylindrical and Elliptical 

Beams or Tubes . 810 

“ Dimensions and Propor¬ 
tions of Wrought - Iron 

Flanged Beams . . 809 

‘ ‘ Elastic Strength of Wrought- 

iron Bars . 808 

“ Elements of Rolled Beams. 807 
“ Flanged Beams, Compara¬ 
tive Strength and Deflec¬ 
tion of. . 809 

“ Flanged IIollow or Annu¬ 
lar Beams of Symmetri¬ 
cal Section . 815 

“ Form and Dimensions of a 
Symmetrical Beam or 

Girder . 825 

‘ ‘ General Deductions . 824 


“ General Formulas for De¬ 
structive Weight of Solid 
Beams of Symmetrical 
and Unsymmetrical Sec¬ 
tion .816, 817 

“ Girders and Beams of Un¬ 
symmetrical Section 810 
“ “ Beams, Lintels , etc., 

822-826 


11 

u 


(( 

a 

u 

u 

<< 

u 

u 


u 

«( 

u 

«( 

<< 

u 


(( 


(( 

<( 


Iron and Steel Rails .812 

Memoranda . 830 

‘ 1 Cast and Wrought 

Iron . 832 

Moment of Resistance .818 

“ of Inertia . 818 

of Cements and Mortars... 596 

of Various Figures of Cast 

Iron . 800 

Materials... 799-801 

“ Metals . 799 

Rectangular, Diagonal , or 
Circular Beam or Shaft. 817 
Relative Stiffness of Mate¬ 
rials . 798 

Rivets . 830 

Solid and Hollow Cylinders 

of Various Materials _ 801 

Steel Bars .817 

“ Plates . 830 


To Compute Centres of Grav¬ 
ity and of Crushing 
and Tensile Strength 
of a Girder or Beam 819 
“ Destructive Weight or 
Loads, Borne by 
Rolled Beams or Gir¬ 
ders or Riveted 

Tubes ..805—807 

“ Inertia, Moment of, of 

a Solid Beam .819 

“ Neutral Axis of a Beam ' 
of Unsymmetrical 
Section .820 


Page 

Strength of Materials, Transverse, 

To Compute Section of 
Flange of a Girder 
or Shaft of Cast 

Iron .817, 818 

“ “ Ultimate Strength of 

Homogeneous Beams 820 
“ Trussed Beams or Girders , 

823, 824 

“ Unequally Loaded Beams.. 810 
“ Wrought - Iron Inclined 

Beams, etc.. 811 
“ “ Plate Girders 811 

“ “ Rectangular 

Girders or 


Tubes . 809 

Working Strength and Factors of 

Safety . 781 

Strength of Models . 644 

“ To Compute Dimensions of a 

Beam, etc. 644 

“ “ Resistance of a Bridge 

from a Model .645 

Stress, Moment of. .621-623 

Stucco.....591 

Suez Canal, Via . 912 

Sugar Cane, and Beet Root. 207 

Sugar-Mill Rollers.911 

Sugar Mills . 903 

Sulphuret of Carbon, Elastic Force 

of Vapor of. . 707 

Sun. 188 


“ Heat of. . 193 

Sunday Cycle or Cycle of the Sun.... 70 

“ or Dominical Letter. 70 

Sun-dial, To Set . 69 

Surcharged Revetments. 699 

Surveying, Useful Humber in . 69 

Suspension Bridges.178, 199, 842 

Sustenance, Human and Animal.... 203 
‘ ‘ Requirements of a Worlc- 

ing-man . 207 


Sweet Potato. 207 

Swimming... 439 


Symbols, Algebraic, and Formulas. 22 


T. 

Tacks, Nails, Spikes, etc., Wrought 

Iron . 154 

Tan.. 482 

Tangential Wheel. 576 

Tangents and Co-tangents.416-426 

“ “ To Compute. 426 

Tannin, Quantity of, in Substances... 190 

Tee and Angle Iron, Weight of. . 130 

Teeth of Wheels.859-861 

“ Involute . 859 

Telegraph Wire, Span of . 179 

Telescopes, Opera-Glasses, etc. 671 

Telford Roads.688, 690 

Temperature. 195 

“ by Agitation . 524 

“ Decrease of,by Altitude. 522 

“ of Enclosed Spaces .526 

“ of Various Localities... 192 

“ To Reduce Degrees of 

Different Scales .523 































































INDEX. 


XXI 


Page 

Temperature, Underground . 519 

Temperatures, Metric . 37 

Tempering Boring Instruments. 197 

Tenacity of Iron Bolts in Woods . 198 

Tensile Strength.784-790 

Terne Plates... 124 

Terra Cotta...602 

Test of Plates of Iron . 749 

Theatres and Opera-Houses. 180 

Thermometers. 523 

Throwing Weights. 439 

Thrust, To Compute Weight of a Body, 

To Sustain . 693 

Tidal Phenomena. 75 

Tide Table, Coast of U. S. . 84 

Tides. 198 

“ of A tlantic and Pacific _191, 198 

“ of Pacific Coast . 85 

“ Rise and Fall of Gulf of Mexico 85 

“ Time of High Water .74, 75 

Tie-rods. 787 

Time, after Apparent Soon, before 

Moon next passes Meridian. 75 

“ Difference of .81-83 

Measures of . 37 

“ New Style . 37 

‘ ‘ Sidereal and Solar . 37 

“ To Compute Difference of be¬ 
tween New York and Grecn- 

■ wich . 83 

“ To Reduce to Longitude . 54 

Timber, Comparative Weight of Green, 217 
“ Measure , and to Compute 

Volume of .61, 62 

“ Waste in Hewing or Sawing , 62 

“ and Board Measure. 61 

“ and Woods.865-870 

“ Impregnation of. . 868 

“ Seasoning and Preserving... 865 

“ Strength of . 870 

Tin. 644 

“ and Lead Pipes, Weight of. . 139 

“ Plate, Marks and Weights . 139 

Tolerance, of Coins . 38 

Tonite. 443 

Tonnage, Approximate Rule . 176 

“ Builder’’s Measurement.... 176 

‘ 1 Corinthian and New Thames 

Yacht Club . 177 

“ Freight or Measurement... 177 

“ of Suez Canal . 177 

“ of Vessels . 175-177 

“ Royal Thames Yacht Club. 177 

“ To Compute . 173 

“ Units for Measurement, 

and Dead Weight Cargoes 176 

“ Weight of Cargo . 177 

Tools. 476 

Torsional Strength.790-797 

Towers and Domes . 180 

TowiDg, Erie Canal and Hudson 

River . 193 

Traction.843-849 

“ Ascending or Descending 

an Elevation . 846 

“ Canal, Slackwater, and 

River . 848 


Page 

Traction, Friction of Roads . 847 

“ Grade . 847 

“ Omnibus . 844 

“ on Common Roads .843, 844 

“ Resistance of a Car .849 

“ “ of a Stage Coach 848 

“ “ of Gravity and 

Grade . 847 

“ “ on Common Roads 

8 43~ 8 45 

“ Results of Experiments on. 843 

“ To Compute Power neces¬ 

sary to Sustain a Vehicle 
on an Inclined Road. 845, 846 

“ Various Roads and Vehi¬ 
cles . 845 

“ Wagon . 845 

Train, Service, of a Quartermaster... 198 

Tramways or Street Rails.435, 848 

Transportation, Canal . 193 

“ of Horses and Cattle. 192 

Transverse Strength.798-841 

Treadmill. 433 

Treenails. 783 

Trees, Large, in California . 184 

Trigonometry, Plane.385-389 

Trim, Change of, in a Vessel . 655 

Tripolith ... 198 

Trotting. 439 

Troy Measure. 32 

Truss, Hon . 178 

Tubes, and Flues ...747, 827 

“ and Pipes, To Compute Weight 

°f . x 47 

“ Brass, Weight of . 142 

“ Copper Drawn, Weight of. 140, 144 

“ Lap Welded Iron Boiler . 137 

“ or Girders... . 809 

“ Seamless Copper .140, 144 

“ Steel Locomotive . 138 

“ Wrought-iron .143,145 

Tubular Bridge... 178 

Tunnels, Lengths of. . 179 

Turbines. 572 

“ Boyden . 574 

“ Comparison with Water¬ 
wheels . 579 

“ Downward-flow . 574 

“ Fontaine . 574 

“ Fourneyron .573 

“ High - Pressure , Operation 

of..... . 574 

“ Inward-flow . 575 

“ Jonval . 575 

“ Low-pressure . 575 

“ Outward-flow . 573 

“ Poncelet . 574 

“ Ratio of Effect to Power... 577 

“ Swain . 575 

u Tremont . 576 

“ Victor . 576 

Turkish Plaster and Mortar..591, 592 

Turning. 477 

“ and Boring Metal . 197 

Turnips. 207 

Turpentine, Elastic Force of Vapor 

of. . 707 





































































































XXII 


INDEX. 


U. Page 

Underground Temperature. 519 

Undershot-Wheel .. 566 

Unguents, Value of . 471 

Uniform Motion. 617 


Value of Coins. 39 

“ and Weight of Foreign Coins . 40 

Vapor in Atmosphere, Volume of ... 68 

“ Weight of . 69 

1 £ Elastic Force of Alcohol, Ether, 
Sulphuret of Carbon, Petroleum, 

and Turpentine .707 

Variable Motion.617 

Variation of Magnetic Needle. 57 

“ Decennial, of Needle . 58 

“ of in U. S. and Canada ... 59 

Varnishes. 876 

Vegetable Marrow. 207 

Vegetables, Analysis of Meat and Fish, 200 
“ and Oil-calce, Nutritious 

Properties of. . 204 

“ Proportion of Starch in.. 205 

Tubers.. .207 

Vegetation, Limit of .. 192 

Velocities, Metric . 37 

Ventilation. 524 

“ of Mines .449 

Vernier Scale... 27 

Vessels, Elements of. . 653 

“ Hulls of. .830 

Veterinary. 186 

Volcanoes, Height of . 182 

“ Power of .9x0 

Volume and Weight of Various Sub¬ 
stances . 216 

Volumes. 193 

W. 

Walking.433, 438 

Wall, Chinese. 179 

Walls and Arches .602 

“ Centre of Gravity of .702 

“ Dams and Embankments .... 700 

“ Elements of . 702 

“ Friction of .698 

“ Moment of. . 701 

“ “ of Pressure .698 

“ of Buildings . 189 

“ Retaining, of Iron Piles . 196 

“ Revetment .694 

“ Stability of. . 702 

Warehouses, Brick Walls for . 603 

Warming Buildings.527-528 

By Hot-air Furnaces or 

Stoves . 528 

“ By Hot Water . 524 

“ By Steam . 527 

“ Coal Consumed per Hour . 527 

“ Furnaces . . . . 528 

“ Illustrations of Heating... 527 

“ Open Fires . 528 

‘ 1 Volume of Air Heated by 
Radiators, Consumption 
of Coal, Areas of Grate! 
and Heating Surface of 
Boiler, etc. 528 


Page 

Washington Aqueduct. 178 

Water.849-852 

“ Boiling-Points of . . 851 

‘ ‘ Density of. . 520 

“ Deposits of. . 852 

“ Expansion of ... 519 

“ Fresh and Sea .849, 851 

“ Inch . 557 

“ Motors, Ratio of Effective 

Power ... 563 

“ Power . 562 

“ Pressure Engines . 579 

“ Rainfall and Volume of.... 850 

“ Resistance of, to an Area of 

One Sq. Foot .646 

“ Velocity of a Falling Stream 

of . 49 6 

“ Volumes of. . 849 

“ Weight of . 852 

‘ 1 Wheels .563, 730 

“ “ Compared with Tur¬ 
bines . 579 

“ “ Overshot . 563 

“ “ Undershot. . 566 

Waterfalls and Cascades.. 184 

Watermelon. 207 

Water Pipes, Cast-Iron . 147 

“ Dimensions, etc. .. .138, 139 
Water-Wheel, Centre of Gyration.. 611 
Water-Wheels, Diameter and Journal 

of a Shaft, etc. 581 

“ Dimensions of Arms .571 

Waves of the Sea.852, 853 

“ Tidal . 853 

“ Velocity of . 853 

Weather-Foretelling Plants. 185 

‘ ‘ Glasses ... .... 430 

“ Indications . 431 

Wedge.630 

Weighing without Scales. 66 

Weight, Avoirdupois ...32, 47 

“ and Diameter of Cast-Iron 

Balls . 153 

“ and Dimensions of Lead 

Balls . 501 

‘ 1 and Dimensions of Wrought- 

iron Bolts and Nuts .. 156-158 

“ and Fineness of U. S. Coins. 38 

“ and Marks of Tin Plates... 139 

“ and Mint Value of Foreign 

Coins . 40 

“ and Strength of Hemp and 

Wire Ropes ... 172 

“ and Strength of Iron Wire, 

etc. 124 

“ and Strength of Stud-link 

Chain Cables . 168 

“ Angle and T Iron .125,130 

“ Apothecaries' .32, 47 

“ Bells . 180 

‘ 1 Brain . . 192 

“ Brass .136,149 

“ “ Plates . 146 

“ “ Sheet and Tubes cor¬ 
responding to Iron. 142 

“ “ Wire .20,121 

“ Cast Iron . 149 


































































































INDEX. 


XX111 


Page 

Weight, Cast-Iron and Lead Balls.. 153 

“ “ Bar or Rod . 131 

“ “ Pipes or Cylinders 132 

“ “ Plates . 146 

“ Composition Sheathing Nails 135 

“ Copper . 136 

“ “ Rods or Bolts . 148 

“ “ Seamless Tubes . 144 

“ “ Sheet... . 135 

“ Corrugated Roof Plates .... 131 

“ Cube Foot of Embankments, 

Walls, etc. 694 

‘‘ Diamond r and Diamonds. 32, 193 

“ Electrical . 34 

“ Flat Rolled Iron .126-128 

“ Foods, to Furnish Nitroge¬ 
nous Matter . 202 

“ Galvanized Sheet Iron .124 

“ Grain . 32 

“ “ and Roots . 34 

“ Green and Seasoned Timber 217 

“ Gun Metal . 149 

“ Hemp and Wire Rope . 166 

“ Hexagonal, Octagonal, and 

Oval . 135 

“ Horses . 35 

“ Ingredients, that of Com¬ 
pound being given .218 

“ Iron, Steel, Copper, and Brass 

Plates .118, 119 
“ “ “ <£ Wire. .120-121 

“ Lead....... . 32 

“ “ Encased Tin Pipes... 151 

“ 11 Pipes 139, 150 

“ “ Plates.. . 146 

“ “ Sheet . 151 

“ Men and Women . 35 

“ Metal by Weight of Pattern. 217 

“ Metals of a Given Sectional 

Area . 149 

“ Molecules, Weight, etc. 194 

“ of Articles of Food Consumed 

in Human System to De¬ 
velop Power of Raising 140 
Lbs. to a Height of 10 000 Ft. 204 

“ of Beef and Cattle . 35 

“ of Cast and Wrought Iron.. 155 

“ “ “ “ Steel, 

Copper, and Brass. 136 

“ of Earths . 33 

“ of Offal . 35 

<£ of Sq. Foot of Slating . 64 

“ Rocks, Earth, etc. 468 

“ Rolled and Bar Steel .... 134,135 

“ Round Rolled Iron . 126 

“ Sheet Iron . 129 

“ Steam .705 

“ Steel ....136,149 

“ “ Plates 134, 146 

“ Tin Pipe 139, 151 

“ To Ascertain, of a Solid or 

Liquid Substance .217 

“ To Compute, of an Elastic 

Fluid . 217 

“ Various Materials . 155 

“ “ Substances in Bulk 217 

“ “ “ per Cube Foot 217 


Page 

Weight, Wire and Hemp Rope . 166 

“ Wood . 33 

“ Wrought and Cast Iron .... 155 
“ “ Iron. .125, 126,136,149 

“ “ Plates . 146 

“ “ Sheet and Hoop... 129 

“ “ Tubes .143-145 

“ “ Tubes and Plates, 

145,146 

“ Zinc Sheets .123,146,151 

“ “ “ and Dimensions 

of..., . 151 

Weights and Measures.26-35 

“ and Volumes of Various 

Substances . 216 

“ English and French . 44 

“ Foreign . 48 

“ Grecian . 53 

“ Hebrew and Egyptian . 53 

“ Measures of .32, 47 

“ Metric..... .33, 36 

“ Miscellaneous . 33 

“ of Steam-Engines. .758, 759, 911 

“ Roman . 53 

Well, Artesian..179, 198 

“ Boring. 197 

Wells or Cisterns, Excavation of, 

etc. 63 

Welding.. 786 

“ Cast Steel, Composition for.. 634 

“ Fluxes for .636 

Wheel and Axle. 626 

“ and Pinion, Combinations or 

Complex Wheel Work . 628 

Wheel Gearing.854-861 

“ Circumference of . 857 

“ General Illustrations .858 

“ Pitch, Diameter , Number of 

Teeth, Velocity, etc.855, 857 

“ Revolutions of . . 858 

“ Spur Gear . 911 

“ Teeth of. . 859 

“ To Compute Diameter of.... 857 

“ “ VP of a Tooth .861 

“ “ Velocities of..... 856,857 

Wheels , Proportions of . 862 

“ Teeth of. .859-862 

Whitewash or Grouting. 594 

Wind, Course of. .675 

* ‘ Effective Impulse of. .665 

“ Force of. .674 

“ Pressure of. . 911 

“ Velocity and Pressure .674 

Winding Engines.476, 862, 863 

“ To Compute Diameter of a 

Drum . 862 

“ “ Number of Revo¬ 
lutions . 863 

Windlass.433 

“ Chinese .627 

Windmills.863-865 

“ Results of Operation of... 865 
u To Compute Elements of .. 864 

Window Glass. 124 

W'ine and Spirit Measures... 45 

Wire Gauge, French. . 123 

“ Standard of Great Britain 122 





























































































XXIV 


INDEX. 


Page 


Wire Ganges.... 122 

‘ ‘ Iron Gauge , Weight and Length 

of . 163 

Wire, Length of. . 124 

“ Rope......161,473 

“ and Equivalent Belt . 167 

“ and Hemp, General Notes .167 

“ Cables, Galvanized Steel . 163 

“ Endless . 167 

“ Fence, Weight and Strength of. 164 
“ Results of an Experiment with 

Galvanized . 161 

“ “ of Experiments on, at 

U. S. Navy Yard . 169 

“ Transmission of Power of.... 167 
“ Ropes, Hemp, Iron, and Steel, 

Relative Dimensions of..... 168 
‘ ‘ and Hemp Rope, Iron and Steel, 

Relative Dimensions of . 172 

“ and Hemp Ropes, Weight and 

Strength of. . 172 

u and Tarred Hemp Rope, Haw¬ 
sers, Cables, Comparison of.. 169 
1 ‘ Rope , Circumference of to Com¬ 
pute, . 169 

“ “ for Standing Rigging, 

Circumference of, to Compute 172 

“ Shrouds ..... 173 

Woods.481, 765, 769, 782, 783 

“ Bituminous or Lignite . 479 

“ Coefficients for Safety . 835 

“ Detrusive Strength of. .782 

“ Floor Beams . 835 

“ Measure. . 47 


“ Relative Value of their Crush¬ 
ing Strength and Stiffness 

combined . 769 

“ Safe Statical Loads for .834 

“ Sawing . 196 

“ Weights of. . 33 

Wood and Timber .865-807 

“ Creosoting, Effects of. .869 

“ Decrease by) Seasoning _869 


Page 

Wood and Timber, Defects of .866 

‘ ‘ Durability of Various .869 

1! Impregnation of. . 868 

‘ 1 Proportion of Water in... 869 

“ Seasoning and Preserving, 

866,868 

“ Selection of Trees . 865 

“ Strength of. .833, 870 

“ Transverse Strength of, to 

Compute .833 

“ Weight of Oak and Yellow 

Pine per Cube Foot .870 

Work.432 

“ Accumulated in Moving Bod¬ 
ies, etc. 619 

“ and Power, Metric . 36 

Works of Magnitude.178, 179 

Wrought Iron. .. .639, 765, 768, 773, 785 
“ Crushing Weight of 

Columns . 768 

“ Deflection of Bars, 

Beams, etc.773-775 

“ “ of Rails . 776 

“ Plates and Bolts.... 749 

“ Plates, Weight of. 118, 119 

“ To Compute Weight 

for . 125 

“ Weight of. . 155 

“ Wire, Weight of.. 120,121 

1 Y. 

Yam. 207 

Year, Bissextile or Leap . 70 

“ Civil . 70 

1 ‘ Solar . 70 

Years of Coincidence. 74 

Z. 

Zinc. 644 

“ Plates, Weight of . . 146 

“ Sheets, Thickness and Weight of, 

123, 151, 152 

Zinc Foil in Steam Boilers . 912 

Zones op a Circle, Areas of _269-271 























































EXPLANATIONS OF CHARACTERS AND SYMBOLS 

Used in Formulas, Computations, etc., etc. 

= Equal to, signifies equality; as 12 inches = 1 foot, or 8 X 8 = 16 x 4. 

+ Plus, or More , signifies addition ; as 4 + 6 + 5 = 15. 

— Minus, or Less, signifies subtraction ; as 15 — 5 = 10. 

X Multiplied by, or Into, signifies multiplication; as 8 X 9 = 72. a X d, 
a.d, or ad, also signify that a is to be multiplied by d. 

-r- Divided by, signifies division; as 72-4-9 = 8. 

: Is to, :: So is, : To, signifies Proportion, as 2:4:: 8:16; that is, as 2 is 
to 4, so is 8 to 16. 

.*. signifies Therefore or Hence , and v Because. 

~ Vinculum, or Bar, signifies that numbers, etc., over which it is 
placed, are to be taken together; as 8 — 2 + 6 = 12, or 3 x 5 + 3 = 24. 

. Decimal point , signifies, when prefixed to a number, that that number 
has some power of 10 for its denominator; as .1 is .15 is etc. 

co Difference, signifies, when placed between two quantities, that their 
difference is to be taken, it being unknown which is greater. 

V Radical sign, which, prefixed to any number or symbol, signifies that 
square root of that number, etc., is required; as Vg, or Va +6. The degree 
of the root is indicated by number placed over the sign, which is termed 
index of the root or radical; as v / , v / , etc. 

> 1 , < L signify Inequality , or greater, or less than, and are put between 
two quantities ; as a 1 b reads a greater than b, and a\_,b reads a less than b. 

() [ ] Parentheses and Brackets signify that all figures, etc., within them 
are to be operated upon as if they were only one; thus, (3 + 2) x 5 = 25; 
T8-2] X 5 = 30 - 

± signify that the formula is to be adapted to two distinct cases, as 
cip v = c, either diminished or increased by v. Here there are expressed 
two values: first, the difference between c and v; second, the sum of c and v. 

In this and like expressions, the upper symbol takes preference of the lower. 

p or 7r is used to express ratio of circumference of a circle to its diameter 
= 3- I 4i6; ^ = .7854, and-^ = .5236. 

0 ' " signify Degrees, Minutes, Seconds, and Thirds. 

' " set superior to a figure or figures, signify, in denoting dimensions, Feet 
and Inches. 

a' a" a"' signify & prime, a second, a third, etc. 

1,2, added to or set inferior to a symbol, reads sub 1 or sub 2, and is used 
to designate corresponding values of the same element, as h, hi, h 2 , etc. 

2 , 3 , 4 , added or set superior to a number or symbol, signify that that num¬ 
ber, etc., is to be squared, cubed, etc.; thus, of means that 4 is to be multi¬ 
plied by 4; 43, that it is to be cubed , as 4 3 = 4 X 4 X 4 = 64. The power, 
or number of times a number is to be multiplied by itself, is shown by the 
number added, as 2 , 3 , 4 , 5 , etc. 






22 


ALGEBRAIC SYMBOLS AND FORMULAS. 


2, etc., set superior to a number, signify square or cube root, etc., of the 

number; as 2 1 signifies square root of 2; also 3 , 3 , a , etc., set superior 

to a number, signify two thirds power, etc., or cube root of square, or square 

or cube root of 4th power, or cube root of sixth power; as 8 3 = V~& or 
= (v / 8) 2 . 


i-7, 3-6, etc., set superior to a number, signify tenth root of 17th power, etc. 
•02 } .059, set superior to a number, signify hundredth root of 2d power, or 
thousandth root of 59th power, the numerator indicating power to which 
quantity is to be raised, and denominator indicating root which is to be ex¬ 
tracted. 


00 signifies Infinite , as - or a quantity greater than any assignable quan- 
a ^ 

tity. Thus, - = 00 signifies that 0 is contained in any finite quantity an in¬ 
finite number of times: - = a, — = 10a, etc. 

1 1 .1 ’ 


cc signifies Varies as. Thus, Mcc DxV signifies that mass of a body in¬ 
creases or diminishes in same ratio as product of its density and volume, or 
S cc t 2 , signifies S varies as t 2 . 


21 . signifies Angle. Peipendicular. A Triangle. □ Square , as □ 
inches; and g] cube, as cube inches. 

Notes. —Degrees of temperature used are those of Fahrenheit. 
g is common expression for gravity = 32.166, 2 g — 64.33, -f 2 # = 8.02 feet. 

£3 signifies Dead Flat, denoting dimensions or greatest amidship section 
of hull of a vessel. 


ALGEBRAIC SYMBOLS AND FORMULAS. 

h' representing h prime, v representing versed sine, 


chord, 
area, 
radius , 


h. 

sin. 

9 


l representing length, 
b “ breadth 

d “ depth, 

h “ height, 

l+b 

— sum of length and breadth divided by depth. 
lb 

=. product of length and breadth divided by depth. 
l—b 

— difference of length and breadth divided by depth. 

1 2 b 3 = product of square of length and cube of breadth. 

= square root of length divided by cube root of breadth. 

VT+b 


h sub, 

sine, 

gravity. 


d 


square root of sum of length and breadth divided by depth. 


3 jhooh, _ cu be roo ^. 0 f difference of h prime and h sub, divided by 

V y 2g 

square root of 2 g. 

V a-\-(c—r) 2 =zx. Add square of difference between the chord and ra¬ 
dius to the area, and extract the square root; the result will be equal to x. 

Note. —It is frequently advantageous to begin interpretation of a formula at its 
right hand, as in the above case. 





ALGEBRAIC SYMBOLS AND FORMULAS. 


23 


7 / (x-\-y) 2 

1 V —-x=2. Divide square of sum of x and y by square of y; 

subtract unity from quotient; extract square root of result; multiply it by 
length, and product will be equal to z. 

2(sin. 

. Divide twice square of sine of the angle of 75° by square 

i-t-^sin. 75 ) 

of sine of the angle of 75° added to unity. 


>. a j 
'2a) 2 ( 


SV2 g (Vh— Vh) +2.303 c. log. \ | 


: t. Multiply 


(SV2g) 2 ( SV ^zgh 

S by the V of 2 g, and this product by difference between square roots of h 
and h prime.; add this to 2.303 times common logarithm of quotient arising 
from dividing product of S into V 2gh diminished by b by product of S 
into V zgh prime diminished by 6, and multiply this sum by the quotient 
of 2d divided by square of product of S into V 2y, which will be equal to t. 

2 a + 3 cos. 98° = 2 a — 3 cos. 82° = twice a diminished by three times 
cosine of 82°. 

Cosine of any angle greater than 90 0 and less than 270 0 is always — or negative, 
but is numerically equal to cosine of its supplement, i. e., remainder after subtract¬ 
ing angle from 180 0 . 

39.127 — .099 82 cos. 2 L = 1 . Assuming L less than 45 0 , as 42 0 , this equation 
becomes 39.127— .099 82 cos. (2 X 42° = 84°) = f; and also, I, greater than 45 0 , as 
50 0 , it becomes 39.127 + .099 82 cos. (180 0 ; —2 X 50° = 8o°) = l. 

L — io° N = L + io° S, as a negative result furnished by a formula in¬ 
dicates a positive result in an opposite direction. 

— —- ^ V ^ - 2 --~ =y. Minus, the fraction B minus &, times v, plus . 

2 times BY, divided by B plus b, is equal to y. 

Sin. _I a;, tan. “ 1 x , cos. _I x, signifies the arc, the sine, tangent or cosine 
of which is x. Thus, if x = .5, this is 30°, as 30° is the arc, the sine of 
which is .5. 

(Sin. x)~ 


: _i_ i = a- 2 = <,- 3 = 4 ,and 4++=S- 

sin. x a, b c r —1 

Raise r to nth power, i. e., multiply r by itself and this result by r, and so 
on, until r appears in result as a factor, as many times as there are units 
in n. Multiply this result by /, diminish this by l; divide remainder by r 
raised to the nth power, diminished by r raised to a power whose exponent 
is n diminished by 1, and quotient = or is value of S. 

n -jJL = r . Divide l by a and extract that root of the quotient, index 
of which is n diminished by 1, and this root is = or value of r. 

Logarithm of a Number is exponent of the power to which a particular 
constant quantity must be raised in order to produce that number. 

Constant Quantity is termed the base of the system. 

Common (or Brigg's) Log. is the logarithm the base of which is 10. 

Hyperbolic Log. is the logarithm the base of which is 2.718 28. 

Com. Log. — Hyp. log. X -434 2 94 * 

Hyp. Log. — Com. log. X 2.302 585052994, ordinarily 2.303 or 2.3026. 










DIFFERENTIAL AND INTEGRAL CALCULUS. 


Illustration.—W hen a number, hyp. log. of which = a given figure or number, 
is required. 

Multiply figure or number (hyp. log.) by ,434294 (modulus of com. log.) = com. log. 
of figure. 

Thus, Required the number, hyp. log. of which = .02. .02 X -434 294 = .00 868 588, 

com. log., and 1.0202 = number. 

Log. ioo-° 59 — 059 X log. of xoo — .059 X 2 = .ii 8; the number corresponding to 
log. .118, is 1.3122; hence, 100- 0 59— 1.3122. That is, if 100 is raised to 59th power, 
and the 1000th root is extracted, the result will be 1.3122. 

In Equation , u = 3 x 2 — 2 x, u is termed a function of x. If it is desired 
to indicate the fact that u thus depends for its value upon the value of x , 
without expressing exact value of u in terms of x, following notation is 
use d : u —f(x), u = F(x), or u = <p ( x ). 

Each of these notations is read, u is a function of x. If in such function 
of x the value of x is assumed to commence with o and to increase uniformly, 
the notation indicating rate of increase is d x, and is read “ the differential 
of x.” 

Differentiation, d is its symbol, and it is the process of ascertaining the 
ratio existing between the rate of increase or decrease of a function of a 
variable and the rate of increase or decrease of the variable itself. If 
y — 3 x 2 , y or its equal 3 x 2 is the function of x, and x is the independent 
variable, while the exponent of the variable or the primitive exponent is 2. 

By the operation of Calculus, such expressions are differentiated by di¬ 
minishing the exponent of the variable by unity, multiplying by the prim¬ 
itive exponent, and attaching the d x. 

Hence, dy — 2 x %xdx — 6 x dx. This indicates the relation between 
the differential of y, the function of x, and the differential of x itself. 

Assume that x increasing at rate of 3 per second becomes 4; that is, x = 4, 
and d x = 3 ; hence dy — 6 X 4 X 3 = 72. That is, if x is increasing at 
rate of 3 per second, at the time that x = 4, the function itself is increasing 
at rate of 72 per second. 

To differentiate an expression of two or more terms, it is necessary to 
differentiate them separately and connect the results with the signs with 
which the terms are connected. 

Thus, differentiating u — 3x 2 — 2x, we have du = d (3X 2 — 2x) = 6xdx 

— 2 d x = (6 x — 2) d x. 

Assuming x = 4 and dx = 3, we have du = { 6x4 — 2) X 3 = 66. This 
indicates that when x = 4, and is increasing at rate' of 3 per second, the func¬ 
tion u, or 3 x 2 — 2 x, is at same instant increasing at rate of 66 per second. 

Integration. Its symbol / was originally letter S, initial of sum, the 
symbol of an operation the reverse of differentiation; and when the oper¬ 
ation of integration is to be performed twice, thrice, or more times, it is 
written //, ///, etc. 

By the operation of Calculus, expressions are integrated by increasing the 
exponent of the variable by unity, dividing by the new exponent, and de-» 
taching the d x. 

Hence, integrating the differential 6 xdx, we have / 6x dx = 3 x 2 . This 
result is the function, the differential of which is 6 xd'x. 

To integrate an expression of two or more terms, it is necessary to inte¬ 
grate the terms separately and connect the results with the signs with which 
the terms are connected. 

Thus, integrating (6x—2) dx, we have /(6x—2) dx=z f (6 xdx—2 dx) 
= 3 x 2 — 2 x. This result is the function the differential of which is (6x 

— 2) dx or (6x — 2x°) dx. ' 

Note.—A quantity with the exponent °, as x° or 3 0 , is equal to unity. 



NOTATION. 


25 


The operation of summation may also be illustrated in use of the sym¬ 
bol / . Assuming x = 4, the former of the preceding results becomes 
f6xdx = ^x 2 — 48, the latter / (6 x — 2) dx — ^x 2 — 2 a? = 40. 

Here x is assumed to commence at o and to continue to increase by in¬ 
finitely small increments of dx until it becomes 4. The summation is the 
addition of all these values of x from o to 4. 

Arithmetically. —The first formula may be written 

6 (x + x" + x"‘ + etc.) d x. If then x is to advance from o to 4 by in¬ 
crements of 1, we have 6 (0+1 + 24-3 + 4) X 1=60, which exceeds 48. 
If d x is assumed to be .5, the result is 54. The correct result is obtained 
only when dx is taken infinitely small. By Arithmetic this is approximated, 
but it is reached by the operations of Calculus alone. 

The second formula may be written 

(6 [x + x" + x'" + etc.] — 2 \x°' +.a 5 °" + x°'" etc.]) dx. Assuming x = 
4, and dx zz: 1, we have (6[i + 2 + 3 + 4j— 2 [i + i + i + i]) x 1=52, 
which exceeds 40. If dx= .25, the result would be 43, and if .125 it would 
be 41.5, ever approaching but never reaching 40, so long as a finite value is 
assigned to d x. 

A, Delta , when put before a quantity, signifies an absolute and finite in¬ 
crement of that quantity, and not simply the rate of increase. 

2 , Sigma , signifies the summation of finite differences or quantities. Thus, 
2 y 2 Ax — (y' 2 + y" 2 + y'" 2 + etc.) A x. Assume y = 6, y" — 8 , y'“ = 4, and 
A x the common increment of x — 5, then hy 2 Ax — (36 + 64+16) X 5 = 
580. 


NOTATION. 


1 = 1. 

20 = XX. 

1 000 = M, or CIO. 

2 = 11. 

30 = XXX. 

2 000 = MM. 

3 = III. 

40 = XL. 

5 000 = Y, or IOO. 

4 — IY. 

50 = L. 

6000 = VI. 

5 = V. 

60 = LX. 

10 000 = X, or CCIOO. 

6 = VI. 

. 70 = LXX. 

50 000 = L, or 1000 . 

7 = VII. 

80 = LXXX. 

60 000 = LX. 

8 = VIII. 

90 = xc. 

100 000 = C, or CCCIOOO. 

9 = IX. 

100 = C. 

1 000 000 = M, or CCCCIOO 

10 = x. 

500 = D, or 10. 

2 000 000 = MM. 


As often as a character is repeated, so many times is its value repeated, 
as CC = 200. 

A less character before a greater diminishes its value, as IY = Y — I. 

A less character after a greater increases its value, as XI = X + 1 . 

For every 0 annexed to 10 the sum as 500 is increased 10 times. 

If C is placed on left side of I as many times as 0 is on the right, the 
number is doubled. 

A bar, thus , over any number, increases it 1000 times. 

Illustration 1.—1880, MDCCCLXXX. 18 560, XVIIIDLX. 

2. —10 = 500. CIO = 500 x 2 = 1000. 100 = 500 x 10= 5000. 

CCI 00 = 5000 X 2=10 000. 1000 = 500 x 10 x 10 = 50 000. CCCIOOO 
= 50 OOO X 2 = IOO OOO. 

c 




26 CHRONOLOGICAL ERAS.-MEASURES AND WEIGHTS. 


CHRONOLOGICAL ERAS AND CYCLES FOR 1884. 

The year 1884, or the 109th, year of the Independence of the United States of America, 
corresponds to 

The year 7392-93 of the Byzantine Era; 

“ 6597 of the Julian Period; 

i: 5644-45 of the Jewish Era; 

“ 2660 of the Olympiads, or the last year of the 665th Olympiad, commenc¬ 

ing in July (1884), the era of the Olympiads being placed at 775.5 
years before Christ, or near the beginning of July of the 3938th 
year of the Julian Period; 

“ 2637 since the foundation of Rome, according to Varro; 

“ 2196 of the Grecian Era, or the Era of the Seleucidte; 

“ 1600 of the Era of Diocletian. 

The year 1301 of the Mohammedan Era, or the Era of the Hegira, begins on the 
7th of February, 1884. 

The first day of January of the year 1884 is the 2,409,178th day since the com¬ 
mencement of the Julian Period. 


Dominical Letters.F, E I Lunar Cyele or Golden Number 

Epact. 3 I Solar Cycle ... 


Roman Indiction was a period of 15 years, in use by the Romans. The precise 
time of its adoption is not known beyond the fact that the year 313 A-D. was a first 
year of a Cycle of Indiction. 

Julian Period is a cycle of 7980 years, product of the Lunar and Solar Cycles and 
the Indiction (19 X 28 X 15), and it commences at 4714 years B C. 

6513 -f- (given year — 1800) = year of Julian Period, extending to 3267. 

Note.— If year of Julian Period is divided by 19, 28, 15, or 32, the remainders will 
respectively give the Lunar and Solar Cycles , the Indiction , and the Year <f the 
Dionysian. 


MEASURES OF LENGTH. 

Standard of measure is a brass scale 82 inches in length, and the 
yard is measured between the 27th and 63d inches of it, which, at tem¬ 
perature of 62°, is standard yard. 


Lineal. 


12 inches = 1 foot. 
3 feet = 1 yard. 
5.5 yards = 1 rod. 
40 rods 
8 furlongs = 1 mile. 


= 1 furlong. 


Inches. Feet. Yftrds. Rods. Furl. 

36 = 3 - 

198= 16.5= 5.5. 

7920= 660 = 220 = 40. 

63 360 = 5 280 = 1 760 = 320 = 8. 


Inch is sometimes divided into 3 barleycorns, or 12 lines. 
A hair’s breadth is .02083 (48th part) of an inch. 

1 yard = .000 568, and 1 inch—..0000158 of a mile. 


Grnnter’s Chain. 

7.92 inches = 1 link. | 100 links = 1 chain, 4 rods, or 22 yards. 

80 chains = 1 mile. 

Ropes and Cak>les. 

1 fathom = 6 feet. | 1 cable’s length =120 fathoms. 


Geograplaical and Nautical. 

1 degree, assuming the Equatorial radius at 6 974 532.34 yards, as given 
by Bessel, = 69.043 statute miles = 364 556 feet. 

1 mile = 2028.81 yards or 6086.44 feet. 

1 league = 3 nautical miles. 








MEASURES AND WEIGHTS. 


27 


Log Lines. 

Estimating a mile at 6086.43 feet, ancl using a 30" glass, 

1 knot = 50 feet 8.64 inches. | 1 fathom = 5 feet .864 inches. 

If a 28" glass is used, and 8 divisions, then 

1 knot = 47 feet 4 inches. | 1 fathom = 5 feet 11 inches. 

The line should be about 150 fathoms long, having 10 fathoms between chip and 
first knot for stray line. 

Note. —This estimate of a mile or knot is that of U. S. Coast Survey, assuming 
equatorial radius of Earth to be 6074 532. 34 yards and a meter to be SQ.S6850535 
inches of the Troughto-n scale at 62°. 

Cloth.. 

1 nail =: 2.25 inches. | 1 quarter = 4 nails. | 5 quarters = 1 ell. 

Pendulum. 

6 points = 1 line. | 12 lines = 1 inch. 

Shoemakers’. 

No. 1 is 4.125 inches, and every succeeding number is .333 of an inch. 
There are 28 numbers or divisions, in two series or numbers—viz., from 1 
to 13, and 1 to 15. 

nVTiscellaneou.s. 

12 lines or 72 points = 1 inch. 1 hand = 4 inches. 

1 palm = 3 inches. 1 span = 9 inches. 

1 cubit = 18 inches. 

• 

“Ve rixier S cal e. 

Vernier Scale is divided into 10 equal parts; so that it divides a scale 
of ioths into iooths when two lines of the two scales meet. 

HMetric, by fAct of Congress of «Tnly 28 , 1866 . 


Unit of Measurement is the Meter, which by this Act is declared to be 39.37 ins. 


Denominations. 

Meters. 

Inches. 

Feet. 

Yards. 

Miles. 

Millimeter. 

. IOO 

•0394 

•3937 

3-937 

39-37 

393-7 




Centimeter. 

. IO 




Decimeter. 

. I 

.328 083 
3.28083 
32.80833 
328.083 33 
3280.833 33 



Meter . 

I. 

I. OQ1 6l 


Reknmeter. 

IO. 

10.93611 
109.36111 
1093.61111 


TTektameter. 

IOO. 


Kilometer. 

I 000. 


.62T *27 

Myriameter. 

IOOOO. 

— 

6.2137 


In Metric system, values of the base of each measure—viz., Meter, Liter, Stere, 
Are, and Gramme—are decreased or increased by following prefix. Thus, 


Milli, 1000th part or .001. I Deci, 10th part or .1. I Hekto, 100 times value. 
Centi, 1 ooth “ .01. I Deka, 10 times value. | Kilo, 1000 “ 

Myria, 10000 times value. 

Note.— The Meter, as adopted by England, France, Belgium, Prussia, and Russia, 
is that determined by Capt. A. R. Clarke, R.E., F.R.S., 1866, which at 32 0 in terms 
of Imperial standard at 62° F. is 39.370432 inches or 1.09362311 yards , its legal 
equivalent by Metric Act of 1864 being 39.3708 inches , the same as adopted in 
France. 

Captain Kater’s comparison, and the one formerly adopted by the U. S. Ordnance 
Corps, was = 39.370 797 1 inches , or 3.28089976 feet , and the one adopted by the 
U. S. Coast Survey, as above noted, is = 39.368 505 35 inches. 



















28 


MEASURES AND WEIGHTS, 


Equivalent Values in Metric Denominations of TJ. S. 


Denominations. 

Value in Meters. 

Denominations. 

Values in Meters. 

Tnrh. 

.0254 
.304 800 6 
.914 401 8 

Rod. 

5.029 209 9 
201.168396 

1609. 347 168 

Foot. 

Furlong.. 

Yard. 

Mile. 


Approximate Equivalents of Old. and IVTetric TJ- S. 
Measures of Length. 


i Chain .. 
i Furlong 


= 20 meters. 

— 200 “ 

5 Furlongs ... = i kilometer. 


i Kilometer .... = .625 mile. 

1 Mile.= 1.6 kilometers. 

1 Pole or Perch . = 5 meters. 

1 Foot.=: 3 decimeters or 30 centimeters. 

1 Metre. = 3.280 833 feet = 3 feet 3 ins. and 3 eighths. 

xi Meters.=12 yards. | 1 Decimeter .. . —finches. 

1 Millimeter .. = 1 thirty-second of an inch. 

To Convert Meters into Inches .— Multiply by 40; and to Convert Inches 
into Meters .— Divide by 40. 

Approximate rule for Converting Meters or parts, into Yards .— Add one 
eleventh or .0909. 

Inches Decimally = Millimeters. 


Inches. 

Milli¬ 

meters. 

Inches. 

Milli¬ 

meters. 

Inches. 

Milli¬ 

meters. 

Inches. 

Milli¬ 

meters. 

.OI 

•25 

.2 

5.08 

.48 

12.2 

.76 

19-3 

.02 

•51 

.22 

5 - 59 

•5 

I2.7 

d 8 

19.8 

•03 

.76 

.24 

6.1 

•52 

13.2 

.8 

20.3 

.04 

1.02 

.26 

6.6 

•54 

I 3-7 

.82 

20.8 

•05 

I. 27 

.28 

7 - n 

•56 

14.2 

.84 

21.3 

.06 

1.52 

•3 

7.62 

•58 

14.7 

.86 

21.8 

.07 

1.78 

•32 

8.13 

.6 

15.2 

.88 

22.4 

.08 

2.03 

•34 

8.64 

.62 

i 5-7 

•9 

22.9 

.09 

2.29 

■36 

9- 1 4 

.64 

16.3 

.92 

23-4 

. I 

2-54 

•38 

9- 6 5 

.66 

16.8 

•94 

23-9 

.12 

3-oS 

•4 

IO. 2 j 

.68 

17-3 

.96 

24.4 

.14 

3-56 

.42 

IO.7 

•7 

17.8 

.98 

24.9 

.l6 

4.06 

.44 

II .2 

.72 

18.3 

I. 

25-4 

.18 

4-57 

.46 

11.7 

•74 

18.8 




Inches. 


3 

4 

5 

6 

7 

8 

9 

10 

n 

12 


Milli¬ 

meters. 


50.8 

76.2 

101.6 

127 

152.4 

177.8 
203.2 

228.6 
254 

279.4 

304.8 
foot 


Inches in Fractions = Millimeters. 


Eighths. 

Six¬ 

teenths. 

Thirty- 

seconds. 

Milli¬ 

meters. 

Eighths. 

Six¬ 

teenths. 

Thirty- 

seconds. 

Milli¬ 

meters. 

Eighths. 

Six- 

1 teenths. 

J 03 

t! c 

— 0 

H % 

Milli¬ 

meters. 

Eighths. 

Six¬ 

teenths. 

Thirty- 

seconds. 

Milli¬ 

meters. 



I 

•79 



9 

7.14 



17 

13-5 



25 

19.8 


I 

— 

i-59 


5 

— 

7-94 


9 

— 

14-3 


13 

— 

20.6 



3 

2.38 



II 

8-73 



J 9 

I 5- 1 



27 

21.4 

I 

— 

— 

3-U 

3 

— 

— 

9-52 

5 

— 

— 

i5-9 

7 

— 

— 

22.2 



5 

3-97 



13 

10.32 



21 

16.7 



29 

23 


3 

— 

4.76 


7 

— 

II. II 


11 

— 

i7-5 


15 


23.8 



7 

5-56 



15 

II.91 



23 

18.3 



3i 

24.6 

2 

— 

— 

6-35 

4 

— 

— 

12.7 

6 

— 

— 

*9 

8 

— 

— 

25-4 


By means of preceding tables equivalent values of inches and millimeters, 
equivalent values of inches in centimeters, decimeters, and meters, may be 
ascertained by altering position of decimal point. 

Illustration.— Take 1 millimeter, and remove decimal point successively by one 
figure to the right; the values of a centimeter, decimeter, and meter become 

In. ' Ins. 

1 millimeter.0394 I 1 decimeter.3.94 I .32 inch = 8.13 millimeters. 

1 centimeter.394 | 1 meter.39.4 | 3.2 inches=;8i.3 “ 










































































MEASURES AND WEIGHTS. 


2 9 


MEASURES OF SURFACE. 

144 square inches = 1 square foot. | g square feet = 1 square yard. 
Architect's Measure , 100 square feet = 1 square. 

Land. 


30.25 square yards = 1 square rod. 

40 square rods = 1 square rood. 

5 square roods ) 

10 square chains J 1 acre * 

640 acres = 1 square mile. 


Yards. Rods. Roods. 

1210 . 

4840 = 160. 

3 097 600 = 102 400 = 2560. 


208.710 326 feet, 69.570109 yards square, or 220 by 198 feet square = 1 Acre. 


Paper. 

24 sheets = 1 quire. | 20 quires = 1 ream. | 21.5 quires = 1 printer’s ream. 
2 reams = 1 bundle. | 5 bundles = 1 bale. 


Drawing. 


Cap. 

13 x 16 inches. 

Columbier .... 

2 3 X 34 inches. 

Demy. 

15 X 20 “ 

Atlas. 

26 x 34 

a 

Medium. 

17 X 22 “ 

Theorem. 

28 X 34 

a 

Royal. 

19 x 23 “ 

Doub. Elephant, 

27 X 40 

u 

Super-royal .. . 

19 x 27 “ 

Antiquarian . . . 

3 i X 53 

u 

Imperial. 

22 X 30 “ • 

Emperor. 

40 X 60 

u 

Elephant. 

23 X 28 “ 

Uncle Sam . . . . 

48 X 120 

u 


Peerless. 

. 18 x 52 inches. 




Tracing. 



Double Crown . .. 


Grand Royal.... 


Double D. Crown 

. . 30 X 40 “ 

Grand Aigle .... 

. . 27 x 40 

u 

Double D. D. Crown, 40 X 60 “ 

Vellum Writing, 18 to 28 ins. in width. 


Mounted on cloth, 38 ins. in width. 


IMIiscellaneons. 

1 sheet = 4 pages. 

1 quarto =8 “ 

1 octavo =16 “ 


1 duodecimo — 24 pages. 
1 eighteenmo = 36 “ 

1 bundle = 2 reams. 


1 piece wall-paper, 20 ins. by 12 yards. 

1 “ u “ French, 4.5 sq. yards. 

Roll of Parchment = 60 sheets. 

Copying. 

100 Words = 1 Folio. 


Metric, Toy AVct of Congress of Jnly 28, 1866. 
Unit of Surface is Are or Square Dekameter. 

A square meter (39.37 s ) = 1549.9969 sq. ins., but by this Act is declared to be 

155° sq. ins. 


Denominations. 

Sq. Meters. 

Sq. Inches. 

Sq. Feet. 

Sq. Yards. 

Acres. 

Centimeter. 

.OOOI 

•65 

.107 638 

— 

— 

Decimeter. 

.OI 

15-50 

— 

— 

Centare or 1 

Square Meter) " " 

I. 

1550 . 

10.763 888 

1.196 

— 

Are. 

IOO. 

— 

1076.388 88 

119.6 

.024 71 

Hectare. 

10 000. 

— 

— 

II 960. 

2.471 


c* 































30 


MEASURES AND WEIGHTS. 


Equivalent "Values irx NEetric Denominations of LI. S. 


Denominations. 

Sq. Meters. 

Denominations. 

Sq. Meters. 

Sq. Hectares. 

Sq. Ares. 

Sq. Inch. 

“ Foot. 

“ Yard. 

“ Rod. 

.000 645 16 
.092 903 23 
.836 12907 
25.292904 

Sq. Chain ... 
“ Rood.... 
“ Acre .... 
“ Mile. 

404.686 47 
1011.716 175 
4046.864 699 

. 404 686 
258.99934 

4.046 865 
10.117162 
40.468 647 

2 5 899-934 074 


Approximate Equivalents of Old. and iVIetric U. S. 
Square Measures. 

6.5 square centimeters = 1 sq. inch. I 1 acre = 1.16 per cent, over 4000 sq. meters. 
1 “ meter =10.75 sq.feet. | x square mile = 259 hectares. 


MEASURES OF VOLUME. 


Standard gallon measures 231 cube ins., and contains 8.3388822 
avoirdupois pounds, or 58 373 Troy grains of distilled water, at temper¬ 
ature of its maximum density (39.i°), barometer at 30 ins. 

Standard bushel is the Winchester , which contains 2150.42 cube ins., 
or 77.627 413 lbs. avoirdupois of distilled water at its maximum density. 

Its dimensions are 18.5 ins. diameter inside, 19.5 ins. outside, and 8 
ins. deep ; and when heaped, the cone must not be less than 6 ins. high, 
equal 2747.715 cube ins. for a true cone. 

A struck bushel contains 1.24445 cube feet. 


Hjicpuici. 


4 gills = x pint. 

2 pints = 1 quart. 
4 quarts = 1 gallon. 


2 pints = 1 quart. 
4 quarts = 1 gallon. 
2 gallons = 1 peck. 

4 pecks = 1 bushel. 


1728 cube inches = 1 foot. 
27 cube feet = 1 yard. 


Dry. 


Cube Ins. 

28.875 

57-75 

231 - 

Cube Ins. 

67.2006 
268.8025 
537-605 
2150.42 


Gills. Pints. 

8 . 

32 = 8. 


Galls. 


Pints. Quarts 

8 . 

16 = 8. 

64 = 32 = 8. 


Cube. 


Inches. 

46656 


Noth.—A cube foot contains 2200 cylindrical inches, or 3300 spherical inches. 


ETuicL. 


60 minims = 1 dram. 
8 drams = 1 ounce. 
16 ounces = 1 pint. 

8 pints = 1 gallon. 


Minims. Drams. Ounces. 

480. 

7 680 = 128. 

6l 240 = IO24 = 128. 


UNTautical. 

1 ton displacement in salt water.=35 cube feet. 

x “ registered internal capacity.=40 “ “ 

Dimensioais of a Barrel. 

Diameter of head, 17 ins.; bung, 19 ins.; length, 28 ins.; volume, 7689 cube ins. 
= 3.5756 bushels. 






















MEASURES AND WEIGHTS 


31 


]Mi s c ellaxx e ou s. 

1 cube foot. 7.480 5 gallons. 

1 bushel. 9-309 18 gallons. 

1 chaldron = 36 bushels, or. 57-244 cube feet. 

1 cord of wood.128 cube feet. 

1 perch of stone. 24.75 cube feet. 

1 quarter = 8 bushels. | 1 load hay or straw = 36 trusses. 

Galls. 

Puncheon of Scotch Whisky, .no to 130 
Puncheon of Brandy 34X52.. no to 120 

Puncheon of Rum.100 to no 

Hogshead of Brandy 28X40.. 55 to 60 

Pipe of Madeira. 92 

Hogshead of Claret. 46 


1 Barrel. 

Galls. 

1 Tierce. 


Butt of Sherry. 


Pipe of Port. 

—34x58 — ns 

Pipe of Teneriffe.... 


Butt of Malaga. 

—33x53— 105 


A Hogshead is one half, a Quarter cask is one fourth, and an Octave is one eighth 
of a Pipe, Butt, or Puncheon. 


TvTetric, Toy xAct of* Congress of J"nly 28, 1866. 
Unit or Base of Measurement is a cube Decimeter or Liter, which is declared to be 

61.022 cube ins. 

Cube Measures. 


Denominations. 

Values. 

Cube Inches. 

Cube Feet. 

Cube Yards. 

Cube Centimeter. 

.001 cube milliliter 
x cube liter. 

.061 022 
61.022 

.035 313657 
3 S- 3 i 3 6 57 

1.308 

“ Decimeter. 

“ Meter. 

Kiloliter or stere.. 


Dry Measures. 


Denominations. 

Values. 

Cube Ins. 

Quarts. 

Pecks. 

Bushels. 

Cube Yards. 

Milliliter . 

1 cube centimeter. 

.061 

_ 

_ 

_ 

_ 

Centiliter. 

10 “ “ 

.6102 

— 

— 

— 

— 

Deciliter. 

.1 “ decimeter.. 

6.1022 

— 

— 

— 

— 

Liter . 

j a u 

61.022 

.908* 

•1135 

— 

— 

Dekaliter . 

10 “ “ 

— 

9.08 

i-i 35 

•28375 

— 

Hektoliter .... 

.1 “ meter . 

— 

— 

n.35 

2-8375! 

H 

U> 

O 

OO 

Kiloliter ( 
or Stere J ''' 

J u u 

— 

— 

— 

28.375 

1.308 


* Or .227 gallon. + 3-531 365 7 cube feet. 

Note. — In practice, term cube Centimeter, abbreviated to cc, is used instead of 
Milliliter, and cube Meter instead of Kilometer. 


Equivalent Walnes in HVLetric Denominations of XJ. S. 


Dry Measures. 


Denominations. 

Centiliters. 

Deciliters. 

Liters. 

Dekaliters. 

Inch. 

— 

_ 

— 

_ 

Pint... 

— 

— 

— 

— 

Quart. 

— 

. no 125 

1.10125 

11.0125 

Gallon. 

— 

•440 5 

4- 405 

44-°5 

Peck. 

.0881 

.881 

8.81 

88 .x 

Bushel. 

•3524 

3-524 

35-24 

352.4 


Diq.ni.cL Measures. 


Denominations. 

Liters. 

Drams. 

Ounces. 

Pints. 

Quarts. 

Gallons. 

Milliliter. 

.OOI 

•27 

— 

— 

— 

— 

Centiliter. 

.OI 

2.7 

•338 

— 

— 

— 

Deciliter. 

. I 

27 

3-38 

.2X1 34 

— 

— 

Liter . 

I 

— 

33-8 

2-113 4 

1.0567 

— 

Dekaliter. 

IO 

— 

— 

21.134 

10.567 

2.6417 

Hektoliter. 

IOO 

— 

— 

— 

— 

26.417 

Kiloliter 1 
or Stere j ' 

IOOO 

— 

— 

— 

— 

264.17 














































































32 


MEASURES AND WEIGHTS. 


Approximate Equivalents of Old. and IVIetric TJ. S. 
Measures of Volume. 


i Gallon.= 4-5 liters. 

i Liter.= .26 gallon. 

1 cube foot.= 28.3 liters. 


1 cube meter.— 1.33 cube yards. 

1 “ yard.= .75 “ meter. 

1 “ kiloliter = 2240 lbs. nearly of water. 


MEASURES OF WEIGHT. 

Standard avoirdupois pound is weight of 27.7015 cube inches of dis¬ 
tilled water weighed in air, at (39.83°) barometer at 30 inches. 

A cube inch of such water weighs 252.6937 grains. 


-A^voirclupois. 
16 drams = 1 ounce. 

16 ounces = 1 pound. 

112 pounds = 1 cwt. 

20 cwt. = 1 ton. 


Ounces. 


Pounds. 


Drams. 

256. 

28 672 = I 792. 

573 440 = 35 840 = 2240. 


1 pound= 14 oz. 11 diets. 16 grs. Troy, or 7000 grains. 
1 ounce = 18 diets. 5.5 grains Troy, or 437.5 grains. 

1 dram = 1 diet. 3.343 75 grains Troy, or 53.5 grains. 
1 stone = 14 pounds. 

Troy. 

24 grains = 1 dwt. 

20 dwt. = 1 ounce. 

12 ounces = 1 pound. 

7000 Troy grains 

437-5 “ “ 

2 7-343 75 Troy grains 
175 Troy pounds 
“ ounces 

u 


175 

I 

I 

I 


ounce 

pound 


Grains. Dwt. 

480. 

5760 = 240. 
= i lb. avoirdupois. 

= 1 oz. “ 

= 1 dram “ 

= 144 lbs. “ 

= 192 oz. “ 

= 480 grs. “ 

,822 857 lb. 


avoirdupois pound = 1.215 278 lbs. Troy. 

.Apothecaries. 

Drams. 


20 grains = 1 scruple. 

3 scruples = 1 dram. 

8 drams = 1 ounce. 

12 ounces = 1 pound. 

45 drops = 1 teaspoonful or a fluid dram. 

2 tablespoonfuls = 1 ounce. 

The pound, ounce, and grain are the same as in Troy weight. 


Grains. Scruples. 

60. 

480 = 24. 

5760 = 288 = 96. 


Diamond. 


1 grain = 16 parts. 
16 parts = .8 grain. 


4 grains = 3.2 grains. 
1 carat = 4 grains. 


15.5 carats = 1 Troy ounce. 

Lead. 

A Fodder of lead = 8 pigs. 

Sheet lead rolls = 6.5 to 7.5 feet in width and from 30 to 35 feet in length. 


Grrain.. 

Standard Weights per Bushel. 
Lbs. I Lbs. I Lbs. I 

Wheat.... 60 1 Corn.... 56 and 58 I Rye. 56 | Oats. 


Lbs. | Lbs. 

32 I Barley.... 48 















MEASURES AND WEIGHTS. 


33 


UMiscellaraeoixs. 

COAL. 

Anthracite.i cube foot = 1.75 broken. 

.50 to 55 lbs. per cube foot. 

. 41 to 45 cube feet == 1 ton broken. 

Bituminous.70 to 78 lbs. per heaped bushel. 

“ .40 to 50 lbs. per cube foot. 

“ Cumberland. 53 “ M “ “ 

“ Cannel.50.3 lbs. per cube foot. 

u Welsh.43 cube feet =. 1 ton. 

“ Lancashire.44 “ “ = 1 “ 

“ Newcastle.45 “ “ = 1 “ 

“ Scotch.43 “ “ =1 “ 

“ R. N. allowance.48 “ “ = 1 “ 

Charcoal, hardwood.18.5 lbs. per cube foot. 

“ pine wood.18 “ “ “ “ 


Virginia pine .. 2700 lbs.: 


WOOD. 

i cord. | Southern pine . 3300 lbs. = 1 cord. 


EARTH. 


River sand .. 21 cube feet= 1 ton. 
Coarse gravel, 23 “ “ = 1 “ 


Marl or Clay, 28 cube feet: 
Mold.33 “ “ : 


: 1 ton. 


UMIetric, "by fAct of Congress of July SB, 1 S 66 . 


Unit of Weight is the Gram, which is weight of one cube centimeter of pure water 
weighed in vacuo at temperature of 4 0 C., or 39.2 0 F., which is about its tem¬ 
perature of maximum density = 15.432 grains. 


Denominations. 

Values. 

Grains. 

Ounces. 

Lbs. 

Ton. 

Milligram. 

1 cube millimeter 

_ 

_ 

— 

— 

Centigram. 

10 “ “ 

•154 32 

— 

— 

— 

Decigram. 

.1 “ centimeter 

1-543 2 

— 

— 

— 

Gram . 


I5-432 

.03527 

— 

— 

Dekagram. 

Hektogram. 

10 “ “ 

1 deciliter. 

_ 

•352 7 
3-527 

. 220 46 

_ 

Kilogram or Kilo .. 

1 liter. 

— 

35-27 

2.204 6 

— 

Myriagram. 

10 “ . 

— 

— 

22.046 

— 

Quintal. 

1 hektoliter. 

— 

— 

220.46 

.oq8 41Q 

Millier or Tonneau. 

1 cube meter. 

— 

— 

2204.6 

.984 196 


Kilogram = 2.679 I 7 ^s. Troy , or 2 lbs. 8 02. 3 dwts. .3072 grain. 


Equivalent. Values in IMietric Denominations of TJ. S. 


Denominations. 

Grams. 

Dekagrams. 

Denominations. 

Grams. 

Kilograms. 


.064 8 
1.296 
1-555 2 

1.771 87 
3.888 


Ounce. 

28.3502 

31.1042 

453.6028 

373-2504 

.02835 

,031 I 

•4536 
•373 25 
1016.057 28 

Scruple. 

Pennyweight. 

Drachm. 

“ (Apoth.).... 

17.7187 

38.88 

“ Troy . 

Pound. 

‘ ‘ Troy. 

Ton. 


Approximate Equivalents of Old. and jST ew TJ. S. 
Measures of Weiglit. 

The ton and the gram are at nearly equal distances above and below the 
kilogram. Thus, 

1 ton ....='1016057.28 grams. | 1 kilogram.= 1000 grams. 

1 gram is nearly 15.5 grains (about .5 per cent. less). 

1 kilogram about 2.2 pounds avoirdupois (about .25 per cent. more). 

1000 kilograms, or a metric ton, nearly 1 Engl, ton (about 1.5 per cent. less). 























































34 


MEASURES AND WEIGHTS, 


Electrical. ( British Association.') 

Resistance.—Unit of resistance is termed an Ohm , which represents resist¬ 
ance of a column of mercury of i sq. millimeter in section and i 0486 rqeters in 
length, at temp. o° C. Equivalent to resistance of a wire 4 millimeters in diameter 
and 100 meters in length. 

1000000 Microhms.= 1 Ohm. 

1 “ .= 10 absolute electro-magnetic units. 

1 Ohm.= 10000000 “ “ ‘‘ 

1000000 Ohms.= 1 Megohm or io r 3 “ “ “ “ 

Electro-motive Force.—Unit of tension or difference of potentials is 
termed a Volt. 

x 000000 Microvolts.= 1 Volt or . 1 of an absolute electro-magnetic unit. 

1 Volt.= 100000 “ “ “ “ “ 

1 Megavolt.=: 1000000 Volts. 

Current. —Unit of current is equal to 1 Weber per second, or the current in a 
circuit has an electro motive force of one Volt and a resistance of an Ohm. 

"Volvime.—Unit of volume is termed an Ampere, and represents that volume 
of electricity which flow's through a circuit having an electro motive force of one Volt 
and a resistance of one Ohm in a second, or it represents a Volt diminished by an Ohm. 


1000000 Microvolts or 100 absolute units of volume.= 1 Ampere. 

1000000 Amperes.= 1 Megaweber 

Capacity-.—Unit of capacity is termed a Farad. 

1000000 Microfarads or 10000000 absolute units of capacity.= 1 Farad. 

1000000 Farads...= 1 Megafarad. 


Heat.—Unit of heat is quantity required to raise one gram of water to i° C. 
of temperature. 

■Weiglits of Grain and Roots. 

Following weights have been fixed by statute in many of the States; and 
these weights govern in buying and selling, unless a specific agreement to 
the contrary has been made. 


ARTICLES. 

| California. 

| Connecticut. 

| Delaware. 

I J < 

as 

*0 

p 

| Indiana. rj 

Il( 

05 

* 

O 

i—i 

Kentucky. ’ll 

in 

ci 

p 

.03 

*<£ 

O 

Hi 

6 

p 

*5 

1 

1 Massachusetts. 

33 

p 

C L 

3 

O 

s 

| Minnesota. cj 

| Missouri. g 1 

| N. Hampshire. 7-* 

| New Jersey. 

| New York. 

O* 

5 

p 

0 

bt 

a> 

O 

| Pennsylvania. 

Rhode Island. 

Vermont. 

p 

0 

fcx 

.£ 

CO 

G 3 

£ 

Wisconsin. 

Bariev. 

5 ° 

— 

— 

48 

48 

48 

48 

32 

— 

46 

48 

48 

48 

— 

48 

48 

48 

46 

47 

_ 

46 

45 

48 

Beans. 

— 

— 

— 

60 

60 

60 

60 






60 

— 

— 

62 




Blue Grass Seed. 

— 

— 

— 

14 

14 

14 

14 

















Buckwheat. 

40 

45 

— 

40 

50 

52 

52 

— 

— 

46 

42 

42 

52 

— 

5 ° 

48 

— 

42 

48 

— 

46 

42 

42 

Castor Beans.... 

— 

— 

— 

46 

46 

46 














Clover Seed. 

— 

— 

— 

60 

60 

60 

60 

— 

— 

— 

60 

60 

60 

— 

64 

60 

60 

60 

— 

_ 

_ 

60 

60 

Dried Apples.... 

— 

— 

— 

24 

25 

24 

— 

— 

— 

— 

28 

28 

24 

— 


— 

— 

28 

— 

_ 

_ 

28 

28 

Dried Peaches .. 

— 

— 

— 

33 

33 

33 

— 

— 

— 


28 

28 

33 

— 

— 

— 

— 

28 

— 

_ 

_ 

28 

28 

Flaxseed . 

— 

— 

— 

56 

56 

5 t> 

56 

— 

— 


— 

— 

56 

— 

55 

55 

56 






56 

Hemp Seed . 

— 

— 

— 

44 

44 

44 

44 
















Corn . 

52 

56 

56 

52 

5 b 

5 b 

5 b 

5 b 

— 

56 

56 

56 

52 

— 

56 

58 

56 

56 

56 

_ 

56 

56 

56 

Corn in ear . 

— 

— 


70 

68 

68 













Corn Meal . 

— 

— 

— 

48 

50 

— 

50 

— 

50 

50 

— 

— 

— 

— 

— 

— 

— 

_ 

— 

5 ° 

— 

— 

— 

Coal . 

— 

— 

— 

80 

70 

— 

— 

— 

— 

— 

— 

— 

80 

— 

— 

— 

— 

— 

—- 


— 

_ 

_ 

Oats . 

32 

28 

— 

32 

32 

35 

33 ! 

32 

30 

30 

32 

32 

35 

30 

30 

32 

32 

34 

32 

—* 

32 

36 

32 

Onions . 

— 

— 

— 

57 

48 

57 

57 











50 


5 ° 


Pease . 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

60 

_ 

_ 

_ 


_ 


. 

Potatoes . 

— 

60 

— 

60 

60 

60 

60 

— 

60 

— 

— 

— 

60 

60 

60 

60 

— 

60 

_ 

60 

60 

60 

60 

Bye . 

54 

56 

— 

54 

56 

56 

56 

60 

— 

56 

56 

56 

5.6 

— 

56 

56 

56 

56 

56 

— 

56 

56 

56 

Rye Meal. 

— 

— 

— 

— 

— 

— 

— 

— 

50 

50 

— 

— 

— 

— 

— 

— 




50 




Salt . 

— 

— 

— 

— 

50 

50 

50 






5 ° 

— 

— 

56 





— 

_ 

Timothy Seed. .. 

— 

— 

— 

45 

45 

45 

45 ' 
















46 

Wheat . 

60 

56 

60 

60 

60 

60 

60 

60 

— 

60 

60 

60 

60 

— 

6c 

60 

6c 

60 

60 

_ 

60 

6c 

60 

Wheat Bran .... 

— 


" 

20 

— 

20 

20 

— 

— 


— 

— 

»20 

— 

— 

— 

— 

— 

— 

— 

— 

— 





































































MEASURES AND WEIGHTS. 


35 


"W eight of Men. and. Women. 

Average weight of 20000 men and women, weighed in Boston, 1864, was 
—men, 141.5 lbs.; women, 124.5 lbs. Average of men, women, and chil¬ 
dren, 105.5 lbs. 


■'W'eigh.t of Horses.—(XT. S.) 
Weight of horses ranges from 800 to 1200 lbs. 


WEIGHT OF CATTLE. 


To Compute Dressed "Weight of Cattle. 


Kule.— Measure as follows in feet : 

1. Girth close behind shoulders, that is, over crop and under plate, 
immediately behind elbow. 

2. Length from point between neck and body, or vertically above 
junction of cervical and dorsal processes of spine, along back to bone at 
tail, and in a vertical line with rump. 


Then multiply square of girth in feet by length, and multiply product 
by factors in following table, and quotient will give dressed weight of 
quarters. 


Condition. 

Heifer, Steer, 
or Bullock. 

Bull. 

Condition. 

Heifer, Steer, 
or Bullock. 

Bull. 

Half fat. 

3-15 

3-36 

Very prime fat... 

3- 6 4 

3-85 

Moderate fat. 

Prime fat. 

3-36 

3-5 

3-5 

3- 6 4 

Extra fat. 

3-78 

4.06 


Illustration.— Girth of a prime fat bullock is 7 feet 2 ins., and length measured 
as above 4 feet 5 ins. 

7' 2"= 7.17, and 7.17 2 51.4, which X 4' 5" and by 3.5 = 794.5 lbs. Exact 
weight was 799 lbs. 

Note.— 1. Quarters of a beef exceed by a little, half weight of living animal. 

2. Hide weighs about eighteenth part, and tallow twelfth part of animal. 


Comparative Weights of Dive Beeves and of Beef. 



Lbs. 

Per cent. 


Lbs. 

Bullocks. 

2800 

72 to 78 
> 70 to 76 

Bullocks. 

1550 

1550 

1260 

Heifers. 


Heifers. 

Bullocks. 

2600 

Bullocks. 

Heifers. 

i 66 to 70 

Heifers. 

1200 

Bullocks. 


Bullocks. 

1050 

IOSO 

Heifers. 

2100 

2100 

1800 

\ 64 to 68 

Heifers. 

Bullocks . 

Bullocks. 

980 

950 

Heifers. 

63 to 66 

Heifers. 


Per cent. 
| 61 to 64 
| 58 to 61 
| 57 to 58 
| 50 to 56 


Weight of OfFals in a Beef and Sheep. 


BEEF. 

Lbs. 

Hide and Hair.... 56 to 98 

Tallow. 42 “ 140 

Head and Tongue . 28 " 49 
Feet. 21 “ 35 

* Including 2 to 6 lbs. for fleece. 


SHEEP. 

Lbs. 

8 to 16* 


5 “ 

6 “ 


14 

lit 

3 


BEEF. SHEEP. 

Lbs. Lbs. 

Kidneys,Heart,). to 6a 6t0lo 

Liver, etc.... f 0 
Stomach, Entrails,etc., 126 “ 196 9 “ 18 

Blood. 42 “ 56 4 “ 6 

t Including 2 to 5 lbs. for horns. 





















































MEASURES, WEIGHTS, PRESSURES, ETC. 


36 


To Compute Equivalents of Old. and. New TJ. S. and. 
of UVEetric Denominations. 

By Act of Congress, July 28, 1866. 

Rule. — Divide fourth term by second, multiply quotient by first 
term, and divide product by third term. 

Or, Ascertain relative ratio of first and second terms, and multiply 
result by ratio of third and fourth terms. 

Note. — When result is required in French or other Metric denominations than 
those of U.S., use exact denominations, as, 61.025 387 for 61.022, 39.370432 for 39.37, 
etc. 

Example i. —If one gallon (1st), per sq. foot, yard, acre, etc. (2d); how many liters, 
(3d), per sq. foot, yard, acre, etc. (4th) ? 


X 231 


Or, — = 1.604, and - 

144 61.022 

Note. —In computing ratios, first term is to be dividedby second, and fourth by third. 
Example 2.—If one ton per cube foot, how many kilograms per cube decimeter? 

022 X 2240 -r- 2.2046 = 35.881 liters , or 35.882 litres. 

1728 


■ 6l.O?2. . . . 

144 


. = 3.7851 liters or 3.7848 litres. 

= 2.3598; hence, 1.604 X 2.3598 = 3.7851 liters. 


MEASURES. 

By Act of Congress of U. S. By Metric Computation. 

1 Liter per sq. foot, etc. = .2642 Gallon per sq.foot, or .264 2 gallon. 

1 Liter per sq. meter . = .0245 Gallon per sq. foot, or .024 5 gallon. 

1 Gallon per sq. foot . =40.746 Liters per sq. meter, or 40.745 4 litres. 

1 Sq. foot per acre ... = .2296 Sq. meters per hectare , or 2.29609 metres. 


WEIGHTS AND PRESSURES. 

By Act of Congress of TJ. S. By Metric Computation. 


Per sq. inch. Per sq. inch. 

1 Centimeter.= .1929 Lb. or .19292 Ih. 

1 Atmosphere .... = 6.6679 Kilograms, or 6.667 8 kilogrammes. 

1 Inch mercury .. = 2.54 Centimeters , or 2.54 centimetres. 

1 Pound.= 453.6029 Grams, or 453.592 6 grammes. 

1 Kilogram.= 3x7.4624 Lbs. per sq.foot, or 317.465 lbs. 


Note. —30 ins. of mercury at 62° = 14.7 lbs. per sq. inch ; hence, 1 lb. =. 2. 0408 ins. 
and a centimeter of mercury = 30 -4- .3937 for U. S. computation, and 30= .393 704 32 
for French or Metric. 


POWER AND WORK. 

1 Horse - power = Cheval or Cheval - vapeur = 4500 k x m =. 33 000 - 4 - 
(4500 X 2.2046 x 39.37 -4- 12) = 1.01388 chevaux. 

1 Cheval or Cheval-vapeur (75 k X m per Second) = horse-power. 

(4500 X 2.2046 x 39.37 -4- 12) -4- 33 000 = .9863 liorse-power. 

By Act of Congress of TJ. S. By Metric Computation. 

1 Foot-pound = Kilogrammeter k x m — 7.233 foot-lbs.; hence, 

1 = (2.2046x3.280 833) = .13826 Kilogrammeter, or .13825 kilogrammetre. 

1 Cube foot per IP .... = .0279 Cube meter per cheval, or .0279 cheval. 

1 Pound “ “ .. = .447 38 Kilogram per cheval, or .447 38 kilogramme. 
1 Cube meter per cheval = 35.8038 Cube feet per IP, or 35.8058 Hh 












PRESSURES, ETC.-MEASURES OF TIME. 


37 


TEMPERATURES. 

i Caloric or French unit = 3.968 Heat-units , and 1 heat-unit = iv 3.968 
= .252 caloric. 

1 U. S. Mechanical equivalent ( 772 foot-lbs. ) = 772 -f- 7.233 — 106.733 
Kilogr ammeters and 106.733 kilogrummet res. 

1 French Mechanical equivalent (423.55 k x to) =3.280833 x 2.2046 x 
423.55 = 3063.505 foot-lbs ., or 3063.566 foot-lbs. Metric. 

1 Heat-unit per pound = .5556 Kilogram , or .5556 kilogramme. 

1 Heat-unit per sq. foot = .2715 Caloric per sq. meter , or .271 g,per sq. metre. 


VELOCITIES. 

1 Foot per second, minute, etc. = .3047 Meter per second , or .3047 metres. 
1 Mile per hour.= .447 “ “ “ or .447 “ 


MEASURES OF TIME. 

60 thirds = 1 second. 60 minutes = 1 degree. 

60 seconds = 1 minute. 30 degrees = 1 sign. 

360 degrees = 1 circle. 

True or apparent time is that deduced from observations of the Sun, 
and is same as that shown by a properly adjusted sun-dial. 

Mean Solar time is deduced from time in which the Earth revolves 
on its axis, as compared with the Sun; assumed to move at a mean 
rate in its orbit, and to make 365.242218 revolutions in a mean Solar 
or Gregorian year. 

Sidereal time is period which elapses between time of a fixed star 
being in meridian of a place and time of its return to that place. 

Standard unit of time is the sidereal day. 

Sidereal day = 23 h. 56 in. 4.092 sec. in solar or mean time. 

Sidereal year , or revolution of the earth, 365 d. 5 h. 48 m. 47.6 sec. in solar 
or mean time = 365.242 24 solar days. 

Solar day , mean = 24 h. 3 m. 56.555 sec. in sidereal time. 

Solar year (Equinoctial, Calendar, Civil or Tropical) =365.242218 solar 
days , or 365 d. 5 h. 48 m. 47.6 sec. 

Civil day commences at midnight. Astronomical day commences at 
noon of the civil day, having same designation, that is, 12 hours later 
than the civil day. 

Marine or sea day commences 12 hours before civil time or 1 day 
before astronomical time. 

New Style was introduced in England in 1752. 


Note. —In Russia days are reckoned by Old Style, and are consequently 12 days 
behind Gregorian record. 


D 






38 


MEASURES OF VALUE. 


MEASURES OF VALUE. 

io mills = i cent. io dimes = i dollar, 

io cents = i dime. io dollars = i eagle. 

Standard of gold and silver is 900 parts of pure metal and 100 of 
alloy in 1000 parts of coin. 

Fineness expresses quantity of pure metal in 1000 parts. 

Remedy of the Mint is allowance for deviation from exact standard 
fineness and weight of coins. 

Nickel cent (old) contained 88 parts of copper and 12 of nickel. 
Bronze cent contains 95 parts of copper and 5 of tin and zinc. 

Pure Gold 23.22 grains = $100. Hence value of an ounce is 
$20.67.183 + . 

Standard Gold, $18.60.465+ per ounce. 


WEIGHT, FINENESS, ETC., OF U. S. COINS. 
C+old. 


Denomination. 

Weigh 

of Coin. 

t 

of Pure 
Metal. 

Denomination. 

Weigh 

of Coin. 

t 

of Pure 
Metal. 

Dollar. 

Quarter Eagle.. 
Three Dollar... 

Oz. 

•OS 3 75 
•134 375 
.161 25 

Grs. 

25.8 

64-5 

77-4 

Grs. 

23.22 

58-05 

69.66 

Half Eagle. 

Eagle. 

Double Eagle... 

Oz. 

.268 75 
•537 5 

1-075 

Grs. 

129 

258 

510 

Grs. 

116.1 

232.2 
464.4 


Silver. 


Dime. 

• 080 375 

38.58 

34.722 

Half Dollar. 

.401 875 

192.9 

20 Cent. 

.16075 

77 -16 

69.444 

Trade Dollar.... 

•875 

420 

Quai'ter Dollar. 

.200937 5 

9^45 

86.805 

Silver Dollar ... 

•859 375 

412.5 


Copper and NLickel. 



Weight. 

Copper. 

Tin and 
Zinc. 


Weight. 

Copper. 

Tin and 
Zinc. 

One Cent.... 
Two Cents... 

Grains. 

48 

96 

Per cent. 
95 

95 

Per cent. 

5 

5 

Three Cents. 
Five Cents.. 

Grains. 

30 

77 -16 

Per cent. 

75 

75 

Per cent. 
25 

25 


Tolerance.-GoW, Dollar to Half Eagle, .25 grains. Eagles, .5 grains. 
— Silver , 1.5 grains for all denominations. — Copper , 1 to 3 cents, 2 grains ; 
5 cents, 3 grains. 

Legal Tenders. — Gold, unlimited. — Silver. Dollars 01412.5 grains 
unlimited ; for subdivisions of dollar, $10. (Trade dollars [420 grains] are 
not legal tender .)—Copper or cents, 25 cents. 

Note. —Weight of dollar up to 1837 was 416 grains, thence to 1873, 412.5. Weight 
of $1000, @ 412.5 gr. =859.375 oz. 

British standards are : Gold, |-| of a pound,* equal to n parts pure gold 
and 1 of alloy ; Silver , §§-§ of a pound, or 37 parts pure silver and 3 of alloy 
= .925 fine. 

A Troy ounce of standard gold is coined into £3 175. iori. zf, and an 
ounce of standard silver into 5s. 6 d. 1 lb. silver is coined into 66 shillings. 

Copper is coined in proportion of 2 shillings to pound avoirdupois. 

£ Sterling (1880) $486.65; hence of this = value of 1 penny = 
2.027 708 33 cents. 


* A pound is assumed to be divided into 24 equal parts or carats, hence the pro¬ 
portion is equal to 22 carats. 







































FOREIGN MEASURES OF VALUE. 


39 


To Compute Value of Coins. 

Rule. —Divide product of weight in grains and fineness, by 480 
(grains in an ounce), and multiply result by value of pure metal per 
ounce. 

Or, Multiply weight in ounces by fineness and by value of pure metal 
per ounce. 

Example i.— When fine gold is $20.67.183-)- per oz., what is value of a British 
sovereign? 

By following tables, p. 40, Sovereign weighs .2567 oz., and .2367 X 480 = 123.216 
grains, and has a fineness of .9165. 

TT 123.216 x■ 916 5 „ „ , . 

Hence, •----X 20.67.183-)- — $4-86.34. 

40O 

Example 2.—When fine silver is $1.15.5 per oz., what is value of U. S. Trade dollar? 
By table, p. 40, Dollar weighs .875 oz. and has a fineness of .900. 

Hence, .875 X .900X 1.15.5 = 90.95625 cents. 

Example 3.—A 4-Florin (Austrian) weighs 49.92 grains and has a fineness of .900. 
What is its value ? 

49 929 °°- X 20.67.183-}- = $1.93.49. 

To Convert XT. S. to British*. Currency and Contrari¬ 
wise. 

Rule i.— Divide Cents by 2.027 7 I_ ~ (2.027 708 33), or, Multiply by 
.493 12— (.493 118 26), and result is Pence. 

2. Multiply Pence by 2.02771 —, or divide by .49312—, and result 
is Cents. 

Example. —What are 100 cents in pence? 

100 X -493 12— = 49.312— pence = 4s. 1.312& 

2. What is a Pound sterling in cents? 

20 X 12 = 240 pence , which X 2.027 7 1 — — $4 86.65. 


FOREIGN MEASURES OF VALUE. 

Weight, Fineness, and Mint Values of Foreign 
Silver* and (Gold Coins* 

By Laws of Congress, Regulations of the Mint, and Reports of its Directors. 

Current Value of silver coins is necessarily omitted, as the value of 
silver is a variable element. Hence, in order to compute current value 
of a silver coin, the price of fine or a given standard of silver being 
known, 

Proceed as per above rule to compute value of coins. 

The price of silver should be taken as that of the London market for 
British standard (925 fine), it being recognized as the standard value, 
and governing rates in all countries. 

Example. _If it is required to determine value of a Mexican dollar in cents. 

Weight 867.5 oz. .903 fine. Value of Silver in London 52.75 pence per ounce = 
106.9616-)- cents. 

Then 867-5 X- 9°3 __ g 4 g 867— and 106.9616 X .846 867 = 90.5822 cents. 

9 2 5 






40 


FOREIGN MEASURES OF VALUE 


Weiglit and Mint Values of Foreign Coins. 
Countries given in Italics have not a National Coinage. 


Country and Denomination. 

Weight. 

Fine¬ 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

Value 

Gc 

U. S. 

Id. 

British. 


Oz. 

rhous’s. 

Grains. 

Cents. 

$ c. 

£ s. d. 

Arabia. 




83.14 



Piastre or Mocha Dollar. 

— 

— 

— 

— 

— 

Argentine Republic. 




50.69 



Dollar = ioo Centisimos.... 

— 

— 

— 

— 

— 

(Employs South American and 
Foreign Coins.) 
Australasia. 

Same as British. 

Australia. 





4 - 85-7 


Sovereign, 1855. 

•256.5 

916 

— 

— 

19 11.5 

Pound, 1852. 

.281 

916.5 

— 

— 

5 - 32-37 

1 1 10.5 

Austria. 







Kreutzer (copper). 

— 

— 

— 

.41 

— 

.2 

Florin, new. 

•397 

900 

171.47 

— 

— 

— 

Dollar. “ . 

• 59 6 

900 

257-47 

— 

— 

— 

4 Florins. 

. 104 

900 

— 

— 

1.93.49 

7 11 

Ducat. 

.112 

986 

— 

— 

2.28.3 

9 4-6 

Souverain . 

• 3 6 3 

900 

— 

— 

6 - 75-4 

1 7 9.1 

Belgium. 

Same as France. 

Bolivia. 







Centena . 

— 

— 

— 

•75 

— 

•37 

Dollar, new. 

.801 

900 

346-03 

— 

— 

— 

Doubloon, 1827-36. 

.867 

870 

362.06 

— 

15 - 59-3 

3 4 1 

Brazil. 

Rei. 

_ _ 

916.66 

. 

•547 

_ 

.27 

Milreis. 

.028.8 

12.67 

— 

- 54-59 

26.92 

Double Milreis. 

.82 

918.5 

393-6 

— 

— 

20 Milreis, 1854-36. 

•575 

9 * 7-5 

— 

— 

10.90.6 

2 4 9.84 

Moidore, 4000 Re:s. 

.261 

9*4 

— 

— 

4.92 

1 0 2.63 

Canada. 







Mil, sterling. 

— 

— 

— 

. I 

— 

•05 

Cent “ 

— 

— 

— 

I-OI 

— 

■5 

20 Cent, currency___ 

•15 

9 2 5 

166.6 

— 

— 


25 “ “ . 

• 187 5 

9 2 5 

83-25 

— 

— 

— 

Penny “ _t. 

— 

— 

— 

1.52 

— 

•75 

Shilling “ . 

— 

-. 

— 

— 

— 


Dollar, sterling. 

— 

— 

— 

— 

I 

4 2 

4 “ =20 shillings, currency 

Pound 11 “ 

— 

— 

— 

— 

3 - 97-43 

16 4 

■— 

— 

— 

— 

3 - 99-97 

16 5-25 

Cape of Good Hope. 

Same as British. 

Central America. 





4 Reals. 

.02 7 

875 

11-34 

— 

— 

— 

Dollar. 

.866 

850 

353-33 

— 

— 

— 

2 Escudos. 

. 209 

8 S 3 -i 


— 

3.68.8 

15 1.88 

Doubloon ante 1834. 

.869 

833 

— 

— 

14.96.39 

3 1 5-97 

Chili. 





Centaro . 

— 

— 

— 

•9 

- 

•45 

Dollar, new...... 

. 801 

900.5 

346.22 


— 


10 Pesos.... 

.492 

'900 

— 

— 

9 - r 5-4 

1 *7 7-45 

Doubloon. 

.867 

870 

— 

— 

15 - 59-3 

3 4 i 

China. 






Cash, Le... 

— 

— 

— 

.14 

— 

•°7 

10 Cents, Leang. 

.087 

901 

37 - 9 8 

— 

— 


Dollar. 

.866 

901 

374-63 

— 

— 

_ 

Cochin China. 





Mas, 60 Sapeks. 

— 

— 

— 

6-75 

— 

3-33 

10 Mas, 1 Quan. 

— 

— 

— 

67.52 

— 

2 9-33 
































































FOREIGN MEASURES OF VALUE 


41 


'W'eiglxt and IVIin. t Values. 


Country and Denomination. 

Weight. 

Fine¬ 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

Value. 
G 0 

U. S. 

Id. 

British. 


Oz. 

rhous’s. 

Grains. 

Cents. 

$ c. 

£ s. d. 

Cuba. 

Same as Spain. 

Colombia. 







Centaro. 

— 

— 

— 

I.OI 

— 

•5 

Peso, new. 

00 

0 

M 

900 

34 6 -°3 

— 

— 


4 Escudos... 

•433 

844 

— 

— 

7 - 55-5 

1 11 0.58 

Doubloon, old. 

.867 

870 

— 

— 

15 - 59-3 

3 4 1 

Costa Rica. 

Same as Mexico. 

Denmark. 







Mark, 16 Skilling. 

— 

— 

— 

8.94 

— 

4-39 

Crown. 

.025 

900 

— 

— 

26.8 

13.22 

2 Rigsdaler. 

.927 

877 

390.23 

— 

— 

— 

10 Thaler. 

.427 

895 

— 

— 

7.90 

1 12 5.6 

East Indies. 

See Hindostan and Japan. 

Ecuador. 






Centaro. 

— 

— 

— 

I.OI 

— 

•5 

Peso. 

.801 

900 

34 6 -°3 

— 

— 

— 

England. 






Penny. 

•304 

— 

— 

2.02-f-* 

— 

I 

Groat.. 

.060.4 

9 2 5 

26.82 

— 

— 

— 

Shilling, new.... 

.182.5 

9 2 4 - 5 

80.99 

— 

— 

— 

‘ 1 average. 

.178 

9 2 5 

79-°3 

— 

— 

— 

Half Crown. 

• 454-5 

9 2 5 

201.8 

— 

— 

— 

Florin. 

• 3 6 3 - 6 

9 2 5 

161.44 

— 

— 

— 

Sovereign or Pound, new ... 

.256.7 

9 i6 -5 

— 

— 

4.86.65 

100 

“ “ average. 

.256.2 

9 i6 -5 

— 

— 

4.85.1 

IOO 

Egypt. 







Piastre, 40 Paras. 

.04 

755 

i 4-5 

— 

° 4-9 

— 

Guinea, Bedidlik. 

•275 

875 

— 

— 

5- 0.52 

1 0 6.84 

Pound. 

•275 

875 

— 

— 

4.97.4 

1 0 5.3 

Purse, 5 Guineas. 

i -375 

875 

— 

— 

25. 2.6 

5 2 10.2 

France. 







Centime. 

.032 

— 

— 

.2 

— 

.1 

Sou, 5 Centimes. 

.l6l 

— 

— 

I.OI 

— 

•5 

Franc, 100 Centimes. 

,l6l 

900 

69-55 

— 

— 

— 

5 Francs. 

.804 

900 

347-76 

— 

— 

— 

20 Francs, Napoleon, new... 

.207.5 

899 

— 

— 

3 - 85-8 

15 10.26 

25 Francs 20 centimes=£1 Stg. 
Germanv, 







Groschen, 10 Pfenning. 

— 

— 

— 

2.38 

— 

i -175 

Mark, 10 Groschen. 

.012.8 

900 

— 

— 

23.8 

11.74 

10 Marks. 

.128 

900 

— 

— 

2.38.24 

9 9-5 

Thaler. 

■595 

900 

257.04 

— 

2.28.38 

9 4-63 

Ducat. 

.112 

986 

— 

— 

Greece and Ionian Islands. 
Same as France. 







Drachma, 100 Lepta. 

.010.4 

900 

— 

— 

i 9-3 

9-5 

5 Drachmas... 

.719 

900 

310.61 

— 

— 

— 

20 Drachmas. 

.185 

900 

— 

— 

44.2 

H i -75 

Pound. 

— 

— 

— 

— 

5. 6.11 

1 0 9.6 

Guatemala. 

Same as Mexico. 







Guiana , British, French, and 







Dutch. 







Same as that of their Countries 







Hanse Towns. 





23.8 


Mark. 

.012.8 

900 

— 

— 

11.74 

Holland. 







Cent..... 

— 

— 

— 

•4 


.2 


* 2.027 71 cents. 

D* 





























































42 


FOREIGN MEASURES OF VALUE 


Weiglit and Mint Values. 


Country and Denomination. 

Weight. 

Fine¬ 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

Vaui 

G 

U. S. 

E. 

0 1 d. 

British. 

Holland. 


Oz. 

Thous’ s . 

Grains. 

Cents. 

$ c. 

£ s. d. 

Florin or Guilder, 

io Guilders. 

Hindostan. 

100 cents. 

.021.6 

.215 

900 

899 

— 

— 

40.49 

3 - 99-7 

1 8 

16 5.11 

Rupee. 

Honduras. 

Same as Mexico. 

Italy. 

Same as France. 


•374 

9 i6 -5 

i 64-53 



1 10.5 

Lira, ioo Centimes. 

. l6 

835 

65.12 

— 

— 

— 

Scudo. 

Indian Empire. 


.864 

900 

373-24 

— 

— 

— 

Pic, nominal. 


— 

— 

— 

■25 

— 

.125 

Anna “ . 


— 

— 

— 

3-°3 

-' ' 

i -5 

Rupee,* 16 Annas 


•375 

916.5 

165 


— 


io Rupees, and 4 Annas .... 

— 

— 

— 

— 

4.86.65 

IOO 

Molaur, 15 Rupees 
Japan. 


•375 

93:6.5 

— 

— 

6.84.36 

I 8 1.5 

Sen. 


— 

— 

— 

I 

— 

.5 

Itzebu, new. 


.279 

890 

119.19 

— 

— 


Yen, 100 Sen. 


.866.7 

900 

374-4 

— 

— 

— 

U U 


•053.6 

900 


— 

99.72 

4 1.18 

Cobang, old. 


.289 

572 

— 

— 

3 - 57-6 

14 8.35 

1 ‘ new. 


.362 

568 

— 

— 

4.44 

18 2.96 

20 Yen. 

Java. 

Same as Holland. 
Liberia. 

U. S. Currency. 

Malta. 


1.072 

900 



19.94.4 

4 1 11.6 

' 12 Scudi — 1 Sovereign. 

Mexico. 

— 

— 

— 

— 

4.86.65 

IOO 

Peso, new. 


.867.5 

9 ° 3 - 

377- 1 7 

— 

— 

4 2 

“ Maximilian. 


. 861 

902.5 

372.98 

— 

— 


Doubloon, new... 


.867.5 

870.5 


— 

15. 6.1 

3 4 i-88 

20 Pesos, Republic 
Morocco. 


1.081 

873 

— 

— 

i 9 - 5 i -5 

4 0 2.4 

Ounce, 4 Blanbeels. 

— 

— 

_ 

_ 

_ 

_ 

10 Ounces, Mitkeel 
Naples. 


— 

— 

— 

— 

— 

— 

Scudo. '. 


.844 

830 

336-25 

— 

— 

_ 

6 Ducati. 

Netherlands. 

Same as Holland. 

New Brunswick. 

Same as Canada. 

Newfoundland. 

Same as Canada. 
New Granada. 


•245 

996 


5 - 4-4 

1 0 8.75 

Dollar, 1857. 


.803 

896 

— 

— 

_ 

. 

Doubloon, Popayan. 

Norway. 

Alike to Denmark. 

.867 

858 



I 5 - 37 - 8 

3 3 3-39 

Mark, 24 Skillingen. 

Nova Scotia. 

Same as Canada. 

Persia. 




21.63 


10.66 

Keran, 20 Shaliis. 


- f 

— 

— 

22.81 

_ 

11.25 

10 Keran, Toman. 

Paraguay. Foreign coins. 

* * .092 76 of a £ 

Stg., nomi 

nal value 

=■ 2 shill 

ngs sterlin 

g- 






















































FOREIGN MEASURES OF VALUE 


43 


Weight and. ALint V 7 "allies. 


Country and Denomination. 

Weight. 

Fine¬ 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

Value. 

Gold. 

U. S. British. 


Oz. 

Thous’ 3 . 

Grains. 

Cents. 

$ c. 

£ a. d. 

Peru. 







Dollar, 1858. 

.766 

900 

341.01 

— 

— 

— 

Sol. 

.802 

900 

346.46 

— 

— 

— 

Doubloon, old. 

.867 

868 

— 

— 

15 - 55-7 

3 3 II -22 

Portugal. 






Coroa, 1838, 10000 Reis. 

.308 

912 

—■ 

— 

5. 80.66 

2 4 5-5 

100 Reis. 

■09s 

912 

— 

— 

10.8 

— 

Roumania. 





2 Lei. 

.322 

83S 

129.06 

— 

— 

— 

Russia. 






Copek. 

— 

— 

— 

•77 

— 

-38 

100 Copek, Rouble. 

.667 

875 

277-73 

— 

— 

— 

5 Roubles. 

.21 

916.6 


— 

3 - 97-6 

16 4.8 

Sandwich Islands. 

U. S. Currency. 

Sardinia. 






Lira.. 

Spain. 

. 16 

835 

65.12 

— 

— 

— 

Centimo. 

— 

— 

— 

.19 

— 

•095 

100 Centimo, Peseta. 

. 16 

835 

64.13 


— 

— 

Dollar, 5 Peseta. 

.8 

9 °° 

345-6 

— 

— 

1 0 4.8 

100 Reals. 

.268 

896 


— 

4.96.4 

10 Escudos . 

.270.8 

896 

— 

— 

5 - i -5 

1 0 7.32 

20 Reals vellons=i U.S. Dollar. 
Sweden. 





Riksdaler, 100 Ore. 

•273 

750 

98.28 

— 

— 

— 

Rixdollar. 

I.O92 

750 

393-12 

— 

— 

— 

Carolin, 10 Francs. 

. 104 

900 

— 

— 

1 - 93-5 

7 11.42 

Switzerland. 

Same as France. 

St. Domingo. 



6-33 



Gomdes, 100 Cents. 

— 

— 

— 

— 

3-125 

Tunis. 







Piastre, 16 Karubs. 

— 

— 

— 

11.83 

— 

5-83 

5 Piastre. 

• 5*1 

898.5 

220.38 

— 

— 

— 

25 Piastre. 

. l6l 

900 

— 

— 

2.99.5 

!2 3-7 

Turkey. 






2.16 

Piastre, 40 Paras. 

-- 

— 

— 

4-39 

— 

20 Piastre. 

•77 

830 

306.77 

— 

4.36.9 

— 

100 Piastre, Medjidie. 

.231 

9*5 

— 

— 

18 0 

Tuscany. 






9 6.1 

Zecchino, Sequin. 

.112 

900 

— 

— 

2 - 31-3 

Tripoli. 





3 0-89 

20 Piastres, Mahbub. 

— 

— 

— 

74-3 

— 

Uruguay. 







Dollar, 100 Centimes. 

— 

— 

— 

— 

— 

— 

West Indies, British. 

Same as England. 

Venezuela. 







Centaro. 

— 

— 

— 

I 

— 

•5 

Bolivar, 1 Franc. 

— 

— 

— 

— 

— 

— 


Memoranda. 

France. —Bronze coins 9.5 coppei', 4 tin, and 1 zinc. 

Hanse Towns.— Monetary system same as that of German Empire. 

Switzerland.— The Centime is termed a Rappe. 

Spain. —25 Peseta piece is 19s. 9.51?. Stg.; Real vellon was 2.5c?. Stg. 

Italy.— All coins same weight aud fineness as those of France. 

Malta.—7 Tari and 4 Grani = 1 Shilling Sterling. 

Egypt.— A Para — .061 5 d. Sterling, and 97.22 Piastres = 1 Sovereign. 

Indian Empire. —1 Lac Rupees=£ioooo Sterling. In Ceylon, Rupee=ioo Cents. 





















































44 ENGLISH AND FRENCH MEASURES AND WEIGHTS. 


ENGLISH AND FRENCH MEASURES AND WEIGHTS. 

MEASURES OF LENGTH. 

English. —Imperial standard yard is referred to a natural standard, 
which is a pendulum 39.1393 ins. in length vibrating seconds in vacuo 
in London, at level of sea; measured between two marks on a brass 
rod, at temperature of 62°. 

Note. — In consequence of destruction of standard by fire in 1834, and difficulty 
of replacing it by measurement of a pendulum, the present standard is field to be 
about 1 part in 17 230 less tfian that of U. S., equal to 3.67 ins. in a mile. 

ALiscellarLeons. 

Land. —Woodland pole or perch or Fen.= 18 feet. 

Forest pole.= 21 “ 

Irish mile = 2240 yards. | Scotch mile.= 1984 yards. 

Sea. —10 cables, or 1000 fathoms, or 6086.44 feet, or 1.152 8 statute miles 
= 1 Admiralty or Nautical mile or knot. 

3 miles = 1 league. 60 Nautical or 69.168 Statute miles or 20 Leagues = 
1 degree. 

Mean length of a minute of latitude at mean level of the sea =1.1508 
statute miles. 

Nautical mile is taken as length of a minute at the Equator. 

Nautical fathom is 1000th part of a nautical mile, and averages about 
.0125 longer than the common fathom. 

French. —Standard Metre or unit of measurement is defined as the 
ten millionth part of the terrestrial meridian, or the distance from the 
Equator to the Pole, passing through Paris. Actual standard is a plat¬ 
inum metre, deposited in the Palais des Archives, Paris. 

NEetric Ifiengtli in. IncL.es, ITeet, etc. 


Denomination. 

Metres. 

Inches. 

Feet. 

Yards. 

Miles. 

x Millimetre. 


•039 37 
•393 7 
3-937 04 




1 Centimetre. 





1 Decimetre. 





1 Metre. 


3. 280 87 



1 Dekametre. 



10.93623 
109.36231 
1093.623 1 
10936.231 


1 Hektometre. 



328.086 9 
3280.869 


1 Kilometre. 



.621 38 
6.21377 

x Myriametre. 

IOOOO 

— 


Note. 


-For length of metre see p. 27. 

Old. Measure 

1 Toise.= 1.949 metres. 

1 Mille.= 1.949 kilometres 

1 Noeud (knot). = 1.855 “ 


1 Terrestrial league = 4.444 kilometres. 
1 Nautical league . = 5.555 “ 

1 Arpent.= 900 sq. toises. 


MEASURES OF SURFACE. 

English. —Same as that of United States of America. 

IVTiscellarLeons. 

Builders. 1 superficial part.= 1 square inch. 

12 parts.= 1 inch. 

12 inches.<.= square f oot. 

Boards. —Boards 7 inches in width are termed battens, 9 inches deals, and 
12 inches planks. 




























ENGLISH AND FRENCH MEASURES AND WEIGHTS. 45 


French. 

HVLetrio Surfaces in. Square Indies, Feet, etc. 
Denomination. Sq. Inches. Sq. Feet. Sq. Yards. Sq. Acres. 


•001 55 
•155003 
15.500309 
1550.030916 


. 107 641 
10.764104 
1076.410358 


1.196 01 
119.601 15 
xi 960.11509 


Square millimetre. 

“ centimetre. 

“ decimetre.. 

“ Metre or Centiare .... 

“ dekametre or are . 

“ hektometre or hectare 

“ kilometre. 

“ myriametre*. 

* Equal 38.6x0 908 sq. miles. 

OlcL System. 

1 square inch = 1.135 8 7 Inches. 

1 toise = 6.394 6 feet. 

1 arpent (Paris) = 900 square toises =: 4089 square yards. 

1 arpent (woodland) = 100 square royal perches = 6108.24 square yards. 


.024 711 
2.471098 
247.109 816 
24 710.981 6 


MEASURES OF VOLUME. 

Imperial gallon measures 277.123 cube ins., but by Act of Parliament 
1825 its volume is 277.274 cube ins., equal to 10 lbs. avoirdupois of 
distilled water, weighed in air, at temperature of 62°, barometer at 30 
inches. 6.2355 gallons in a cube foot. 

Imperial bushel , 18.5 ins. internal diameter, 19.5 external, and 8.25 
in depth, contains 2218.192 cube ins., and when heaped in form of a 
right cone, at least .75 depth of the meas-ure, must contain 2815.4872 
cube ins. or 1.6293 cube feet. 

Grain. —1 quarter = 8 bushels or 10.2694 cube feet. 

Vessels. — 1 ton displacement = 35 cube feet; 1 ton freight by measure¬ 
ment = 40 cube feet. 

1 ton internal capacity = 100 cube feet, and 1 ton ship - builders = 94 
cube feet. 

English standard No. 5 is .008 grain heavier than the pound, and U. S. pound is 
.001 grain lighter than English. 


Wine and Spirit Measures. 

4 quarts (231 cube ins.).= .8333 Imperial gallon. 

.= 1 anchor. 

.= 1 runlet. 

.= 1 barrel. 

.= 1 tierce.' 

.= 1 hogshead. 

.= 1 puncheon. 

.= 1 pipe or butt. 


10 gallons 
18. “ 

(15 imperial) 

3 i -5 “ 

26.25 “ 

42 

35 

63 “ 

52.5 “ 

84 “ 

70 “ 

126 “ 

105 “ 


2 pipes or ) 

3 puncheonsJ 


= 1 tun. 


ATe and Beer Measures. 


Imp’l gall’s. 

4 quarts (28c cube ins.) . . = 1.017 

9 gallons = 1 firkin.= 9.153 

2 firkins = 1 kilderkin . . . = 18.306 


Imp’l gall’s. 

2 kilderkins = 1 barrel = 36.612 
54 gallons == 1 hogshead = 54.918 
108 “ = 1 butt . . . . = 109.836 



























46 ENGLISH AND FRENCH MEASURES AND WEIGHTS. 


Apothecaries’ or UTuid. Measures. 

4 drachms.= 1 tablespoon. 

2 ounces (875 grains) = 1 wineglass. 


1 drop.= x gram. 

60 drops.= 1 drachm. 


Coal Measures. 


50 pounds 

88 “ . 

9 bushels ... .; 

80 or 84 pounds: 

90 or 94 “ : 

93 pounds . . . .: 
3 heaped bush.: 
10 sacks.: 


: 1 cube foot. 

: i bushel. 

: 1 vat. 

f 1 London or 
l Newcastle bushel. 
1 Cornish “ 

1 We/sh bushel. 

: 1 sack. 

: 1 ton. 


12 sacks.: 

1 chaldron.: 

5.25 chaldrons . .: 
1 London chaldron: 
1 Newcastle u : 

1 ton.: 

1 room.: 

21 chaldrons.: 

1 barge or keel.. = 


: 1 chaldron. 

: 58.6548 cube ft. 
: 1 room. 

126.5 crots. 

■ 53 “ 

144.5 cube feet. 

: 7 tons. 

: 1 score. 

: 21.2 tons. 


Nliscellaireous. 


1 last corn. 

1 ton water . .. . 
1 dicker hides . . 
1 last hides 
1 barrel tar 
6 bushels wheat 

1 clove. 

1 score . 

1 sack flour 
1 truss straw .. . 


: 80 bushels. 
135.9 cube feet. 

: 10 skins. 

: 20 dickers. 

: 26.5 gallons. 

1 sack flour. 

: 7 pounds. 

: 20 “ 

: 28.2 “ 

: 36 “ 

35.9 cube feet 


1 truss old hay.= 50 pounds. 

1 “ new “ .= 60 “ 

1 bushel oats.= 40 “ 

1 “ barley ... . = 47 “ 

1 “ wheat.= 60 “ 

1 cube yard new hay = 84 “ 

1 “ “ old “ = 126 “ 

1 quintal.— 100 “ 

1 boll.= 140 “ 

1 sack wool.== 364 “ 

; 1 ton water. 


Liquid. 


1 wine gallon = 231 cube ins. 

1 beer “ = 282 “ “ 

1 litre.= .22009 gallon . 

1 gallon . . . . = 4.544 litres. 

1 cube foot.. =: 6.2321 gallons. 

1 anker-= 8.333 “ 


1 hogshead wine . . = 52.5 gallons. 
1 “ beer ... = 54-918 “ 

1 puncheon wine . . = 70 “ 

1 pipe or butt wine = 105 “ 

1 “ “ “ beer = 109.836 “ 

1 tun.= 2x0 “ 


1 ton water 62° = 224 gallons. 
Builders. 


1 solid part.. = 12 cube ins. 

12 “ parts.= 1 “ inch." 

12 “ inches .= 1 cube foot. 

1 load timber, rough = 40 “ feet. 
1 “ “ hewn =50 11 “ 

1 “ lime = 32 bushels. 

1 “ sand.= 36 “ 


1 square.= 100 sq.feet. 

1 bundle laths.= 120 laths. 

1 rod brickwork . . . = 306 cube feet. 
1 rood masonry . . . = 648 “ “ 

Batten, in section . . = 7'x 2.s ins. 
Deal, l£ ..= 9x3 “ 

Plank, “ “ .. = 11x3 “ 


Metric Volumes in. Cube Indies, ICeet, etc. 


Denominations. 

Litres. 

Gills. 

Pints. 

Quarts. 

Gallons. 

Bushels. 

Quarters. 

Centilitre. 


.0704 

•7°43 

7.0429 

.0176 

1761 

1.7607 





Decilitre. 






Litre* . 



.2201 

2.2009 

22.OO91 

220.0908 



Dekalitre. 


8.8036 

.275 XI 
2.751 13 
27-5ii 35 


Hectolitre. 


/ 


-3439 

3 - 43^9 

Kilolitre. 

IOOO 

— 

— 

— 


* Equal 61.025 24 cube ins. 


























































ENGLISH AND FRENCH MEASURES AND WEIGHTS. 47 


Wood Measiire. 

1 Stere or cube metre = 35.3150 cube feet or 1.308 cube yards. 

1 Yoie de bois (Paris) = 70.6312 cube feet ; 1 voie de charbon (charcoal) 
— 7.063 cube feet; 1 corde = 4 cube metres = 141.26 cube feet. 


MEASURES OF WEIGHT. 

British.— 1 Troy grain = .003 961 cube inches of distilled water. 

1 Troy pound =22.815 689 cube inches of water. 

1 Avoir, drachm = 27.343 75 Troy grains. 


_A_voirdxxpois. 


16 drachms, or } 
437.5 grains J 
16 ounces, or ^ 
7000 grains J 


.= 1 ounce. 

.= 1 pound. 

20 hundredweights 


8 pounds 

.. = 1 stone (for meat). 

14 « 

.. = 1 stone. 

28 “ 

.. = 1 quarter. 

112 “ 

. . = 1 cwt. 


= 1 ton 


The grain , of which there are 7000 to the pound avoirdupois, is same as 
Troy grain, of which there are by the revised table 7000 to the Troy pound. 
Hence Troy pound is equal with the Avoirdupois pound. 

In Wales, the iron ton is 20 cwt. of 120 lbs. each. 


Troy. 


24 grains.= 1 dwt. 

20 pennyweights, or | _ 

437-5 grains j 


= 1 ounce. 


16 ounces.= 1 pound. 

25 pounds.= 1 quarter. 

4 quarters, or 100 pounds = 1 cwt. 


By this are weighed gold, silver, jewels, and such liquors as are sold by 
weight. 

The old Troy ounce to the Avoirdupois ounce was as 480 grains, the 
weight of the former, to 437.5 grains, weight of the latter; or, as 1 to .9115. 


Apothecaries.* 

437.5 grains = 1 ounce. | 16 ounces = 1 pound. 


French. 

Nletric Weights in Avoirdupois. 


Denominations. 

Grammes. 

Grains. 

Ounces. 

Pounds. 

Ton. 

Milligramme . 

OOI 

01543 

— 

— 

— 

Centigramme. 

OI 

•154 32 

— 

— 

— 

Decigramme. 

. I 

1 543 23 

— 

— 

— 

Gramme . 

I 

I 5-432 35 

— 

— 

— 

Dekagram me. 

IO 

154.323 49 

•3527 

— 

— 

Hektogramme . 

IOO 

1 543-234 87 

3-5274 

. 220 46 

— 

Kilogramme t . 

I OOO 

15432.348 74 

35-2739 

2.204 62 

.- 

Myriagramme. 

IO OOO 

— 

— 

22.O46 21 

— 

Quintal. 

IOO OOO 

— 

— 

22O.462 12 

— 

Millier or Ton. 

I OOO OOO 

— 

— 

2204.621 25 

.9842 


f Kilogramme = 2 lbs. 3 oz. 4 drachms, 10.4734 grains. 


Note. — For the values of the prefixes, as Milli, Centi, etc., see p. 27. 

OlcI System. 

1 grain . . = 0.8188 grains Troy. 1 ounce = 1.0780 oz. Avoirdupois. 
1 gross . . = 58.9548 “ 1 livre = 1.0780 lbs. 


* As by revised Pharmacopoeia. 


































48 


FOREIGN MEASURES AND WEIGHTS. 


FOREIGN MEASURES AND WEIGHTS. 

It being wholly impracticable to give all the denominations of measures 
and weights of all countries, the following cases are selected as essential and 
as exponents. 

With parent countries, as England, France, etc., their denominations ex¬ 
tend to their colonies and dependencies. Thus, the denominations of England 
extend to Canada, a large portion of the East and West Indies, and parts of 
South America, and those of France to a part of the West Indies, Algiers, etc. 


Abyssinia. 

Pic, Stambouili. 26.8 ins. 

“ geometrical. 30.37 44 

Madega. 3-466 bush 

Ardeb. 34-66 “ 

“ Musuah. 83.184 “ 

Wakea.400 grains. 

Mocha. 1 Troy oz. 

Rottolo. 10 u “ 

Also, same as in Egypt and Cairo. 

Africa, Alexandria, Cairo, 
and Egypt. 

Cubit.20.65 i ns - 

Derah.25.49 “ 

Pic, cloth.26.8 “ 

“ geometrical.29.53 “ 

Kassaba, 4.73 Pics.11.65 ft- 

Mile.2146 yds. 

Feddan al-risach.552 48 acre. 

Roobak. 1.684 galls. 

Ardeb. 4.9 bush. 

Rottol.9821 lb. 

Distances are measured by time. 

A Maragha = 15 Dereghe or 1 hour. 

Aleppo and Syria. 

Dra Mesrour.21.845 ins. 

Pic.26.63 14 

Road Measures are computed by time. 
Algeria. 

Rob, Turkish. 3. n ins. 

Pic, “ .24.Q2 “ 

“ Arabic.18.89 “ 

Also Decimal System. 

Alicante. 

Palmo. 8.908 ins. 

V ara.35.632 <l 

Amsterdam. 

Voet.11.144 ins. 

El.21.979 “ 

Faden. 5.57 ft. 

Lieue. 6.383 yds. 

Maat. 1.6728 acres. 

Morgen... 2.0095 “ - 

Vat.. 40 cub. ft. 

Also Decimal System. 

Antwerp. 

Fuss.11-275 ins. 

Elle, cloth.26.94 

Corde.24.494 cub. ft. 

Bonnier. 3.2507 acres. 

Also Decimal System. 


Arabia, Bassora, and 

NEoclia. 

Foot, Arabic. 1.0502 ft. 

Covid, Mocha.19 ins. 

Guz, “ 25 “ 

Kassaba.12.3 ft. 

Mile, 6000 feet.2146 yds. 

Baryd, 4 farsakh.21 120 “ 

Feddan.57 600 sq. ft. 

Noosfia, Arabic.138 cub. ins. 

Gudda. 2 galls. 

Maund. 3 lbs. 

Tomand.168 “ 

Other Measures like those of Egypt. 

Argentine Confederation, 
Paraguay, and Uruguay. 

Fanega. 1.5 bush. 

Arroba. 25.35 lbs. . 

Quintal.101.4 “ 

Also Decimal System in Argentine Con¬ 
federation and Paraguay. 

Australasia. 

Land Section.80 acres. 

Other Measures same as English. 
Austria. 

Zoll. 1.0371 ins. 

Fuss. 1 0371 ft. 

Meile.24000 ft. 

Klafter, quadrat..35-854 sq. yds. 

1 Jochart. 6.884 “ 

Cube Fuss. 1.1155 cub. ft. 

Aehtel. 1.C92 galls. 

Eimer.12.774 44 

Viertel. 3-1143 “ 

Metze. 1.6918 bush. 

Unze. .8642 grains. 

Pfund (1853, 500grammes), 1.2347 lbs. 

Centner.123.47 44 

Also Decimal System. 

Babylon. 

Pachys Metrios.18.205 > ns - 

Baden. 

Fuss.11.81 ins. 

Klafter. 5.9055 ft. 

Ruthe. 9.8427 tc 

Stunden.4860 .yds. 

Morgen.8896 acre. 

Stutze. 3.3014 galls. 

Malter. 4.1268 bush. 

Pfund. 1.1023 lbs. 

Also Decimal System. 















































































FOREIGN MEASURES AND WEIGHTS. 


49 


Bagdad. 

Guz.31-665 ins. 

Bai-Dary States. 

Pic, Tunis linen.18.62 ins. 

“ “ cloth..26.49 u 

“ Tripoli.21.75 “ 

Batavia. 

Foot.....,. 

Covid.. 

El.... 


12.357 ms. 
27 u 
27-75 “ 


Bavaria. 

Fuss.n-49 ins - 

Klaf'ter. 5-745 36 ft. 

Ruthe,.. 3.1918 yds. 

Meile. 8060 “ 

Ruthe, quadrat.10.1876 sq.yds. 

Morgen or Tagwerk.8416 acre. 

Flatter, cube. 4.097 cub. yds. 

Eimer.15.05856 galls. 

Schelfel. 6.119 u 

Metze. 1 0196 bush, 

Pfund.8642 grains. 

Also Decimal System. 

Belgium. 

Meile...2.132 yds. 

Also Decimal System. 

Benares. 

Yard, Tailor’s.33 ins. 

Bengal, Bombay, and Cal¬ 
cutta. 

Moot... 3 ins. 

Span. 9 “ 

Ady, Malabar.10.46 ins. 

Hath...18 u 

Guz, Bombay.27 “ 

“ Bengal.36 u 

Corah, minimum. 3-417 ft. 

Coss, Bengal. 1,136 miles. 

“ Calcutta.. 1.2273 “ 

Kutty. 9.8175 sq. yds. 

Biggah, Bengal.3306 acre. 

u Bombay.8114 “ 

Seer, Factory.68 cub. ins. 

Covit, Bombay.12.704 cub. ft. 

Seer, Bombay. 1-234 pints. 

Parah. 4.4802 galls. 

Mooda.112.0045 ‘‘ 

Liquids and Grain measured by weight. 

Bohemia. 

Foot, Prague.n.88 ins. 

“ Imperial.12.45 “ 

Also same as Austria. 

Bolivia, Cdili, and Bern. 

Vara. 33-333 ins. 

Fanegada. 1.5888 acres. 

Gallon.74 gall. 

Fanega. 1-572 ,l 

Libra. 1.014 lbs. 

Arroba.25.36 “ 

Originally as in Spain ; now Decimal 
System in Chili and Peru. 


Brazil. 

Palrno, Bahia.8.5592 ins. 

Vara.3.566 ft. 

Braca.7-132 “ 

Geora ..i.1-448 acres. 

Also same as Portugal , and sometimes 
as ip England. 

Buenos Ayres. 

Vara..._;..'_2.84 ft; 

Legua.3.226 miles. 

Suertes de Estancia.... 27 000 sq. varas. 
Also same as Spain. 

Burmah. 

Paulgat.. 1 inch. 

Dain.. 4.277 yds. 

Viss. 3.6 lbs. 

Taint. 5.5 “ 

Saading.22 “ 

Also same as England. 

Canary Isles. 

Onza.927 inch. 

Pic, Castilian.11.128 ins. 

Altitude..... .... .0416 acre. 

Fanegada.5 “ 

Libra. 1.0148 lbs. 

Also same as Spain. 

Cape of Good Hope. 

Foot.11 616 ins. 

Morgen. 2.116 54 acres. 

Also same as in England. 

Ceylon. 

Seer.1 quart. 

Parrah.5.62 galls. 

Also same as in England. 

China. 

Li.486 inch. 

Chili, Engineer’s. 12.71 ins. 

“ or Covid.. 13-125 “ 

“ “ legal. 14.1 “ 

Chang.131-25 “ 

“ legal.141 “ 

Pu. 4.05 ft. 

Chang, fathom. 10.9375 ft. 

Li...486 yds. 

Pu or Rung. 3.32 sq. yds. 

King, 100 Mau. 16.485 acres. 

Tau. 1.13 galls. 

Tael. 1.333 oz. 

Catty.. 1-333 ibs. 

Cochin China. 

Thuoc or Cubit...19.2 ins. 

Sao.64 sq. yds. 

Mao. 1.32 acres. 

Hao. 6.222 galls. 

Sliita.12.444 “ 

Nen.8594 lb. 

Colombia and “Venezuela. 

Libra. 1.102 lbs. 

Oncha.25 “ 

Also Decimal System. 


E 



























































































50 


FOREIGN MEASURES AND WEIGHTS. 


Denmark,* Greenland, Ice¬ 
land, and Norway. 

Tomme. 1.0297 ins. 

Fod. 1.0297 ft. 

Favn, 3 Aleu. 6.1783 “ 

Mil. 4.68055 miles. 

“ nautical. 4.61072 “ 

Anker. 8.0709 galls. 

Skeppe.478 bush. 

Fjerdingkar.9558 “ • 

Fund. 1.1023 lbs. 

Lispund.17-367 “ 

Centner.110.23 “ 

* Also Decimal System. 

Ecuador. 

Decimal System. 

Genoa, Sardinia, and 
Turin. 

Palmo. 9.8076 ins. 

Piede, Manual, 8 oncie... 13.488 “ 

“ Liprando, 12 “ ...20.23 “ 

Trabuco orTesa.. 10.113 ft. 

Miglio. 1-3835 miles. 

Starello.9804 acre. 

Giomaba.9394 “ 

Germany. 

The old measures of the different States 
differ very materially ; generally , how¬ 
ever, 

Foot, Rhineland.12.357 ins. 

Meile. 4.603 miles. 

Decimal System made compulsory in 1872. 

Greece. 

Stadium.6155 mile. 

Also Decimal System. 

Guinea. 

Jachtan.12 ft. 

I I am la n rg. 

Puss.11.2788 ins. 

Klafter. 5.6413 ft. 

Morgen. 2.386 acres. 

Cube Fuss.831 1 cub. ft. 

Tehr..99,73 ‘ <■ 

Viertel. 1.594 7 galls. 

Pfund (500 grammes)... 1.102 32 lbs. 

Ton • ..2135.8 lbs. 

Also Decimal System. 

Hanover. 

Fuss..... ins. 

Morgen.6476 acre. 

Hin dost an. 

Borrel.. 1.211 j ns . 

Gerah... 2.387 

Haut.19.08 “ 

E°Be...29.065 

Boss. 3 63 miles. 

P uc t a . 1.184 cub. ft. 

Candy.14.209 “ 


Hungary. 

Fuss.12.445 ins. 

Elle.30-67 “ 

Meile. 9.139 yds. 

Also as in Vienna. 

Indian Empire. 

Guz.27.125 ins. 

Cowrie. 1 sq. yd. 

Sen.61.02539 cub. ins. 

“ ... 2.204737 lbs. 

Uniform standard of multiples of the Sen 
adopted in 1871. 

11 a 1 y. 

IMilan and Wen ice. 
Decimal System. 

The Metre is termed Metra; the Are, Ara; 
the Stere, Stero; the Litre, Litro; the 
Gramme, Gramma, and the Tonneau, 
Tonnelata de Mare. 

Naples and Two Sicilies. 

Palmo.10.381 ins. 

Canna. 6.921 ft. 

Miglio. 1.1506 miles. 

Migliago.7467 acre. 

Moggia.86 “ 

Pezza, Roman.6529 “ 

Roman States. 

Old Measure. 

Foot.11-592 ins. 

“ Architect’s.n.73 “ 

Braccio. 30-73 “ 

Palmo.. 8.347 “ 

Miglio.1628 yds. 

Quarta.. 1.1414 acres. 

Lucca and Tuscany. 

P> e .11.94 ins. 

Palmo.u. 49 “ 

Braccio.22.98 “ 

Passetto.3.829 ft. 


Passo 


5-74 


Miglio...1.0277 miles. 

Quadrate.8413 acre. 

Saccato. 1-324" “ 

Japan. 

Sun, .30303 Metre - 1.193* ins. 

Shaku, 3.0303 Metres-n.9305* ins. 

Jo, 30-303 “ - 9.9421* ft. 

Ken, 5.5 u . ... 5-9653* u 

Ri, 11 880 “ .... 2.4403 miles. 

Kai-ri.6080 feet.t 

PPtro. 4-971* feet. 

Momme.3-756521 7 grammes Fr. 

Hiyaku-me.828 17 lbs. 

Kwam-me. 8.28171 “ 

Hiyak kin.132.50722 (< 

Man’s load. 57-972 “ 

.331.26831 u 

Hij-ak-koku. 33126.8308 “ 

* These are as equivalent as they are practi¬ 

cable of reduction. 

t Admiralty knot. 













































































FOREIGN MEASURES AND WEIGHTS, 


51 


Java,. 

Duim. 1.3 ins. 

Ell...27.08 “ 

Djong. 7.015 acres. 

Kan.328 galls. 

Tael. 593-6 grains. 

Sach. 61.034 16s. 

Pecul. 122.068 “ 

Catty. 1-356 “ 

IMadi-as. 

Ady. 10.46 ins. 

Covid.,.18.6 “ 

Guz...33 “ 

Culy.20.92 ft. 

League.3472 yds. 

Puddy. .338 galls. 

Marcal. 2.704 “ 

Tola.180 grains. 

Seer.625 lbs. 

Viss. 3.086 “ 

Maund.24.686 “ 

Malabar. 

Ady.10.46 ins. 

jVIalacca. 

Hasta or Covid.18.125 ins. 

Depa.. 6 ft. 

Orlong.80 yds. 

Malta. 

Palmo.10.3125 ins. 

Pie.11.167 “ 

Canna.82.5 “ 

Salma. 4.44 acres. 

Also as in Sicily. 

Moldavia. 

Foot. 8 ins. 

Kot, silk.24.86 ins. 

Fathom. 8 ft. 

Molucca Islands. 

Covid.18.333 ins. 

Morocco. 

Totnin. 2.81025 ins. 

Cadee.20.34 ins. 

Cubit.21 “ 

Muhd. 3.081 35 galls. 

Kula, oil. 3-356 “ 

Rotal or Artal. 1.12 lbs. 

Liquids other than oil are sold by weight. 

Mysore. 

Angle. 2.12 ins. 

Haut.19.1 “ 

Guz.38.2 “ 

Candy.500 lbs. 

HSTetlierlands. 

File. 39-37 0 432 ins. 

Decimal System since 1817. 

Persia. 

Gereh. 2.375 ins. 

Gueza, common.25 

“ Monkelrer.37.5 “ 


Archin, Schah.31.55 ins. 

“ Arish.38.27 “ 

Parasang.6076 yds. 

Chenica.80.26 cub. ins. 

Artaba. 1.809 bush. 

Mi seal.71 grains. 

Ratel. 2.1136 lbs. 

Batman Maund. 6.49 “ 

Liquids are measured by weight. 

Poland. 

Trewice.14-03 ins. 

Precikow.17 ins. 

Pretow. 4.7245 yds. 

Mile, short.6075 yds. 

Morgen. 1.3843 acres. 

Portugal and Mozambique. 

Foot. 13 ins. 

Milha. 1.2788 miles. 

Almude. 3.7 galls. 

Fanga. 1.488 bush. 

Alguieri. 3.6 “ 

Libra. 1.012 lbs. 

Also Decimal System. 


Prussia. 


Fuss. 



Ruthe. 



Meile. 


.. 24 000 feet. 

Quadrat Fuss. 



Morgen. 



Cube Fuss.... 



Scheffel. 



Anker.. 


. 7-559 S alls - 

Pound. 



Zollpfund. 

,.. ,.. 


Centner. 




Russia. 

Vershok. 



Foot. 



Arschine. 


.28 « 

Rhein Fuss... 



Sajene . 


. 7 ft- 

Verst. 



Mila. 



Dessatina. 



Vedro. 



Tschel-werha. 



Pajak. 



Tschetwert... 


. 5 - 7704 “ 

Pound . 



Funt. 




Decimal System adopted in 1872. 


Siam. 

K’up. 9.75 ins. 

Covid...18 ins. 

Ken.39 “ 

Jod.09848 mile. 

Roeneng. 2.462 miles. 

Silesia. 

Fuss.. n. 19 ins. 

Ruthe. 4.7238 yds. 

Meile.7086 yds. 

Morgen. 1-3825 acres. 







































































































52 


FOREIGN MEASURES AND WEIGHTS 


Singapore. 


Hasta or Cubit.18 ins. 

Dessa. 6 ft. 

Orlong.80 yds. 


Smyrna. 


Pic.26.48 ins. 

Inclise..24.648 “ 

Berri..1828 yds. 


Spain, Cuba, Malaga, Ala- 
nil la, Guatemala, Hondu¬ 
ras, and. AEeXico. 


Pie. 

Vara.. 

Mi 11a. 

Legua, 8000 varas 

Fanegada. 

Vara, cubo. 

Cuartilla... 

Arroba, Castile... 

Fanega. 

Libra. 

Tonelada. 


,. 11.128 ins. 

■ • 33-384 “ 

.865 mile. 

.. 4.2151 miles. 
,. 1.6374 acres. 
,. 21.531 cub. ft. 
.. .888 gall. 

3-554 S aIls - 
.. 1.5077 bush. 

.. 1.0144 lbs. 

2028.2 lbs. 


Also Decimal System. 


Stettin. 

Fuss. 

Foot, Rhineland. 

File... 

Morgen. 


n. 12 ins. 
12-357 “ 

25.6 ins. 
1.5729 acres. 


Sumatra 

Jankal or Span. 

File.. 

Hailoh. 

Fathom. 

Tung.— 


. 9 ins. 
. 18 “ 


36 “ 

6 ft. 

4 yds. 


Surat. 

Tussoo, cloth. 

Guz, “ . 

Hath. 

Covid. 

Biggah . 

Sweden 

Fot. 

Ref.. 

Faden. 

League... 

Meile. 


... 1.161 ins. 
... 27.864 ‘ ■ 

... 20.9 “ 

...18.5 “ 

_ .51 acre. 

11.6928 ins, 

32.4703 yds. 
5.845 ft. 
3.3564 miles. 
6.6417 “ 


Tunnland. 1.2198 acres 

Anker. 8.641 galls. 

Spann. 1.962 bush. 

Centner.112.05 lbs. 

Also Decimal System. 

Switzex-land. 

Fuss, Berne...11.52.ins. 

“ - 11-54 “ 

Vaud.11.81 “ 

Klafter.. 5.77 ft. 

Meile.. 4.8568 miles 

Juchart, Berne..85 acre. 

Maas.:. 2.6412 pints. 

Eimer. 8.918 galls. 

Malter... 4.1268 bush. 

Pfund.. 1.1023 lbs. 

Also Decimal System. 

Tripoli. 

Pik, 3 palmi.26.42 ins. 

Almud.319.4 cub. i ns - 

Killow.,..2023 “ “ 

Barile...14.267 galls. 

Temer. .7383 bush. 

Rottol.7680 grains. 

Olce. 2.8286 lbs. 


Turkey. 


Pic, great.27.9 ins. 

“ small.27.06 u 

Berri. 1.828 yds. 

Alma... 1.154 galls. 

Also Decimal System. 

“W" urtemberg. 

Fuss.n.29 ins. 

File. 2.015 ft- 

Meile.. 8146.25 yds. 

Morgen.7793 acre. 

Cube Fuss............. . 830 45 cub. ft. 

Eimer .. 64.721 galls. 

Scheff'el. 4.878 bush. 

Pound.7217 grains. 


Fuss. 

File. 

Klafter.. 

Meile. 

Jachart. 

Cube Klafter 


ZuricH. 

.11.812 ins. 

.23.625 “ 

. 5.9062 ft. 1 

... 4.8568 miles. 

.808 acre. 

.144 cub. ft. 


LENGTHS OF ENGLISH RACE-COURSES. 


Course. 

Miles. 

Course. 

Miles. 

Course. 

NEWMARKET. 

Across the Flat . 

1.292 
A. 206 

DONCASTER. 

Circular. 


GOODWOOD. 
Cup Course . .. 

Beacon... 

Fitzwilliam. 


LIVERPOOL. 

New Course... 

Cambridgeshire. 

Cesarewitch. 

1.136 
2.266 

Red House. 

St. Leger. 

.711 

1.825 

2.634 

1. 25 

Round . 

3*579 

1.009 

2 

Cup Course . 

New Castle .. 

Rowley Mile . 

Summer Course .... 

ETSOM. 

Craven . 

Oxford . 

YORK. 

Two-year old, new .. 

. .702 

Derby and Oaks. ... 

i -5 

Stakes Course. 

Yearling . 

.277 

Metropolitan . 

2.25 

Two-mile . 


Miles. 


2-5 


i -5 

!. 79 6 


r -75 

1.923 













































































































SCRIPTURE MEASURES.—ANCIENT WEIGHTS. 


53 


SCRIPTURE AND ANCIENT LINEAR MEASURES. 
Scripture. 

Digit .912 inch. I Span, 3 palms. io . 9 44 ins. 

Palm, 4 o digits. 3.648 ins. | Cubit. 2 spans.21.888 “ 

Fathom, 4 cubits.7 feet 3.552 ins. 

Hebrew and Egyptian. 

Nahud cubit.1.475 feet. 

Royal “ .1.7216 “ 

Egyptian finger.061 4 5 “ 

Hebrew sacred cubit, 


Babylonian foot.i.i 4 ofeet. 

Hebrew “ .1.212 “ 

“ cubit. . .^817 “ 

.2.002 feet. 


Grecian 

Digit...755 4 inch. 

Pous (foot).1-0073 feet. 

Cubit.....1,1332 

Pythic or natural foot. 8i 4 foot. 

Attic or Olympic “ .1.009 feet. 


Ancient Greek foot 1 

(16 Egyptian fingers)).984 1 f° ot - 

Arabian foot.1-095 feet. 

Stadium.604.0375 u 

Olympic stadium.606.29 “ 


Mile, 8 stadium. 4 8 35 feet. 

Alexandrian or Phileterian stadium (600 Phil, feet) = 708.65 feet. 

Volume .—Keramion or Metretes. 8. 4 88 gallons. 

Jewish. 

Cubit. i .82 4 feet. I Mile, 4 ooo cubits.7296 feet. 

Sabbath day’s journey_ 36 4 8 “ | Day’s journey.. 33. i6 4 miles. 

Roman Eong Measures. 


Digit.72575 ins. 

Uncia (inch).967 “ 

Pes (foot). ii.6o 4 “ 


Cubit.1-4505 feet. 

Passus.4-835 “ 

Mile, milliarium. 4 8 4 2 “ 


ANCIENT WEIGHTS. 


Hebrew and Egyptian. 


Attic obolus. 


Troy grains. 
) 8.2* 

• I 9-it 


1 51 - 9 * 

“ drachma.i54-6f 

(6 9 t 

Lesser mina. 3-892 

Greater mina. 5.46 

Egyptian mina. 8.326* 

Ptolemaic “ . 8.985* 

Alexandrian “ . 9.992* 

Obolus.. 4.63 


Troy grains. 

Denarius, Roman.f 

I 62.57 

“ Nero. 54$ 

Shekel. 92.62 

(415-1* 

Ounce. ' 437.2! 

[431.2% 

Drachm. 146.5 

Libra.. 4086.1 

Pound.12 Roman ounces. 

Talub.581.71 ounces. 


Talent (60 min®).56 lbs. avoirdupois. 


Grecian. 
Troy grains. 

Obolus, ancient. 8.33 

r “ IX -57 

Drachma.50.01 

* “ great...69.47 


Troy ounces. 

Mina. 10.41 

“ great. 14-472 

Talent.625.19 

“ Attic.868.32 


Ounce. 


Eoman. 

416.82 grains. | Pound 


10.41 ounces. 


* Christiani. 


t Arbuthnot. 

E* 


t Paucton. 






































































54 


GEOGRAPHIC MEASURES AND DISTANCES. 


GEOGRAPHIC MEASURES AND DISTANCES. 


To Reduce Toiagitu.de into Time. 

Rule.—M ultiply degrees, minutes, and seconds by 4, and product is 
the time. 

Example.— Required time corresponding to 50° 31'. 50° 31' X 4 = 3^- 22 m. 4s. 


To Reduce Time into Longitude. 

Rule.—R educe hours to minutes and seconds, divide by 4, and quo¬ 
tient is the longitude. Or , Multiply them by 15. 

Example. — Required longitude corresponding to 5 h . 8 m . it. 2 s . 

5 h. 8m. 11.25. = 308m. 11.25., which - 4 - 4 = 77 0 2' 45.5”. 

Or, multiplying by 15: 5 li. 8m. 11.2s. X 15 — 77 0 2' 45.5”. 


T'aDle of Departures for a Distance run of 1 IVIile. 


Course. 

Departure. 

Course. 

Departure. 

Course. 

Departure. 

3.5 points. 

•773 

4.5 points. 

•634 

5.5 points. 

.471 

4 

• 7°7 

5 

•556 

6 “ 

•383 


Thus, if a vessel holds a course of 4 points, that is without leeway, for distance 
of 1 mile, she will make .707 of a mile to windward. 

Or, a vessel sailing E.N.E. upon a course of 6 points for 100 miles will make 38.3 
(100 x -383) miles of longitude. 


Degrees, JVTinutes, and. Seconds of eacli Point of tHe 
Compass ■wit-li Meridian. 


North. 


N. 


South. 


N. by E. 

N. by W. 


N.N.E. 

N.N.W. 


N.E. by N. .. 
N. VC by N... 


N.E. 

N.W. 


N. E. by E. .. 
N.W. by W... 


E.N.E. 

W.N.AV. 


E. by N. 

W. by N. 

East or West. 


S. 


S.by E. 

S. by W. 


5.5. E. 

5.5. W. 


S.E. by S. ... 
S.W. by S.... 


S.E. 

S.W. 


S.E. by E.... 
S.W. by W... 


E. S.E. 

W. S. W. 


E. by S. 

W. by S. 

East or West. 


Points. 

0 * n 

Sin. A* 

| Cos. A.* 

Tan. A.* 

•25 

2 48 45 

.0489 

.9988 

.0491 

•5 

5 37 30 

.5398 

' -'9952 

.0985 

■75 

8 26 15 

.1467 

.9891 

.1484 

I 

u i 5 

•195 

.9808 

•1989 

1.25 

i 4 3 45 

.2429 

•97 

•2504 

i -5 

16 52 30 

•2903 

•9569 

•3034 

i -75 

19 4 i 15 

.3368 

• 94 i 5 

•3578 

2 

22 30 

.3827 

• 9 2 39 

.4142 

2.25 

25 18 45 

•4275 

• 9°4 

•4729 

2-5 

27 7 30 

.4714 

.8819 

•5345 

2-75 

30 56 13 

•5141 

•8577 

•5994 

3 

33 45 

•5556 - 

•8315 

.6682 

3-25 

36 33 45 

•5957 

.8032 

.7416 

3-5 

39 22 30 

• * * S. 6 344 

•773 

. 8207 

3-75 

42 11 15 

• 6 7 i 5 

.7409 

•9063 

4 

45 . 

.7071 

.7071 

I 

4-25 

47 48 45 

.7404 

•6715 

1-103 

4-5 

50 37 30 

•773 

.6344 

1.218 

4-75 

53 26 15 

.8032 

•5957 

1.348 

5 

56 15 

8315 

•5556 

1.497 

5-25 

59 3.45 

•8577 

•5141 

1.668 

5-5 

61 52 30 

.8819 

• 47 i 4 

1.871 

5-75 

64 41 15 

.904 

•4275 

2.114 

6 

67 30 

•9239 

.3827 

2,4x4 

6.25 

70 18 45 

• 94 i 5 

•3368 

2 -795 

6-5 

73 7 30 

••9-569 

•2903 

3.296 

6-75 

75 56 15 

•97 ■ 

.2429 

3 - 94 i 

7 

78 45 

.9808 

•i 95 

5.027 

7-25 

81 33 45 

.9891 

.1467 

6.741 

7-5 

84 22 30 

•9952 

.098 

10.153 

7-75 

87 11 15 

.9988 

.0489 

20-555 

8 

90 

I 

.OOOO 

00 


* A, representing course or points from the meridian. 









































GEOGRAPHIC LEVELLING. 


55 


GEOGRAPHIC LEVELLING. 

Cnrvatnre and. Refraction. 

Correction for Curvature of Earth, to be subtracted from reading of 
a levelling-staff, is determined as follows : 

Divide square of distance in feet from level to staff, by Earth’s Equa¬ 
torial diameter—viz., 41 852 124 feet. 

Or, Two thirds of square of distance in statute miles equal the cur¬ 
vature in feet. 

Correction for Ref raction is to be added to reading, and as a mean 
may be taken at about one sixth of that for curvature. 

Correction for Curvature and Refraction combined, is to be subtracted 
from reading 011 staff. 


Formulas of Capt. T. J. Lee , U. S. Engineers. 

D 2 D 2 

—— =z correction for curvature , -r- m~ correction for refraction , and 
2 R R 

D 2 

(1 — 2 m) —- 7 = correction for curvature and refraction. D representing 

2 It 

distance , R radius of earth , and rn a coefficient of refraction — .075, all 
in feet. 

Illustration. — A distance is 3 statute miles, what is correction for cuiwature 
and refraction ? 


C280 ^ o 

(1 — 2 X .075) —--= .85 X 5-996 = 5-097 fiet. 

41 O52 124 


Approximately , — D 2 = curvature infect. 


Re veiling loy Roiling IPoint of Neater. 

To Compute Heiglit Above or Below Level of Sea. 

517 (212 0 — T) -f- (212 0 — T) 2 = Height. 

Illustration.— What is height of an elevation, when boiling point of water is 182° ? 
517 X 212 0 —182° -(-212 0 —182° = 517 X 30-)- 30 2 — 16 410 feet. 


Corrections for Temperature to he made in Connection with Formula. 


Temp. 

Correc¬ 

tion. 

Temp. 

Correc¬ 

tion. 

Temp. 

Correc¬ 

tion. 

Temp. 

Correc¬ 

tion. 

Temp. 

Correc¬ 

tion. 

1 

Temp. 

Correc¬ 

tion. 

O 

•93 6 

O 

18 

.972 

O 

36 

1.008 

54 

1.046 

O 

72 

1.083 

9° 

I. 12 . 

2 

•94 

20 

.976 

38 

I. 012 

56 

1.05 

74 

1.087 

92 

I. 124 

4 

•944 

22 

.98 

40 

1.016 

58 

1.054 

76 

I.O9I 

94 

1.128 

6 

.948 

24 

.984 

42 

1.02 

60 

1.058 

78 

1.096 

96 

1-132 

8 

•952 

26 

.988 

44 

I.O24 

62 

1.062 

80 

I. I 

98 

1.136 

# IO 

•956 

28 

.992 

4 6 

1.028 

64 

1.066 

82 

I. 104. 

IOO 

1.14 

12 

.96 

3 ° 

.996 

48 

1.032 

66 

I. 07 I 

84 

1.108 

102 

I. I44 

14 

• 9 6 4 

3 2 

I 

So 

1.036 

68 

i -°75 

86 

1.112 

IO4 

1.148 

l6 

.968 

34 

1.004 

52 

1.041 

70 

1.079 

88 

1.116 

106 

1-152 


Illustration. —Assume temperature in preceding illustration to have been 8 o°- 
Then 16 410 X 1.1 = 18051 feet 































56 


GEOGRAPHIC LEVELLING AND DISTANCES. 


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of Curvi 
Refra 

Land. 

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g mco vo m co cn 0 on n m cn m onoo vo m co m on ov n m h o cn 

ta ’ ’ h ci co 4 in mvd t^oo onO 0 h w co 4 »o mvo t^.oo on d h m 

mmmmmmhmmmmmCNCNCN 

H E I 

of 

Curvature 

above 

Land. 

CN 

VO O'. CO 00 t-s hxVO m co CN H HH NOnNOnW COHCO COH h NONW 

gvo OnOnOnOnOnOnOnOnOnOnOn On 00 COCOOOOOOOOOOOOOOOOOVO 

* h cn cd 4- mvd t^co on o h cn cd 4- idvd r^cd on o h cn cd 4- on 

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of Line of 
Sight. 

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r ® . 

H CN CO Ht- mvo t^OO ON 0 H CN CO ^ mvo t^OO On 0 H CN CO Hi- m o 

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^ h h h cn cn cn cocciccion444tt4icuoioinwioio mvd vo 


rt* h co Hf -t On COOO m 
CN ^ m N On O m CO t}-vO 


"Tt- t^CO CN 
t^CO ON M 


Note i.—H eight or elevation in second column of table is also curvature of Earth at Ocean. 

2.—Refraction is greater and more variable at sunrise and sunset, and comparatively stationary between hours of io a. m. and 4 p. 



































GEOGRAPHIC LEVELLING.-MAGNETIC VARIATION. 57 


Illustration. —Curvature of Earth independent of refraction is computed at 
.667 foot= 8.004 ' ns - f° r 1 geographical mile, and as refraction on land is taken as 
.104 foot or 1.248 ins., and on ocean at .099 foot or 1.188 ins., relative visible dis¬ 
tances of an object, including curvature andheffaction, for an elevation of 

.667 foot is 1.09 miles on land, and 1.08 miles at sea. 

1 “ “ 1.33 “ “ “ “ 1.32 “ “ “ 

9 feet “ 4 “ “ “ “ 3.98 , “ “ “ 

1 mile <£ 104.03 “ u “ “ 103.54 “ “ 

Difference between two levels in feet is as square of their distance in 
miles. 

Illustration. —At what elevation can an object be seen, at surface of ocean, when 
it is 2 miles distant? 

i 2 : 2 2 :: .568 : 2.272 feet = 2 feet 3.25-f- ins. 

Difference between two distances in miles is as square root of their heights 
in feet. 

Illustration i. —At an elevation of 9 feet above level of sea, at what distance 
can an object be seen upon its surface? 

-^.568 = .754 : 1 :: y/g : 3.98 miles. 

2.—If a man at the fore-topgallant mast-head of a vessel, 100 feet from water, sees 
another and a large vessel “hull to,” how far are the vessels apart? 

A large vessel’s bulwarks are at least 20 feet from water. 

Then, by table, 100 feet.= 13.27 

20 u .= 5-93 

Distance. 19.20 miles. 

When an observation for distance is taken from an elevation, as from 
a light-house, a vessel’s mast, etc., of an object that intervenes between 
observer and horizon, or contrariwise, observer being at a horizon to 
elevated object, distance of observer from intervening object can be 
determined by ascertaining or estimating its elevation from horizon, and 
subtracting its distance from whole distance between observer and 
point from which observation is taken, and remainder will give distance 
of object from observer. 

Illustration.— Top of smoke-pipe of a steamer, assumed to be 50 feet above sur¬ 
face of water, is in range with horizon from au elevation of 100 feet; what is dis¬ 
tance to steamer from elevation ? 

100 feet...•..== 13.27 

50 “ .. — 9-3 8 

Distance. 3.89 raiVes. 

Approximately .—Curvature less Refraction = .566 D 2 for land and .563 D 2 for sea 
D representing distance in miles. 


MAGNETIC VARIATION OF NEEDLE. 

America .—Needle reached a Westerly maximum in 1660, and then 
varied to East until 1800, when it reversed to West. 

London (Eng.).—From 1576 to 1815 variation ranged from n° 15' 
East to 24 0 27' West, when it receded gradually to 21 0 in 1865. 

Jamaica (W. I.).—No variation from year 1660. 

Diurnal Variation .—There is a small diurnal variation, being greatest 
in summer (15'), and least in winter (7' 30"), added to which a change 
of temperature affects a needle. 











58 


MAGNETIC VARIATION OF NEEDLE, 


Variation in U. S. — Professor Loomis concludes that the Westerly 
variation is increasing and Easterly diminishing in every part of United 
States; that this change occurred between 1793 and 1819, and that 
present annual change is about 2' in Southern and Western States, from 
3' to 4' in Middle States, and 5' to 7' in Eastern States. 

Rules for computation of variation are empirical, except in each 
particular locality, as the annual and diurnal variations of the needle, 
added to local attraction, render it altogether unreliable. 


Decennial "Variation of NTeedle. 
Mr . Schott , U . S . Coast and Geodetic Survey . 
From , January 1,1790, to January 1,1880. 


Location. 


Halifax, N. S. 

Quebec, Can. 

Portland, Me. .... 
Burlington, Vt.... 
Newburyport, M’s. 
Portsmouth, N. H. 

Rutland, Vt. 

Salem, Mass. 

Boston, Mass. 

Cambridge, Mass.. 
Providence, R. I. . 
Hartford, Conn. .. 
New Haven, Conn. 
New York, N. Y. . 
Philadelphia, Pa.. 
Baltimore, Md.... 
Albany, N. Y. 

Buffalo, N. Y. 

Erie, Pa. 

Cleveland, 0. 

Detroit, Mich. 

Washington, D. C. 

Acapulco, Mex.... 
Charleston, S. C... 
Havana, Cuba.... 

Kingston, W. I_ 

San Diego, Cal.... 

Savannah, Ga. 

Mobile, Ala. 

Key West, Fla.... 
Monterey, Cal. ... 

Mexico, Mex. 

New Orleans, La.. 
San Bias, Mex. ... 
San Francisco, Cal. 

Sitka, Alaska. 

Vera Cruz, Mex... 


1790. 

1800. 

1810. 

1820. 

1830. 

1840. 

1850. 

| i860. 

1870. 

1880. 

w. 

W. 

W. 

W. 

W. 

W. 

W. 

W. 

W. 

W. 

0 

O 

O 

0 

0 

0 

0 

0 

0 

0 

i 5 -1 

i 5-9 

16.7 

17.4 

18.1 

18.7 

i 9-3 

19.8 

20.1 

20.3 

— 

— 

11.2 

12.3 

13-4 

14.4 

15-3 

16 

16.4 

— 

8-5 

8.9 

9.4 

10 

10.6 

n.23 

11.82 

i 2 -35 

12.8 

I 3 -I 5 

7-7 

7-52 

7-39 

7-58 

8.17 

8.94 

9.62 

10.21 

10.97 

11.97 

7.2 

7-4 

7.8 

8.4 

9 

9.6 

10.23 

10.83 

11 -4 

11.8 

7.8 

8 

8.4 

8.8 

9-35 

9.94 

io -55 

ii-i 5 

n -7 

12.2 

6-5 

6.2 

0.14 

6-39 

6.9 

7.64 

8-53 

9-53 

10.54 

11.49 

6.2 

6.2 

6-5 

7 

7 - 8 

8.7 

9.8 

10.9 

11.9 

12.8 

6.7 

7 

7-4 

7-9 

8-43 

9-°5 

9.69 

10.32 

10.9 

11.41 

6.9 

7 -i 

7-5 

8 

8.64 

9-33 

10.03 

10.67 

11.21 

11.63 

6.24 

6-37 

6-45 

6-73 

7-43 

8.31 

9.09 

9- 6 5 

10.21 

10.94 

5-2 

5.16 

5-24 

5-46 

5-8 

6.24 

6.77 

7-36 

7-99 

8.62 

4.8 

4'7 

4.8 

5 

5-43 

5-99 

6.67 

7.41 

8.18 

8.9 

4.29 

4.28 

4-3 

4-47 

4 - 9 1 

5-59 

6-34 

6.96 

7-43 

7.84 

2.4 

2.1 

2.1 

2.28 

2.71 

3-33 

4.11 

4-99 

5-89 

6.76 

— 

— 

.6 

.8 

1.2 

*~7 

2.4 

2.9 

— 

— 


E. 

5-4 

5-79 

6.32 

6.97 

7-7 

8.47 

9.2 

9.9 

• r 4 

.01 

•°5 

•3 

•74 

i -33 

2.05 

2.85 

3-68 

4.49 

E. 


E. 

E. 

E. 





•03 

•35 

•49 

•43 

•17 

• 2 5 

• 8 3 

i -5 

2.23 

2.96 






E. 

E. 



2.2 

2 

1.8 

i -5 

1.05 

.6 

.14 

• 3 i 

•72 

1.07 








E. 

E. 


v s 

3 -11 
W. 

w? 

2-55 

W. 

2.og 

W. 

1.56 

W. 


.41 

W. 

•13 

.1 

— 

•3 

.6 

1 

1.49 

i -99 

2.47 

2.9 

3.26 


E. 

E. 

E. 

E. 

E. 

E. 

E. 

E. 

E. 

7.2 

7.8 

8-3 

8.68 

8.88 

8.91 

8.79 

8-5 

8.06 

7-5 

5 i 

4.9 

4-5 

4.04 

3-44 

2.78 

2.12 

1.52 

1 

.62 

— 

6.2 

6.26 

6.22 

6.12 

5-94 

5 - 7 i 

5-44 

5 -i 

— 

6 -3 

6 

5-7 

5-4 

5 

4.6 

4.2 

3-8 

3-4 

— 

11 

11.1 

n -3 

11.6 

11.9 

12.2 

12.54 

12.88 

13.2 

13-5 

— 

— 

4.9 

4.8 

4-5 

4.14 

3-65 

3.08 

2.48 

1.89 

— 

7 -i 

7.2 

7-3 

7.2 

7 -i 

7 

8.8 

— 

6.3 

— 

— 

— 

6.9 

6.52 

6.03 

5-47 

4.86 

4.24 

3-65 

11.4 

12 

12.6 

13-3 

*3 9 

14.44 

14-95 

15.42 

15-79 

16.08 

7 -i 

7-7 

8-3 

8.6 

8.8 

8.9 

8.76 

8.48 

8.04 

7.46 

7 

7-5 

7-9 o 

8.1 

8.2 

8.14 

7-94 

7.61 

7 -i 5 

6.62 

7.41 

7.88 

8.28 

8.61 

8.84 

8.97 

9.9 

8.91 

— 

12.8 

i 3-4 

13-9 

14.42 

14.92 

I 5-38 

15-78 

16.11 

16.36 

16.52 

— 

26.12 

27.11 

27.89 

28.48 

28.88 

29.08 

29.08 

28.88 

28.5 

8-37 

8-95 

9 -32 

9.48 

9.42 

9.14 

8.66 

7.98 

7 -i 5 

— 


For variation in other locations in United States and North America, 
see Treatises of J. B. Stone, C.E., New York, and Heller and Brightly! 
Philadelphia, 1878. 6 3 ' 

























MAGNETIC VARIATION OF NEEDLE. 


59 


TaLle for Reducing Observed Daily "Variation of Needle 
to Mean "Variation of the Day. 

U. S. Coast and Geodetic Survey , 1878. 



Needle East of Mean Mag- 


Needle West of Mean Magnetic 


Season. 


netic Meridian. 





Meridian. 





A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

NOON. 

P.M. 

P.M. 

P.M. 

P.M. 

P.M. 

P.M. 


h. 

h. 

h. 

h. 

h. 

h. 


h. 

h. 

h. 

h. 

h. 

h. 


6 

7 

8 

9 

10 

II 

Noon. 

I 

2 

3 

4 

5 

6 


r 

/ 

/ 

/ 

/ 

t 


t 

t 

/ 

/ 

/ 

t 

Spring. 

3 

4 

4 

3 

I 

I 

4 

5 

5 

4 

3 

2 

I 

Summer. 

4 

5 

5 

4 

I 

2 

4 

6 

5 

4 

3 

2 

I 

Autumn. 

2 

3 

3 

2 

— 

2 

3 

4 

3 

2 

I 

I 

— 

Winter. 

1 

I 

2 

2 

I 

— 

2 

3 

5 

2 

I 

I 

— 


Variation of Needle 
U.S. 


Augusta, Me. 
Bangor, Me. 


Batavia, N. Y. 


Belfast, Me. 

Bridgeport, Conn... 

Calais, Me. 

Concord, N. H. 

Dover, Del. 

Fall River, Mass.... 

Hamilton, Can. 

Harrisburg, Pa.. 

Hudson, N. Y. 

Lewiston, Me. 

Lowell, Mass. 

Montpelier, Vt. 

Montreal, Can. 

New Bedford, Mass. 
New London, Conn. 
Newark, N. J. 


at Locations in United States and 
Canada, 1 BTC 3 . 

Coast and Geodetic Survey. 

EAST. 

Variation. 


Location. 

Variation. 

Location. 

* 

Astoria, W. T. 

O ' 

21 QO 

Montgomerv. Ala. 

Augusta, Ga. 

2 28 

Natchez, Miss. 

Austin. Tex. 

Q I ^ 

Nebraska, Neb. 

Bismarck, Dak. 

16 6 

New Orleans. La. 

Chicago, Ill. 

C 

Olympia, W. T. 

Cincinnati, 0 . 

D 

2 

Omaha, Neb..... 

Colorado Springs, Col. 

14 18 

Oregon City, Or. 

Columbia, S. C. 

I 4 ^ 

Paducah, Kan. 

Columbus, 0 . 

1 8 

Portland, Or. 

Deadwood, Dak. 

16 20 

Port Townsend, W. T. 

Denver, Col. 

14 44 

Sacramento, Cal. 

Detroit, Mich... 

Q 

Salt Lake City, Utah. 

Duluth, Min. 

IO 12 

San Antonio, Tex. 

Galveston, Tex. 

8 13 

Santa Barbara, “ . 

Green Bay, WiS. 

6 

Santa Fe, N. Mex. 

Houston, Tex. 

27 

Springfield, 111 . 

Indianapolis, Ind. 


St. Augustine, Fla. 

Jackson, Miss. 

7 

St Louis, Mo. 

Jacksonville, Fla. 

*3 

St. Paul, Minn. 

Kansas, Kan. 

O 

Q 20 

Tallahassee, Fla. 

Keokuk, la. 

7 * 5 *! 

Toledo, 0 . 

Little Rock, Ark. 

8 4 

Topeka, Kan. 

Louisville, Ky. 

A 

Vincennes, Ind. 

Milwaukee, Wis. 

5 48 

Yazoo, Miss. 


WEST. 


14 34 

Newburgh, N. Y. 

l 6 

Newport, R. I. 

4 4 ° 

Norfolk, Va. 

15 22 

Ogdensburgh, N. Y. 

8 12 

Oswego, N. Y. 

18 

Ottawa, Can. 

11 42 

Pittsburgh, Pa. 

4 12 

Raleigh, N. C. 

10 30 

Richmond, Va. 

2 55 

Rochester, N. Y. 

4 18 

Saratoga, N. Y. 

8 48 

Stamford, Conn. 

14 

Syracuse, N. Y. 

11 15 

Toronto, Can. 

12 5 

Trenton,N. J. 

12 20 

Troy, N. Y. 

10 30 

Utica, N. Y. 

9 

Wilmington, Del. 

7 18 

Wilmington, N. C. 


0 

/ 

5 

2 

7 

26 

11 

20 

6 

50 

22 

8 

11 


20 

55 

6 

2 

21 

4 

23 


17 

4 

17 


9 

17 

14 

58 

13 

18 

6 

3 

2 

55 

6 

30 

10 

30 

4 

i 4 

z 

2 

10 

12 

5 


7 

2 

8 


10 

4 

2 

35 

9 

25 

6 

8 

9 

38 

1 

28 


24 

1 

48 

5 

20 

9 

40 

8 


7 


3 

50 

6 

8 

9 

25 

8 


4 

52 


18 

































































































































60 GEOGRAPHIC LEVELLING.—BASE LINE.-SOUNDINGS. 


Dip of Horizon. 

Approximate, 57.4 3/ II — dip in seconds , varying with temperature of 
air. H representing height of, observer’s eye in feet. 

.667?a 2 = H :' .498 s 2 =H: 1.42-/II — s: 1.23 /H — n. 

n representing distance in geographical miles and s in statute. 

HVLeasurement of Dleiglxts with. a Sextant. 


Multi¬ 

plier. 

Angle. 

Multi¬ 

plier. 

Angle. 

Multi¬ 

plier. 

Angle. 

Multi¬ 

plier. 

Angle. 

Multi¬ 
plier. • 

Angle. 

I 

0 - / 

45 0 

2-5 

O / 

68 11 

4 

0 / 

75 58 

5-5 

O / 

79 42 

8 

O / 

82 52 

i -5 

5618 

3 

71 34 

4-5 

77 29 

6 

80 32 

9 

83 40 

2 

63 26 

3-5 

74 4 

5 

78 41 

7 

81 52 

IO 

84 17 


Operation . — Sot sextant to any angle in table, and height will equal distance 
multiplied by number opposite to it. 


Illustration; —When sextant is set at 8o° 32', and horizontal distance from ob¬ 
ject in a Vertical line is 100 feet, what is its height? 

100 x 6 = 600 feet. * 

By Trigonometry: 1 : 100 :: 5.997 (tan. angle) : 599.7 feet. 


To Reduce a Somacliiag to Dow ‘'W'atex*. 

- |iq: cos. —^-J==/G h representing vertical rise of tide , and It sound¬ 
ing or depth at loiv icater , both in feet; t time between high and low ivate.r , and 
t! time from time of sounding to low \ water , in hours. — cos. when <90°, 
and + cos. when >90°. 

Illustration. —Low water occurring at 3.45, and high water at 10.i£ p.m., a 
sounding taken at 5^30 p.m. was 18.25 feet; what was depth at low water, vertical 
rise being 10 feet? 

h^=i 10 feet; V = 5 h. 30 m. —3 h. 45 m. = ih. 45m.1.75 hours, 
t = ioh. 15 m. — 3/1. 45m. — 6 h. 20m. =6.5 hours. 

180X1.75^ 


Then — (1 q= cos. 


6-5 


-^=5(1 — 48° 27' 24”) = 5 X (1 — .663 186) 1.68407 feet. 
Sounding 18.25 feet — Reduction 1.684 07 feet=- 16.565 93 feet. 

Dengths of a Degree of Dongitu.de oa pax-allels of Dati- 
fu.de, for each of its Degrees from Equator to Dole. 


Lat. 

Allies. 

Lat. 

Aides. 

Lat. 

Miles. 

Lat. 

Miles. 

| Lat. 

Allies. 

Lat. 

Aides. 

i° 

59-99 

16° 

57-67 

3 i° 

51-43 

46° 

41.68 

6i° 

29.O9 

76° 

14.52 

2 

59-96 

17 

57-38 

32 

50.88 

47 

40.92 

62 

28.17 

77 

13-5 

3 

59 - 9 2 

l8 

57.06 

33 

50.32 

48 

40.15 

63 

27.74 

78 

12.48 

4 

59- 8 5 

19 

56.73 

34 

49-74 

49 

39-36 

64 

26.3 

79 

n -45 

5 

59-77 

20 

56 .38 

35 

49- 1 5 

50 

38.57 

65 

25.36 

80 

10.42 

6 

59- 6 7 

21 

56.01 

36 

48.54 

5 i 

37 d 6 

66 

24.4 

81 

9-38 

7 

59-55 

22 

55-63 

37 

47.92 

52 

36.94 

67 

23-44 

82 

8-35 

8 

59-42 

23 

55-23 

38 

47.28 

53 

36.11- 

68 

22.48 

83 

7-31 

9 

59.26 

24 

54.81 

39 

46.63 

54 

35.27 

69 

21.5 

84 

6.27 

IO 

59-°9 

2 5 

54-38 

40 

45-96 

55 

34 - 4 i 

70 

20.52 

85 

5-23 

II 

58.89 

26 

53-93 

4 i 

45.28 

56 

33-45 

7 i 

19-53 

86 

4.18 

12 

58.69 

27 

53-46 , 

42 

44-59 

57 

32.68 

72 

18.54 

87 

3- I 4 

13 

58.46 

28 

52-97 1 

43 

43.88 

58 

31.79 

73 

I 7-54 

88 

2 

14 

58.22 

29 

52.48 i 

44 

43.16 

59 

3 o -9 

74 

16.54 

89 

1.05 

15 

57-95 

30 

51-96 i 

45 

42-43 

60 

30 

75 

15-53 

90 

.OO 

Note. — Degrees 

of longitude are to each other in 

length 

as Cosines of their 


latitudes. 

















































FIGURE OF EARTH.-BOARD AND TIMBER MEASURE. 6 1 


Elements of* Figure of tlie Earth. 
Capt. A. R. Clarice , 1866. 


Feet. Miles. 

Major semi-axis of Equator (longitude 15 0 34' E.) .20926350 3963.324. 

Minor “ “ “ “ ( “ i°5°34 , E.).20919972 3962.115. 

Polar “ “ .20853429 3949.513. 

Equatorial semi-axis.20926062 3963.269. 

Circumference, mean. . 24898.562. 

Diameter, “ ...- - 7916. 


BOARD AND TIMBER MEASURE. 

BOARD MEASURE. 

In Board Measure , all boards are assumed to be 1 inch in thickness. 
To Compoate Measure or Surface. 

When all Dimensions are in Feet. 

Rule. —Multiply length by breadth, and product will give surface in 
square feet. 

When either of Dimensions are in Inches. 

Rule. —Multiply as above, and divide product by 12. 

When all Dimensions are in Inches. 

Rule. —Multiply as before, and divide product by 144. 

Example. — What are number of square feet in a board 15 feet in length and 16 
inches in width ? 

15 x 16 = 240, and 240-7-12 =20 feet. 


TIMBER MEASURE. 


To Compute Volume of Round Tiixfber. 

When all Dimensions are in Feet. 

Rule.— Add together squares of diameters of greater and lesser ends, 
and product of the two diameters; multiply sum by .7854, and product 
by one third of length. 


Or, a + a'-fa" X - = V, and c 2 3 + c' 2 -f- c x c' X .079 58 X - = V. a and 

a! representing areas of ends , cl' area of mean proportional , l length , and c 
and d circumference of ends. 

Note.— Mean proportional is square root of product of areas of both ends. 
Illustration.— Diameters of a log are 2 and 1.5 feet, and length 15 feet. 


2 2 -f-1.5 2 —4 -}~ 3- 25 d - 2 X 1.5 = 9.25, which X -7854 and — = 36.32 cube feet. 

3 


When Length in Feet , and Areas or Circumferences in Inches. 
Rule. —Proceed as above, and divide by 144. 


When all Dimensions are in Inches. 


Rule. —Proceed as before, and divide by 1728. 

Note. — Ordinary rule of Hutton, Ordnance Manual of U. S., and Molesworth, of 
_ 2 

IX c- 4-4, gives a result of about .25 less than exact volume, or what it would be 
if the log was hewn or sawed to a square, c representing mean circumferences. 

F 














62 


BOARD AND TIMBER MEASURE. 


To Compute ‘Volizm.e of Squared Timber. 

When all Dimensions are in Feet. 

Rule. —Multiply product of breadth by depth, by length, and product 
will give volume in cube feet. 

When either Dimension is in Inches. 

Rule. —Multiply as above, and divide product by 12. 


When any two Dimensions are in Inches. 

Rule. —Multiply as before, and divide by 144. 

Example.— A piece of timber is 15 inches square, and 20 feet in length 5 required 
its volume in cube feet. 

144 

Allowance is to be made for bark, by deducting from each girth from 
.5 inch in logs with thin bark, to 2 inches in logs with thick bark. 

Measures of Timber.- ( English .) 

100 superficial feet \ _ _ __„„„ 

of planking f ^ 

120 deals ...= 1 hundred. 


50 cube feet of squared )_ V 

timber J ~ 1 l0act * 

40 feet of unhewn timber = 1 load. 


600 superficial feet of inch planking = 1 load. 


Deals. 

Deals. — Boards exceeding 7 ins. in width, and if less than 6 feet in 
length, are termed deal ends. 

Battens are similar to deals, but only 7 inches in width. 

Balk .—Roughly squared log or trunk of a tree. 

Planks are boards 12 ins. in width. 


Local Standards. 


Country. 

Long. 

Eroad. 

Thick. 

Volume. 

Country. 

Long. 

Broad. 

Thick. 

Volume. 


Ft. 

Ins. 

Ins. 

Cub. ft. 


Ft. 

Ins. 

Ins. 

Cub. ft. 

Russia and 





Norway.. 

12 

9 

3 

2.25 

Prussia.. 

12 

II 

1,5 

1-375 

Christiana 

II 

9 

1.25 

•859 

Sweden... 

14 

9 

3 

2.625 

Quebec... 

12 

11 

2-5 

2.292 


100 Petersburg!! standard deals equal 60 Quebec deals. 


SPARS AND POLES. 

Pine and Spruce Spars, from 10 to 4.5 inches in diameter inclusive, 
are to be measured by taking their diameter, clear of bark, at one third 
of their length from abut or large end. 

Spars are usually purchased by the inch diameter; all under 4 inches 
are termed Poles. 

Spars of 7 inches and less should have 5 feet in length for every 
inch of diameter, and those above 7 inches should have 4 feet in length 
for every inch of diameter. 


Loss or 'Waste in Hewing or Sawing of Timber. 
tC . Mackraw .) 


Oak, English.200 per cent. 

“ African...100 “ “ 

“ Dantzic...... 50 “ “ 

“ American...... 10 “ “ 


Yellow Pine from planks.. 10 per cent 

Teak.... 15 “ “ 

Elm, English.200 “ “ 

“ American. 15 “ “ 



























CISTERNS.-SHINGLES. 


63 


CISTERNS. 

Capacity of Cisterns in Cube ITeet and. Gallons. 


For each 10 Inches in Depth. 


Diam. 

Cub. ft. 

Gallons. 

Diam. 

Cub. ft. 

Gallons. 

Diam. 

Cub. ft. 

Gallons. 

Feet. 



Feet. 



Feet. 



2 

2.618 

19.58 

9-5 

59.068 

441.8 

17 

189.15 

1414.94 

2-5 

4.091 

30.6 

10 

65.449 

489.6 

I 7-5 

200.432 

1499-33 

3 

5-89 

44.07 

10.5 

72.158 

539-78 

18 

212.056 

1586.28 

3-5 

8.018 

59-97 

11 

79.194 

592.4 

19 

236.274 

1767-45 

4 

10.472 

78-33 

n-5 

86.558 

647-5 

20 

261.797 

1958.3 

4-5 

I3-254 

99.14 

12 

94.248 

705 

21 

288.632 

2159.11 

5 

16.362 

122.4 

12.5 

102.265 

764.99 

22 

316.776 

2369.64 

5-5 

19.798 

148.x 

13 

no.61 

827.4 

23 

346.23 

2589.97 

6 

23.562 

176.24 

i3-5 

119.282 

892.29 

24 

376.992 

2820.09 

6-5 

27.652 

206.84 

14 

128.281 

959-6 

25 

409.062 

3059-8 

7 

32.07 

239.88 

14-5 

137.608 

1029.38 

26 

442.44 

3309.67 

7-5 

36.816 

275-4 

i5 

147.262 

1101.6 

27 

47I-I3 

3569-17 

8 

41.888 

3I3-33 

15-5 

157-243 

1176.26 

28 

513.126 

3838.44 

8-5 

47.288 

353-72 

16 

167.552 

1253-37 

29 

550-432 

4ii7-5i 

9 

53-oi4 

396-55 

16.5 

178.187 

I332-93 

30 

589.048 

4406.08 


Excavation and. Lining of 'Wells or Cisterns. 

For each 10 Inches in Depth. 


2 

G 

.2 

05 

Bricks. 

Masonry. 

<0> 

.2 

ci 

Bricks. 

Masonry. 

s 

05 

O 

Num- 

Laid 

8 inches 

1 foot 

I 

C5 

c 3 

Num- 

Laid 

8 inches 

1 foot 

5 

w 

ber. 

dry. 

thick. 

thick. 

5 

H 

ber. 

dry. 

thick. 

thick. 

Feet. 

Cub. ft. 


Cub. ft. 

Cub. ft. 

Cub. ft. 

Feet. 

Cub. ft. 


Cub. ft. 

Cub. ft. 

Cub. ft. 

3 

12.29 

126 

5-24 

6.4 

10.47 

8-5 

63.29 

356 

14.83 

l6 

24.87 

3-5 

15.29 

147 

6. ii 

7.27 

11.78 

9 

69.89 

377 

I 5 - 7 I 

16.87 

26.18 

4 

18.62 

168 

6.98 

8.14 

13.09 

9-5 

76.81 

398 

16.58 

17-75 

27.49 

4-5 

22.27 

188 

7-85 

9.02 

14.4 

IO 

84.07 

419 

17-45 

18.62 

28.8 

5 

26.25 

209 

8-73 

9.89 

I 5 - 7 1 

10.5 

91.65 

440 

* 8-33 

19.49 

3°. 11 

5-5 

30-56 

230 

9.6 

10.76 

17.02 

II 

99-56 

461 

19.2 

20.36 

31.42 

6 

35-2 

251 

10.47 

11.64 

18.33 

12 

116.36 

503 

20.94 

22 . II 

34-03 

6-5 

40.16 

272 

n -34 

12.51 

19.63 

13 

134.46 

545 

22.69 

23.85 

36-65 

7 

45-45 

293 

12.22 

13-38 

20-94 

14 

153-88 

586 

24-43 

25.6 

39-27 

7-5 

51-07 

3*4 

13.09 

14.25 

22.25 

15 

174.61 

628 

26.18 

27-34 

41.89 

8 

57.02 

335 

13.96 

i 5 -i 3 

23.56 

l6 

196.64 

670 

27.92 

29.O9 

44 - 5 i 


Number of bricks and width of curb are taken at dimensions of ordinary 
brick—viz., 8 by 4 by 2.25 ins. = 72 cube ins. 

In computing number of bricks required, an addition of 5 per cent, should 
be added for waste. It is to be considered, also, that diameter of excavation 
necessarily exceeds that of masonry. 


SHINGLES. 

Usually of white Cedar and Cypress; 27 inches in length and 6 to 7 
inches in width, dressed to light .25 inch at point and .3125 inch at 
abut. 

Laid in three thicknesses and courses of about 8 inches, so that less 
than .33 of a shingle is exposed to air, or about 2.25 shingles are re¬ 
quired per square foot of roof. 

Shingles, alike to Slates, are laid upon boards or battens. 
































6 4 


SLATES AND SLATING. 


SLATES AND SLATING. 


A Square of Slate or Slating is ioo superficial feet. 

Gauge is distance between the courses of the slates. 

Lap is distance which each slate overlaps the slate lengthwise next 
but one below it, and it varies from 2 to 4 inches. Standard is assumed 
to be 3 inches. 

Margin is width of course exposed or distance between tails of the 
slates. 

Pitch of a slate roof should not be less than 1 in height to 4 of length. 

To Compute Surface of a Slate "when, laid., and USTnm— 
ber of Squares of Slating. 

Rule. — Subtract lap from length* of slate, and half remainder will 
give length of surface exposed, which, when multiplied by width of 
slate, will give surface required. 

Divide 14 400 (area of a square in inches) by surface thus obtained, 
and quotient will give number of slates required for a square. 

Example. — A slate is 24 x 12 inches, and lap is 3 inches; what will be number 
required for a square ? 

24 — 3 = 21, and 2i -i- 2 = 10.5, which X 12 = 126 inches ; and 14 400 '- 4 -126 = 
114.29 slates. 

Dimensions of Slates. 


Ins. 

Ins. 

Ins. 

American, 

Ins. 

•] 

Ins. 

Ins. 

I Ins. 

14 X 7 

14 X 8 
14 x 9 

14 X 10 
16 x 8 
16 x 9 

16 X JO 
18 X 9 
18 X 10 

18 X II 

18 X 12 
20 X IO 

20 X II 
20 X 12 
22 X II 

22 X 12 
22 X 13 
24 X 12 

24 X 13 
24 X 14 
24 x 16 


English. 



J Ins. 


Ins. 


Ins. 

Doubles. 

I3XIO 

r 

i 

12 X 8 

Marchioness .. 

22X22 

u 

I3X 7 

14X 8 

Duchess. 

24XI2 

Small doubles . 

IIX 6 

Ladies. 

1 

14X12 

Imperial. 

30X24 

il 

10 X 5 

I5X 8 

Rags. 

36X24 

Plantations.. | 

12x10 

1 

l 

i6 X 8 

Queens. 

36 x 24 

13 X10 

16X10 

Empress ..... 

26 x 15 

Viscountess ... 

18X10 

Countess. 

20 X10 

Princess. 

24 X 14 


Thickness of slates ranges from .125 to .3125 of an inch, and their weight 
varies from 2 to 4.53 lbs. per sq. foot. 


Weiglit of One Square Foot of Slating. 

.125 in. thick on laths.4.75 lbs. I . 25 in. thick on laths. 9.25 lbs- 

“ “ “ “ 1 in. boards.. 6.75 “ “ “ “ “ 1 in. boards.. 11.25 “ 

.1875 in. thick on laths.7 “ .3125 in. thick on laths.n.15 “ 

“ “ “ “ 1 in. boards. 9 “ | “ “ “ “ 1 in. boards, 14.10 “ 


Slate weighs from 167 to 181 lbs. per cube foot, and in consequence, of 
laps, it requires an average of nearly 2.5 square feet of slate to make one of 
slating. 

Weights per 1000 and Number Required to Cover a Square. 


Doubles 
Ladies . 


13 x 6 

Lbs. 

1680 

No. 

480 

Countess. 

.. 20 x 10 

Lbs. 

6720 

15 X 8 

2800 

240 

Duchess . 

. . 24 x 12 

4480 


* Length of a slate is taken from nail-hole to tail. 








































SHOT AND SHELLS.-FRAUDULENT BALANCES. 65 


PILING OF SHOT AND SHELLS. 

To Compute Number of Shot. 

Triangular Pile . Rule.— Multiply continually together, number of shot 
in one side of bottom course, and that number increased by 1, and again by 
2, and one sixth of product will give number. 

Example. —What is number of shot in a triangular pile, each side of base contain¬ 
ing 30 shot? 

30 X 3° + 1 X 3°~t~ 2 29760 

---— —^—- = 4960 shot. 

o o 

Square Pile . Rule.— Multiply continually together, number in one side 
of bottom course, and that number increased by 1, double same number in¬ 
creased by 1, and one sixth of product will give number. 

Example.— How many shells are there in a square pile of 30 courses? 

30 X F+Lx 3 °-* ! ±i = S 6 | 3 g = 9455 

Oblong Pile . Rule.—F rom 3 times number in length of base course sub¬ 
tract one less than number in breadth of it; multiply remainder by number 
in breadth, and again by breadth, increased by 1, and one sixth of product 
will give number. 

Example.— Required number of shells in an oblong pile, numbers in base course 
being 16 and 7? 

.6X3-7yTx7 x7±I = shells. 

6 6 

Incomplete Pile . Rule.— From number in pile, considered as complete, 
subtract number conceived to be in that portion of pile which is wanting, 
and remainder will give number. 


fraudulent balances. 

To Detect Them . _After an equilibrium has been established between 

weight and article weighed, transpose them, and weight will preponder¬ 
ate If article weighed is lighter than weight, and contrariwise if it is 
heavier. 

To Ascertain True Weight . Rule.— Ascertain weight which will produce 
equilibrium after article to be weighed and weight ha\e been transposed , 
reduce these weights to same denomination^ multiply them together, anc. 
square root of their product will give true weight. 

Example. — If first weight is 32 lbs., and second, or weight of equilibrium after 
transposition, is 24 lbs. 8 oz., what is true weight ? 

24 lbs. 8 oz. =24.5 lbs. 

Then 32 X 24.5 = 784, and 7/784 == 28 lbs. 

Or when a represents longest arm , I A greatest weighty and 

b “ shortest arm, | B least weight. 

Then Wa = Ab, and W& = Ba; multiplying these two equations, W 2 a& = ABa&, 
or W 2 = AB, and W = -/ AB - 

Illustration. -A = 32; B = 24.5; W = 28. Assume length of longest arm = 10. 

Then 32 : 28 :: 10 : 8. 75. 

Hence, a — 10, b = 8.75, or 28 2 == 32 X 24- 5, and V 32 X 24.5 = 28. 

F* 














66 WEIGHING WITHOUT SCALES.—PAINTING. 


'W'eigbdng witlioiat Scales. 

To Ascertain "W eight of a Bar, Beam, etc.. Toy- .A-id. of 

a known AVeight. 

Operation. —Balance bar, etc., over a fulcrum, and note distance between 
it and end of its longest arm. Suspend a known weight from longest arm, 
and move bar, etc., upon fulcrum, so that bar with attached weight will be 
in equilibrio; subtract distance between the two positions of fulcrum from 
longest arm first obtained; multiply this remainder by weight suspended, 
divide product by distance between f ulcrums, and quotient will give weight. 

Example.— A piece of tapered timber 24 feet in length is balanced over a fulcrum 
when 13 feet from less end; but when the body of a man weighing 210 lbs. is sus- 
pended'from extreme of longest arm, the piece and weight are balanced when ful¬ 
crum is 12 feet from this end. What is weight of the timber? 

13 —12 = 1, and 13 — 1 —12 feet. Then 12 X 210-T-1 = 2520 lbs. 


PAINTING. 


1 pound of paint will cover about 4 square yards for a first coat and about 
6 yards for each additional coat. 

Proportions of Colors for ordinary Paints.—By- “Weight. 


Colors. 

White 

Lead. 

Lamp¬ 

black. 

Red 

Lead. 

Red 

Ochre. 

■d . 

U 03 

o» *r 
> u 

Spanish 
Brown. 

Colors. 

I White 

Lead. 

Lamp¬ 

black. 


IOO 







98 


Black. 

IOO 

_ 

_ 

_ 


Red. 

_ 

Green. 

25 

— 

— 

— 

75 

— 

Chocolate.. 

— 

4 




1 

■S a 

_d 

rs a 


a £ 

0 > q) 

A-* -1 

a> 0 

0 ) •- 
> u 

WM 

50 

So 

_ 

z 

— 

— 

— 

96 


These are the colors alone, to which boiled linseed oil, litharge, Japan varnish, 
and spirits turpentine are to be added according to the application of the paint. 

Lamp-black and litharge are ground separately with oil, then stirred into the 
load and oil. 

Thus for black paint: Lamp-black 25 parts, litharge 1, Japan varnish 1, boiled lin¬ 
seed oil 72, and spirits turpentine 1. 


Tar Baint.—Coal tar 9 gallons, slaked lime 13 lbs., turpentine or naphtha 2 
or 3 quarts. 


A Gallon of Paint will cover 

Superficial 

feet. 

A Gallon of Paint will cover 

Superficial 

feet. 

On stone or brick, about. 

190 to 225 

On well-painted surface or iron 

600 

On composite, etc., from. 

3 °° “ 375 

One gallon tar, first coat. 

90 

On wood, from. 

375 “ 525 

“ “ “ second coat... 

160 


Boiled. Oil—Raw linseed oil 91 parts, copperas 3, and litharge 6. 

Put litharge and copperas in a cloth bag and suspend in middle of a kettle. Boil 
oil four hours and a half over a slow fire, then let it stand and deposit the sediment. 


'White Baint. 

Inside work. Outside work. Inside work. 

White lead, in oil.. 80 . 80 Raw 7 oil. — 

Boiled oil . 14.5 . 9 Spirits turpentine. 8 

New wood work requires 1 lb. to square yard for three coats. 


Outside work. 
... 9 

... 4 


Coats for 100 Square Yards New White Pine. 


Inside. 

White 

lead. 

Raw 

oil. 

Turpen¬ 

tine. 

Drier. 

Outside. 

White 

lead. 

Raw 

oil. 

Boiled 

oil. 

Turpen¬ 

tine. 


Lbs. 

Pts. 

Pts. 

Lbs. 


Lbs. 

Pts. 

Pts. 

Pts. 

Priming.... 

l6 

— 

6 

•25 

Priming.... 

18.5 

2 

2 


2d coat. 

15 

3-5 

i -5 

•25 

2d and 3d ) 





3 d “ . 

13 

2-5 

i -5 

•25 

coats ) 

J 5 

2 

2 

•5 


.1 lb. of drier with priming and coating for outside. 




































































HYDROMETERS. 


67 


HYDROMETERS. 

U. S. Hydrometer (Tralle’s) ranges from o (water) to 100 (pure spirit); 
it has pot any subdivision or standard termed “Proof,” but 50, upon 
stem of instrument, at a temperature of 6o°, is basis upon which com¬ 
putations of duties are made. 

In connection with this instrument, a Table of Corrections, for differences in tem¬ 
perature of spirits, becomes necessary; and one is furnished by the Treasury De¬ 
partment, from which all computations of value of a spirit are made. 

Illustration. —A cask contains 100 gallons of whiskey at 70 0 , and hydrometer 
sinks in the spirit to 25 upon its stem. 

Then, by table, under 70° and opposite to 25, is 22.99, showing that there are 22.99 
gallons of pure spirit in the 100. 

Commercial Hydrometer (Gendar’s) has a “ Proof ” at 6o°, which is 
equal to 50 upon U. S. Instrument and its gradations, run up to 100 
with it, and down to 10 below proof, at o upon U. S. Instrument; or o 
of the Commercial Instrument is at 50 upon U. S. Instrument, from 
which it progresses numerically each way, each of its divisions being 
equal to two of latter. 

In testing spirits, Commercial standard of value is fixed at proof; 
hence any difference, whether higher or lower, is added or subtracted, 
as case may be, to or from value assigned to proof. 

A scale of Corrections for temperature being necessary, one is fur¬ 
nished with a Thermometer. 

Application of Thermometer. —Elevation of the mercury indicates correction to 
be added or subtracted, to or from indication upon stem of hydrometer. 

When elevation is above 6o°, subtract correction; and when below, add it. 

Illustration. —A hydrometer in a spirit indicates upon its stem 50 below proof, 
and thermometer indicates 4 above 6o° in appropriate column. 

Then 50 — 4 = 46 = strength below proof. 

To Compute Strength, of a Spirit, or Volume of its Pure 

Spirit, by Commercial Hydrometer, and. Convert it to 

Indication of a U. S. Hydrometer. 

When Spirit is above Proof. Rule.— Add 100 to indication, and divide sum by 2. 

When Spirit is below Proof Rule. — Subtract indication from 100, and divide 
remainder by 2. 

Example. — A spirit is n above proof by a Commercial Hydrometer; what pro¬ 
portion of pure spirit does it contain? 

11 —f-100 -4- 2 = 55.5 x>er cent. 

To Compute Strength, etc., by a U. S. Hydrometer. 

When Spirit is above Proof. Rule. —Multiply indication by 2, and subtract 100. 

When Spirit is below Proof. Rule. — Multiply indication by 2, and subtract it 
from 100. 

Example.—A spirit is 55.5; what is its per centage above proof? 

55-5 X 2 —100 = 11 per cent. 

Commercial practice of reducing indications of a hydrometer is as follows: 

Multiply number of gallons of spirit by per centage or number of degrees above 
or below proof, divide by 100, and quotient will give number of gallons to be added 
or subtracted, as case may be. 

Illustration.— 50 gallons of whiskey are n per cent, above proof. 

Then 50 X n -r-100 = 5.5, which added to 50 — 55.5 gallons. 



68 


HYGROMETER. 


HYGROMETER. 

Dew-point.—When air is gradually lowered in its temperature at a 
constant pressure, its density increases, and ratio of increase is sensibly 
same for the vapor as for the air with which it is combined, until a 'point is 
reached at which the density of the vapor becomes equal to the maximum 
density corresponding to the temperature. 

This temperature is termed cletv-point of given mass, and any further re¬ 
duction of it will induce the condensation of a portion of the vapor in form 
of dew, rain, snow, or frost, according as temperature of surface is above or 
below freezing point. 

Mason’s or like Hygrometer. 

To Ascertain Dew-point. 

Rule. —Subtract absolute dryness from temperature of air, and remainder is 
dew-point. 

Example.— Temperature of air 57 0 , and absolute dryness 7 0 . 

Hence 57 0 — 7 0 — 50 0 dew-point. 

To Ascertain Absolute Existing Dryness. 

Rule.— Subtract temperature of wet bulb from temperature of air, as indicated 
by a dry bulb, add excess of dryness from following table, multiply sum by 2, and 
product will give absolute dryness in degrees. 

Example.— Temperature of air 57 0 , wet bulb 54°- 

Then 57 0 — 54 0 == 3 0 , and 3 0 +. 5 0 (from table) X 2 = 7 0 absolute dryness. 


Observedl Excess of 
Dryness. | Dryness. 

Observed 

Dryness. 

Excess of 
Dryness. 

Observed;Excess of 
Dryness. jDryn ess. 

Observed 

Dryness. 

Excess of 
Dryness. 

Observed 

Dryness. 

Excess of 
Dryness. 

O 

O 

O 

O 

O 

0 

O 

O 

O 

O 

•5 

.083 

5 

•833 

9-5 

1-583 

T 4 

8-333 

18.5 

3 -o 83 

I 

.166 

5-5 

• 9 i6 5 

IO 

1.666 

* 4-5 

2.4165 

J 9 

3.166 

i*5 

•2495 

6 

I 

10.5 

1-7495 

*5 

2-5 

* 9-5 

3-2495 

2 

•333 

6-5 

1.083 

II 

1-833 

* 5-5 

2-583 

20 

3-333 

2-5 

• 4 i6 5 

7 

1.166 

**•5 

1-9*65 

l6 

2.666 

20.5 

3 - 4*65 

3 

•5 

7-5 

1.2495 

12 

2 

16.5 

2-7495 

21 

3-5 

3-5 

•583 

8 

i -333 

12.5 

2.083 

*7 

2-833 

21.5 

3-583 

4 

.666 

8-5 

1.4165 

*3 

2.166 

* 7-5 

2.9165 

22 

3.666 

4-5 

•7495 

9 

i -5 

13-5 

2.2495 

18 

3 

22.5 

3-7495 


To Compute "V”olrirne of Yapor in Atmosphere. 

By a Hygrometer. 

When temperature of atmosphere in shade, and of dew-point are given. —If temper¬ 
ature of air and dew-point correspond, which is the case when both thermometers 
are alike, and air consequently saturated with moisture, then in table* opposite to 
temperature will be found corresponding weight of a cube foot of vapor in grains. 

Illustration.— Assume temperature of air and dew-point 70 0 . Then opposite 
temperature weight of a cube foot of vapor^r 8.392 grains. 

But if temperature of air is different from dew-point, a correction is necessary to 
obtain exact weight. 

Illustration.— Assume dew-point 70 0 as before, but temperature of air in shade 
8o°, then the vapor has suffered an expansion due to an excess of io°, which re¬ 
quires a correction. 

In table of corrections for io° is 1.0208. Then divide 8.392 grains at dew-point— 
viz., 70 0 by correction corresponding to degrees of absolute dryness—viz., io°. 

— ‘ 39* 1 — 8.221 grains of existing vapor, which, subtracted from weight of vapor 

corresponding to temperature of 8o°, will give number of grains required for satu¬ 
ration at that temperature. 

n.333 grains at temperature of 8o° —8.221 contained in the air = 3.112 required 
for saturation. 


* For table, see Mason’s as published by Pike & Sons, New York, and compared with Sir John 
Le.lie’s and Professor Daniel’s. 



























HYGROMETER.—SUN-DIAL.—CHAINING. 


69 

To ascertain relations of these conditions on natural scale of humidity (complete 
saturation being 1000), divide weight of vapor at dew-point by weight at tempera¬ 
ture of air, and quotient will give degrees of saturation. 

Illustration.— Dew-point = 70 0 , weight = 8.392. 

Then 8.392-1-11.333 (at 8o°) = .7405 degrees of humidity; saturation = 1000. 

To Compute Weight of Vapor in a Cube Foot of Air. 

See Pressures, Temperatures, Volumes, and Density of Steam, p. 708. 

Thus, Required weight of vapor in a cube foot of saturated air at 212 0 . 

At a temperature of 212 0 density or weight of 1 cube foot of air = .038 lb. 

If density is required for any temperatures not in table, see rule, p. 706. 

Humidity. —Condition of air in respect to its moisture involves amount of 
vapor present in air and ratio of it to amount which would saturate it at its 
temperature, and it is this element which is denoted by term humidity , and 
it is expressed as a per centage; thus, if weight of vapor present is .7 of that 
required for saturation, the humidity is 70. 

Dry Air is air, humidity of which is below zero, but it is customary to 
term it dry when its humidity is below the average proportion. 

Note.— Air in a highly heated space contains as much vapor (when weight of it 
is equal) as a like volume of external air, but it is drier as its capacity for vapor 
is greater. 

SUN - DIAL. 

To Set a Sum-dial. 

Set column on which dial is to be placed perpendicular to horizon. Ascertain by 
spirit level that upper surface is perfectly horizontal; screw on plate loosely by means 
of centre screw, and bring gnomon as nearly as practicable to its proper direction. 

On a bright day set dial at 9 a.m. and 3 p.m. exactly, with a correctly regulated 
watch; observe difference between them, and correct dial to half difference. Pro¬ 
ceed in same manner till watch and dial are found to agree perfectly. Then fix 
plate firmly in that situation, and dial will be correctly set. 

This is obvious; for, if there were any defects, the Sun’s shadow would not agree 
with time indicated by watch, both before and after he passed meridian. Take 
care, however, to allow for equation of time, or you may set dial wrong. Best day 
in the year to set a dial is 15th of June, as there is no equation to allow for, and no 
error can arise from change of declination. A dial may be set without a watch, by 
drawing a circle around centre, and marking spot where top of shadow of an upright 
pin or piece of wire, placed in centre, just touches circle in a.m., and again in p.m. 
A line should be drawm from one spot to the other, and bisected exactly; then a 
line drawn from centre of dial through that bisection will be a true meridian line, 
on which the XII hours’ mark should be set. 


CHAINING OVER AN ELEVATION. 


I C = L, and 0 = cos. angle. 

I representing length of line chained , C cos. angle of elevation with horizon , 
and L length of line reduced to horizontal. 

Illustration.— Length of an elevation at an angle of 30 0 17' is 100 feet; what is 
horizontal distance ? 

By Table of Cosines, 30 0 17' = .86334. Hence,. 100 X .863 54 = 86.354/^. 

To set out a ILiglit Angle with. a Chain, Tape-line, etc. 


Take 40 links on chain or feet of line for base, 30 links or feet for perpendicular, 
and 50 for hypothenuse, or in this ratio for any length or distance. 

Useful Numbers in Surveying. 


For Converting 

Multiplier. 

Converse. 

Feet into links.. 

I- 5 I 5 

.66 

Yards “ “ .. 

4-545 

.22 


For Converting 

Multiplier. 

Converse. 

Square feet into acres.. 
Square yards “ “ .. 

.OOO 022 9 

. 000 206 6 

43 56o 

4 840 












70 


CHRONOLOGY, 


CHRONOLOGY. 

Solar day is measured by rotation of the Earth upon its axis with respect 
to the Sun. 

Motion of the Earth, on account of ellipticity of its orbit, and of perturba¬ 
tions produced by the planets, is subject to an acceleration and retardation. 
To correct this fluctuation, timepieces are adjusted to an average or mean 
solar day (mean time), which is divided into hours, minutes, and seconds. 

In Civil computations day commences at midnight, or A.M., and is divided into 
two portions of 12 hours each. 

In Astronomical computations and in Nautical time day commences at M., or 
12 hours later than the civil day, and it is counted throughout the 24 hours. 

Solar Tear , termed also Equinoctial, Tropical, Civil, or Calendar Tear, is the 
time in which the Sun returns from one Vernal Equinox to another; and its average 
time, termed a Mean Solar Tear, is 365.242218 solar days, or 365 days, 5 hours, 48 
minutes , and 47.6 seconds. 

Tear is divided into 12 Calendar months, varying from 28 to 31 days. 

Mean Lunar Month , or lunation of the Moon, is 29 days, 12 hours, 44 minutes, 
2 seconds, and 5.24 thirds.* 

Bissextile or Leap Tear consists of 366 days; correction of one year in four is 
termed Julian ; hence a mean Julian year is 365.25 days. 

In year 1582 error of Julian computation of a year had amounted to a period of 
10 days, which, by order of Pope Gregory VIII., was suppressed in the Calendar, and 
5th of October reckoned as 15th. 

Error of Julian computation, .007 76 days, is about 1 day in 128.79 y ears , and adop¬ 
tion of this period as a basis of intercalation is termed Gregorian Calendar, or New 
Style ,f Julian Calendar being termed Old Style. 

Error of Gregorian year (365.2425 days) amounts to 1 day in 3571.4286 years. 

New Style was adopted in England in 1752 by reckoning 3d of September as 14th. 

By an English law r , the years 1900, 2100, 2200, etc., and any other 100th year, ex¬ 
cepting only every 400th year, commencing at 2000, are not to be reckoned bissex¬ 
tile years. 

Dominical or Sunday Letter is one of the first seven letters of alphabet, and is 
used for purpose of determining day of week corresponding to any given date. In 
Ecclesiastical Calendar letter A is placed opposite to 1st day of year, January 1st; 
B to second; and so on through the seven letters; then the letter which falls oppo¬ 
site to first Sunday in year will also fall opposite to every following Sunday in that 
year. See table, p. 73. 

Note. —In bissextile years two Dominical letters are used, one before and the other 
after the intercalary day. 

In Ecclesiastical Tear the intercalary day is reckoned upon 24th of February; 
hence 24th and 25th days are denoted by same letter, the dominical letter being set 
back one place. 

In Civil Tear the intercalary day is added at end of February, the change of letter 
taking place at 1st of March. 

Dominical Cycle is a period of 400 years, when the same order of dominical letters 
and days of the week will return. 

Cycle of the Sun, or Sunday Cycle, is the 28 years before same order of Dominical 
letters return to same days of month, and it is considered as having commenced 9 
years before the era of Julian Calendar. 

To Compute Cycle of tlae Sun. 

Rule— Add 9 to given year; divide sum by 28; quotient is number of cycles that 
have elapsed, and remainder is number or years of cycle. 

Note.— -Use of this computation is determination of dominical letter for any given 
year of Julian Calendar for each of the 28 years of a cycle. 


* Ferguson. 


t Now adopted in every Christian country except Russia and Greece. 




CHRONOLOGY. 


7 1 


By adoption of Gregorian Calendar , order of the letters is necessarily interrupted 
by suppression of the century bissextile years in 1900, 2100, etc., and a table of Do¬ 
minical letters must necessarily be reconstructed for following century. 

Lunar Cycle , or Golden Number , is a period of 19 years, after which the new 
moons fall on same days of the month of Julian year, within 1.5 hours. 

Year of birth of Jesus Christ is reckoned first of the Lunar Cycle. 

To Compute Dumar Cycle, or Gf olden. Number. 

Rule. —Add 1 to given year; divide sum by 19, and remainder is Golden Number. 

Note. —If o remain, it is 19. 

Example. —What is Golden Number for 1879? 

18794- 1 -f-19 = 98, and remainder = 18 = Golden Number. 

Epact for any year is a number designed to represent age of the moon on 1st day 
of January of that year. See table, p. 73. 

To Compute tlie Roman Indiction. 

Rule.— -Add 3 to given year; divide sum by 15, and remainder is Indiction. 

Note. —If o remain, Indiction is 15. 

Number of Direction is the number of days that Easter-day occurs after 21st of 
March. 

Easter-day is first Sunday after first full moon which occurs upon or next after 
21st of March; and if full moon occurs upon a Sunday, then Easter-day is Sunday 
after, and it is ascertained by adding number of direction to 21st of March. It is 
therefore March N 4 ~ 21, or April N —10. 

Illustration. — If Number of Direction is 19, then for March, 194-21 = 40, and 
40 — 31 = 9 = 9 th of April; 

again for April, 19 —10 = 9 = 9 th of April. 

Note. —Moon upon which Easter immediately depends is termed Paschal Moon. 

Full Moon is 14th day of moon, that is, 13 days itfter preceding day of new moon. 

Days of tlie Roman Calendar. 

Calends were the first 6 days of a month, Nones following 9 days, and Ides remain¬ 
ing davs. 

In March, May, July, and October, Ides fell upon 15th and Nones began upon 7th. 
In other months Ides commenced upon 13th and Nones upon 5th. 

For Roman Indiction and Julian Period see p. 26. 


B Clxvonology. 

4004. Creation of World (according to Julius Africanus, Sept. 1, 5508; Samaritan 
Pentateuch, 4700;'Septuagint, 5872; Josephus, 4658; Talmudists, 5344; Sca- 
liger, 3950; Petavius, 3984; Hales, 5411). 


2348 Deluge (according to Hales, 3154). 

2247. Bricks made and Cement first used. 
Tower of Babel finished. 

2203. Chinese Monarchy. 

2090. First Egyptian Pyramid and Canal. 

1920. Gold and Silver Money first intro¬ 
duced. 

1891. Letters first used in Egypt. 

1822. Memnon invents the Egyptian Al¬ 
phabet. 

1490. Crockery introduced. 

1240. Axe, Wedge, Wimble, Lever, Masts 
and Sails invented by Daedalus 
of Athens. 

1180. Troy destroyed. 

1120. Mariner’s Compass discovered in 
China. 

753. Foundation of Rome. 

640. Thales asserts Earth to be spherical. 

605. Geometry, Maps, etc., first intro¬ 
duced. 


576. Money coined at Rome. 

562. First Comedy performed at Athens. 

480. First recorded Map by Aristagoras. 

420. First Theatre built at Athens. 

336. Calippus calculates the revolution of 
Eclipses. 

320. Aristotle writes first work on Me¬ 
chanics. 

310. Aqueducts and Baths introduced in 
Rome. 

306. First Light-house in Alexandria. 

289. First Sun-dial. 

267. Ptolemy constructs a Canal from the 
Nile to the Red Sea. 

224. Archimedes demonstrates the Prop¬ 
erties of Mechanical Powers and 
the Art of measuring Surfaces, Sol¬ 
ids, and Sections. 

219. Hannibal crossed the Alps. 

219. Surveying first introduced. 

202. Printing introduced in China. 



72 


CHRONOLOGY. 


B. C. 

198. Books with leaves of vellum first 
introduced by Attalus. 

170. Paper invented in China. 

168. An eclipse of the Moon which was 
predicted by Q. S. Gallus. 

162. Hipparchus locates the first degree 
of Longitude and the Latitude at 
Ferro. 


B.C. 

159. Clepsydra, or Water-clock, invent¬ 
ed. 

146. Carthage destroyed. 

70. First Water-mill described. 

51. Caesar invaded Britain. 

45. First Julian Year by Caesar. 

8. Augustus corrects the Calendar. 


A. D. 

69. Destruction of Jerusalem. 

79. Destruction of Herculaneum and 
Pompeii. 

214. Grist-mills introduced. 

622. Year of Hegira, commencing 16th 
July; Glazed windows first intro¬ 
duced into England in this cent’y. 

667. Glass discovered. 

670. Stonebuildiugs introduced intoEng- 
land. 

842. Lands first enclosed in England. 

933. Printing said to have been invented 
by the Chinese. 

991. Arabic Numerals introduced. 

1066. Battle of Hastings. 

mi. Mariner’s Compass discovered. 

1180. Destruction of Troy. Mariner’s 
Compass introduced in Europe. 

1368. Chimneys first introduced into 
Rome from Padua. 

1383. Cannon introduced. 

1390. Woollens first made. 

1434. Printing invented at May^ice. 

1460. Wood-engraving invented and First 
Almanac. 

1471. Printing in England by Caxton. 

1477. Watches first introduced at Nurem¬ 
berg. 

1492. America discovered. 

1497. Vasco de Gama discovers passage 
to India. 

1500. Variation of Mariner’s Compass ob¬ 
served. 

1522. F. de Magellan circumnavigates the 
Globe. 

1530. Incas conquered by Pizarro. 

1545. Needles first introduced. 

1586. Potato introduced into Ireland from 
America. 

1590. Telescopes invented by Jansen and 
used in London in 1608. 

1616. Tobacco first introduced into Vir¬ 
ginia. 

1620. Thermometer invented by Drebel. 

1627. Barometer invented. 

1629. First Printing-press in America. 

1639. First Printing-office in America at 
Cambridge. 

1647. Otto Van Gueriche constructed first 
electric machine. 

1650. Railroads with wooden rails intro¬ 
duced near Newcastle. 

1652. First Newspaper Advertisement. 

1704. First Newspaper in America. 

1705. Blankets first made at Bristol, Eng¬ 

land. 

* Witnessed 


A.D. 

1752. Benjamin Franklin demonstrated 
identity of the electric spark and 
lightning, by aid of a kite. 

1752. New Style, introduced into Britain; 

Sept. *3 reckoned Sept. 14. 

1753. First Steam-engine in America. 
1769. James Watt—First design and pat¬ 
ent of a Steam-engine with sepa¬ 
rate vessel of condensation. 

1772. Oliver Evans—Designed the Non¬ 
condensing Engine. 1792. Ap¬ 
plied for a patent for it. 1801. 
Constructed and operated it. 

1774. Spinning-jenny invented by Robert 
Arkwright. 

1776. Iron Railway at Sheffield, England. 
1783. First Balloon ascension, and Vessel’s 
bottoms coppered. 

1790. Water-lines first introduced in mod¬ 
els of Vessels in the U. S. 

1797. John Fitch—Propelled a yawl-boat 
by application of Steam to side- 
wheels, and also to a screw-propel¬ 
ler, upon Collect Pond, New York. 
1807. Robert Fulton — First Passenger 
Steamboat. 

1824. Compound marine steam-engines 

first introduced by James P. Al¬ 
lan, New York. 

1825. Introduction of steam towing by 

Mowatt, Bros. & Co., of New York, 
by steam-boat “ Henry Eckford,” 
New York to Albany.* 

1826. Voltaic Battery discovered by Alex. 

Volta, and First Horse-railroad. 

1827. First Railroad in U. S., from Quincy 

to Neponset. 

1829. First Lucifer Match and first Loco¬ 

motive in America. 

1830. Liverpool and Manchester Railroad 

opened. First Steel Pen and first 
Iron Steamer. 

1832. S. F. B. Morse invents the Magnetic 
Telegraph. 

1836. Robert L. Stevens first burned An¬ 
thracite Coal in furnace of boiler 
of steamboat “Passaic.” 

1840. First steam-boiler constructed for 
burning Anthracite Coal in steam¬ 
boat “North America,” N. Y. 
1844. Telegraph line from Washington to 
Baltimore, Md. , 

1846. First complete Sewing-machine. 

Elias Howe, inventor. 

1866. Submarine Telegraph laid from 
Valencia to Newfoundland, N.S. 

by author. 




CHRONOLOGY. 


73 


Dates of Day of Week, corresponding to Day deter¬ 
mined by following Table. 


February, 

March, 

November. 

February,* 

August. 

May. 

January, 

October. 

January,* 

April, 

July. 

September, 

December. 

June. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

n 

12 

13 

14 

15 

16 

17 

18 

r 9 

20 

21 

22 

23 

24 

25 

26 

27 

28 

2 9 

30 

3 1 






Thus, if Monday is the day determined by the year given, the following dates are 
the Mondays in that year. 


Epacts, X)omiriieal Letters, and. an Almanac, from 
1SOO to 1901. 

Use op Table. —To ascertain day of the week on which any given day of the 
month falls in any year from 1800 to 1901. 


Illustration. —The great fire occurred in New York on 16th of December, 1835; 
what was day of the week ? 

Opposite 1835 is Sunday; and by following table, under December, it is ascertained 
that 13th was Sunday; consequently , 16 th was Wednesday. 


Years. 

Days. 

Dom. 

Let¬ 

ters. 

CS 

W ’ 

Years. 

Days. 

Dom. 

Let¬ 

ters. 

| Epact. 

Years. 

Days. 

Dom. 

Let¬ 

ters. 

[ Epact. 

1800 

Saturday. 

E 

4 

1834 

Saturday. 

E 

20 

1868 

Sunday.* 

ED 

6 

1801 

Sunday. 

D 

15 

1835 

Sunday. 

D 

1 

1869 

Monday. 

c 

17 

1802 

Monday. 

C 

26 

1836 

Tuesday. * 

CB 

12 

1870 

Tuesday. 

B 

28 

1803 

Tuesday. 

B 

7 

1837 

Wednesd. 

A 

23 

1871 

Wednesd. 

A 

9 

1804 

Thursda}\* 

AG 

18 

1838 

Thursday. 

G 

4 

1872 

Friday.* 

GF 

20 

1805 

Friday. 

F 

29 

I §39 

Friday. 

F 

i5 

i8 73 

Saturday. 

E 

1 

1806 

Saturday. 

E 

11 

1840 

Sunday.* 

ED 

26 

1874 

Sunday. 

D 

12 

1807 

Sunday. 

D 

22 

1841 

Monday. 

C 

7 

1875 

Monday. 

C 

23 

1808 

Tuesday.* 

CB 

3 

1842 

Tuesday. 

B 

18 

1876 

Wednesd.* 

BA 

4 

1809 

Wednesd. 

A 

*4 

!843 

Wednesd. 

A 

29 

1877 

Thursday. 

G 

15 

1810 

Thursday. 

G 

25 

1844 

Friday.* 

GF 

11 

1878 

Friday. 

F 

26 

1811 

Friday. 

F 

6 

i8 45 

Saturday. 

E 

22 

1879 

Saturday. 

E 

7 

1812 

Sunday.* 

ED 

17 

1846 

Sunday. 

D 

3 

1880 

Monday. * 

DC 

18 

1813 

Monday. 

C 

28 

1847 

Monday. 

C 

14 

1881 

Tuesday. 

B 

29 

1814 

Tuesday. 

B 

9 

1848 

Wednesd.* 

BA 

25 

1882 

Wednesd. 

A 

11 

1815 

Wednesd. 

A 

20 

1849 

Thursday. 

G 

6 

1883 

Thursday. 

G 

22 

1816 

Friday.* 

GF 

1 

1850 

Friday. 

F 

17 

1884 

Saturday.* 

FE 

3 

1817 

Saturday. 

E 

12 

1851 

Saturday. 

E 

28 

1885 

Sunday. 

D 

14 

1818 

Sunday. 

D 

23 

1852 

Monday.* 

DC 

9 

1886 

Monday. 

C 

25 

1819 

Monday. 

C 

4 

1853 

Tuesday. 

B 

20 

1887 

Tuesday. 

B 

6 

1820 

Wednesd..* 

BA 

15 

1854 

Wednesd. 

A 

1 

1888 

Thursday. * 

AG 

17 

1821 

Thursday. 

G 

26 

1855 

Thursday. 

G 

12 

1889 

Friday. 

F 

28 

1822 

Friday. 

F 

7 

1856 

Saturday.* 

FE 

23 

1890 

Saturday. 

E 

9 

1823 

Saturday. 

E 

18 

!857 

Sunday. 

D 

4 

1891 

Sunday. 

D 

20 

1824 

Monday.* 

DC 

29 

1858 

Monday. 

C 

IS 

1892 

Tuesday. * 

CB 

1 

1825 

Tuesday. 

B 

11 

1859 

Tuesday. 

B 

20 

1893 

Wednesd. 

A 

12 

1826 

Wednesd. 

A 

22 

i860 

Thursday.* 

AG 

7 

1894 

Thursday. 

G 

23 

1827 

Thursday. 

G 

3 

1861 

Friday. 

F 

18 

1895 

Friday. 

F 

4 

1828 

Saturday.* 

FE 

*4 

1862 

Saturday. 

E 

29 

1896 

Sunday. * 

ED 

15 

1829 

Sunday. 

D 

25 

1863 

Sunday. 

D 

11 

1897 

Monday. 

C 

26 

1830 

Monday. 

C 

6 

1864 

Tuesday.* 

CB 

22 

1898 

Tuesday. 

B 

7 

1831 

Tuesday. 

B 

17 

1865 

Wednesd. 

A 

3 

i8 99 

Wednesd. 

A 

18 

1832 

Thursday.* 

AG 

28 

1866 

Thursday. 

G 

14 

1900 

Thursday. 

G 

29 

1833 

Friday. 

F 

9 

1867 

Friday. 

F 

25 1 

1901 

Friday. 

F 

11 


* In leap-year, January and February must be taken in columns marked *. 

G 








































74 CHRONOLOGY.-MOON’S AGE.—TIDES. 

To -Ascertain. YTear or Years of Coincidences of a given 
Day of tlie Week with. a given Day of a ATontli. 

Look in preceding table and ascertain day of week opposite to year .of 
occurrence, and every year in which same day is given will be year of coin¬ 
cidences required. 

Illustration.— If a child was born on Saturday, 19th Sept. 1829, when could and 
can his birthdays be celebrated, that occurred or are to occur on same day of week 
and date of month ? 

Opposite to 1829 is Sunday, and in preceding table the Sundays for September of 
that year were 6th, 13th, 20th; hence, if 20th was Sunday, the 19th was Saturday. 

Hence, every year in table opposite to which is Sunday are the years of the coin¬ 
cidence required, as 1835,1840,1846, 1857,1863,1868,1874, 1885, etc. 


moon’s AGE. 

To Compute Moon’s Age. 

Rule. —To day of month add Epact and Number of month; from sum 
subtract 29 days, 12 hours, 44 min. and 2 sec., as often as sum exceeds this 
period, and result will give Moon’s age approximately at 6 o’clock a.m. in 
United States, east of Mississippi River. 




IN umbers 

of 

tlie HVIontlxs. 





d. h. 


d. 

h. 


d. h. 


d. 

h. 

January... 

5 

April. 

I 

21 

July. 

• 4 7 

October.... 

•• 7 

l6 

February.. 


May. 

2 

8 

August. 

• 518 

November.. 

.. 9 

4 

March. 

9 

June. 

3 

19 

September. 

• 7 5 

December.. 

.. 9 

15 


Example. —Required age of Moon on 25th February, 1877 ? 

Given day 25 -f- epact 15 -)- number of month 1.22 = 41 d. 22 h. — 29 d. 12 h. 44 m. 
2 sec. == 12 d. 9 h. 15 min. 58 sec. 

In Leap-years add 1 day to result after 28th February. 

To Compute Age of IVIooii at NLeaix JSTooix at any otlier 
Location tlxan tliat Griven. 

Rule. —Ascertain age, and add or subtract difference of longitude or time, 
according as place may be West or East of it, to or from time given. 

Or, when time of new Moon is ascertained for a location, and it is required 
to ascertain it for any other, add difference of longitude or time of the place, 
if East, and subtract it if it is West of it. 

Moon's Southing, as usually given in United States Almanacs, both Civil and Nau¬ 
tical, is computed for Washington. 


To Compute Time of High-water by Aid. of American 
jN auitical Almanac. 

Rule. —Ascertain time of transit of Moon for Greenwich, preceding time 
of the high-water required. 

For any other location (west of Greenwich), multiply the time in column 
“ diff. for one hour ” by longitude of location west of Greenwich, expressed 
in hours, and add product to time of transit. 

Note.— It is frequently necessary to take the transit for preceding astronomical 
day, as the latter does not end until noon of day under computation. 

Example.— -Required time of high-water at New York on 25th of August, 1864. 

Longitude of New York from Greenwich = 4 h. 56 m. 1.65 sec., which, multiplied 
by 2.17 min., the difference for 1 hour = 10.71 min. for correction to be added to time 
of transit, to obtain time of transit at New York. 
















TIDES.-MOON’S SOUTHING. 



Time of transit, 18 h. 38.8 to.; then 18 h. 38.8 m. -|-10.71 to. = 18 hours 49.51 min. 
Time of transit at New York, 24 d. 18 h. 50 m. 

Establishment of the Port, 8 13 

25 d. 3 h. 3 m. z= time of high-water. 

Note. — Time of 25th at 3 h. 3 m. Astronomical computation = 25th at 3 h. 3 m. 
p.m. Civil Time. 

To Compute Time of High-water at Wiill and. Change 

of Moon. 

Time of High-water and Age of Moon on any Day being given. 

Rule.—N ote age of Moon, and opposite to it, in last column of following 
table, take time, which subtract from time of high-water at this age of 
Moon, added to 12 h. 26 m., or 24 h. 52 m ., as case may require (when sum to 
be subtracted is greatest), and remainder is time required. 

Example.— What is time of high-water at full and change of Moon at New York? 

Time of high-water at Governor’s Island on 25th of Jan. 1864, was 9 h. 20 m. a.m. 
civil time. Age of Moon at 12 m. on that day was 16 d. 8 h. 59 m. 

Opposite to 16 days, in following table, is 13 h. 28 to., and difference between 16 d. 
and 16 d. 12 h. — (16.5 —16, or 13.53 —13-28) is 25 m. ; hence, if 12 h. —25 m. , 16 d. 
8 h. 59 m. —16 d. — 8 h. 59 m. -- 18.71 or 19 m., which, added to 13 h. 28 m. = 13 h. 
47 m. 

Then 9 h. 20 m. -f-12 h. 26 to. (as sum to be subtracted is greater than time) —13 h. 
47 m. — 21 h. 46 to. —13 h. 47 m. = 7 h. 59 to. 

This is a difference of but 13 minutes from Establishment of Port. 

Time after apparent Noon "before UNfoon next 
passes Meridian, .Age at iSToon "being given. 

(S. H Wright, A. 31 ., Ph.D.) 


Age of 
Moon. 

Moon at 
Meridian. 

Age of 
Moon. 

Moon at 
Meridian. 

Age of 
Moon. 

Moon at 
Meridian. 

Age of 
Moon. 

Moon at 
Meridian. 

Age of 
Moon. 

Moon at 
Meridian. 

Days. 

H. M. 

P. M. 

Days. 

H. M, 

P. M. 

Days. 

H. M. 

P. M. 

Days. 

H. M. 

A. M. 

Days. 

H. M. 

A. M. 

.O 

O 

6 

5 

3 

12 

10 6 

18 

IS 8 

24 

20 II 

•5 

25 

6-5 

5 28 

12.5 

10 31 

18.5 

15 34 

2 4-5 

20 37 

I 

50 

7 

5 53 

13 

10 56 

*9 

15 59 

25 

21 2 

i -5 

1 16 

7-5 

6 19 

13-5 

II 21 

19-5 

16 24 

2 5-5 

21 27 

2 

x 41 

8 

6 44 

14 

ii 47 

A. M. 

20 

16 49 

26 

21 52 

2.5 

2 6 

8-5 

7 

9 

i 4-5 

12 12 

20.5 

17 15 

26.5 

22 17 

3 

2 31 

9 

7 34 

i 5 

12 37 

21 

17 40 

27 

22 43 

3-5 

2 57 

9-5 

7 59 

i 5-5 

13 2 

21.5 

18 5 

2 7-5 

23 8 

4 

3 22 

IO 

8 25 

16 

13 28 

22 

18 30 

28 

23 33 

4-5 

3 47 

10.5 

8 50 

16.5 

13 53 

22.5 

18 56 

28.5 

23 5 8 

5 

4 12 

II 

9 *5 

W 

14 28 

2 3 

19 21 

29 

24 24 

5-5 

4 3 8 

n -5 

9 4 ° 

I 7-5 

14 43 

2 3-5 

19 46 

2 9-5 

24 48 


Tidal Phenomena. 

The elevation of a tidal wave towards the Moon slightly exceeds that of the op¬ 
posite one, and the intensity of it diminishes from Equator to the Poles. 

The Sun by its action twice elevates and depresses the sea every day, following 
the action of the Moon, but with less effect. 

Spring Tides arise from the combined action of the Sun and Moon when they are 
on both sides of the Earth. 

Neap Tides are the consequence of the divided action of the Sun and Moon, when 
they are on opposite sides of the Earth, and the greatest elevations and depressions 
do not occur until the 2d or 3d day after a full or a new Moon. 

When Sun and Moon are in conjunction, and the time is near to the Equinoxes, 
the tides are fullest. The mean effect of the Moon on the tidal wave is 4.5 times 
that of the Sun. If, therefore, the Moon caused a tide of 6 feet, the Sun will cause 
one of 1.33 feet; hence a spring tide will be 7.33 feet, and a neap tide 4.67 feet. 

Particular locations as to contour of shores, straits, capes, and rivers, lengths and 
depths of channels, shoals, etc., disturb these general rules. 






















76 


LATITUDE AND LONGITUDE, 


LATITUDE AND LONGITUDE. 

Latitiade and Longitude of ^Principal Locations 
and Observatories. 

Compiled from Records of U. S. Coast and Geodetic Survey and Topograph¬ 
ical Engineer Corps , Imperial Gazetteer , and Bowditch's Navigator. 

Longitude computed from Meridian of Greenwich. 

A., represents Academy; Az., Azimuth ; A. S., Astronomical Station; C., College ; 
Cap., Capitol; Ch., Church; C. H., City Hall; C. S., Coast Survey; Ct., Court-house; 
Cy., Chimney; F. S., Flagstaff; G. S., Geodetic Station; Hos., Hospital; L. Light¬ 
house ; Ohs..Observatory; S.H., State-house; Sp. Spire; Sq., Square; S.S., Signal 
Station ; T., Telegraph ; T. H., Town Hall; U., University; Un., Union ; B., Baptist; 
Con., Congregational; E., Episcopal; I’., Presby. ; and M.Ch., Math. Churches. 


Location. 

Latitude. 

Longitude. 

Location. 

Latitude. 

| Longitude. 

NORTH AND SOUTH 

N. 

w. 

NORTH AND SOUTH 

N. 

w. 

AMERICA. 


O / 

n t " 

AMERICA. 

n f 


Acapulco. 

Mex. 

i"6 50 19 

99 49 9 

Canandaigua_N.Y. 

42 54 9 

77 17 

Albany, P.Ch_ 

N. Y. 

42 39 3 

73 45 24 

Cape Ann, S. L. .Mass. 

42 38 11 

70 34 10 

Ann Arbor. 

Mich. 

42 16 48 

83 43 3 

Cape Breton.Va. 

45 57 

59 48 5 

Annapolis. 

. Md. 

38 58 42 

76 29 6 

Cape Canaveral.. .Fla. 

28 27 30 

80 33 

Apalachicola, F.S. Fla. 

29 43 30 

84 59 

Cape Cod, L.P.L...Ms. 

42 2 

70 9 48 

Astoria, F.S. 

..Or. 

46 11 19 

123 49 42 

Cape Fear.N.C. 

33 48 

77 57 

Atlanta, C. H_ 

..Ga. 

33 44 57 

84 23 22 

Cape Flattery, L..W.T. 

48 23 15 

124 43 54 

Auburn. 

N.Y. 

42 5.5 

76 28 

Cape Florida, L... Fla. 

25 39 54 

80 9 2 

Augusta. 

..Ga. 

33 28 

81 54 

Cape Hancock, Colo. R. 

46 16 35 

124 1 45 

Augusta, B.Ch.. 

. Me. 

44 18 52 

69 46 37 

Cape Hatteras, L., N. C. 

35 i 5 2 

75 30 54 

Austin. 

.Tex. 

30 16 21 

97 44 12 

Cape Henlopen, L., Del. 

38 46 6 

75 4 7 

Balize. 

.. La. 

29 8 5 

89 1 4 

Cape Henry, L.... Ya. 

36 55 30 

76 0 2 

Baltimore, Mon’t 

. Md. 

39 17 48 

76 36 59 

Cape Horn, S. Pt., Her- 

S. 


Bangor,Tho's Hill. Me. 

44 48 23 

68 46 59 

mit’s Island. 

55 59 

67 16 

Barbadoes, S.Pt. 

. W.I. 

13 3 

59 37 


N. 


Barnegat, L. 

.N. J. 

39 46 

74 6 

Cape May, L.N. J. 

38 55 48 

74 57 18 

Bath, W.S.Ch... 

..Me. 

43 54 55 

69 49 

Cape Race.N. S. 

46 39 24 

53 4 3 

Baton Rouge... 


30 26 

91 18 

Cape Sable.N.S. 

43 24 

65 36 

Beaufort, Ct. 

.N.C. 

34 43 5 

76 39 48 

Cape Sable, C.S...Fla. 

25 6 53 

81 15 

Beaufort, E.Ch.. 

.S.C. 

32 26 2 

80 40 27 


S. 


Belfast, M.Ch... 

.. Me. 

44 25 29 

69 19 

Cape St. Roque, Brazil 

5 28 

35 17 

Benicia, Ch. 

. Cal. 

38 3 5 

122 Q 23 


N. 


Benington. 

..Yt. 

42 40 

73 18 

Carthagena.N.G. 

10 26 

75 38 

Bismarck. S. S .. 

.Neb. 

46 48 

100 38 

Castine.Me. 

44 22 30 

68 45 

Boston, L. 

Mass. 

42 19 6 

7 ° 53 6 

Cedar Keys, Depot Isl. 

29 7 3 ° 

83 2 45 

Boston, S. H. 

(1 

42 21 30 

71 3 3 ° 

Chagres.N.G. 

9 20 

80 1 21 

Brazos Santiago. 

.Tex. 

26 6 

97 12 

Charleston, C. Ch., S.C. 

32 46 44 

79 55 39 

Bridgeport. 

Conn. 

41 10 30 

73 11 4 

Charlestown, Mon., Ms. 

42 22 36 

71 3 18 

Bristol. 

.R. I. 

41 40 11 

71 16 5 

Cheboygan, L.. .Mich. 

45 4 ° 9 

84 24 37 

Brooklyn, C.H.. 

N. Y. 

40 41 31 

73 59 2 7 

Chicago, C.Ch.Ill. 

4 i 53 48 

87 37 47 

Brownsville, S.S. 

.Tex. 

26 

97 3 ° 

Chickasaw.Miss. 

35 53 3 ° 

88 6 25 

Brunswick, A... 


31 8 51 

81 29 26 

Cincinnati, Obs. 0 . 

39 6 26 

84 29 45 

Brunswick, C. Sp 

. Me. 

43 54 29 

69 57 24 

Cleveland, Hos.“ 

4 i 3 ° 25 

81 40 30 

Buffalo, L. 

N.Y. 

42 50 

78 59 

Colorado Springs..Col. 

38 5 ° 

104 49 8 

Burlington. 

N. .J. 

40 4 52 

74 52 37 

Columbia, S.H_S.C. 

33 59 58 

81 2 3 

Burlington, C.... 

. .vt. 

44 28 52 

78 10 

Columbus, Cap. 0 . 

39 57 4 ° 

82 59 40 

Burlington,Pub. Sq.,Ia. 

40 48 22 

91 6 25 

Concord, S. H_N. H. 

43 12 2Q 

71 29 

Bushnell. 

Neb. 

4 1 13 54 

103 52 57 

Corpus Christi.. .Tex. 

27 47 18 

07 27 2 

Cairo. 

. .Ill. 

36 sq 48 

89 11 14 

Council Bluffs. .Neb.T. 

41 3 ° 

95 48 

Calais, C.S. Obs.. 

. Me. 

45 n 5 

67 16 5 

Crescent City, L.. .Cal. 

4 i 44 34 

124 II 22 



S. 


Cumberland.Md. 

39 39 i 4 

78 45 25 

Callao, F.S. 

Peru 

12 4 

77 13 

Darien, W.H.Ga. 

3 i 21 54 

81 25 39 



N. 


Davenport, S. S_la. 

4 i 32 

9 ° 38 

Cambridge, Obs.,Mass. 

42 22 52 

7 i 7 43 

Dayton. 0 . 

39 44 

84 11 

Camden. 

.S.C. 

34 17 

80 33 

Deadwood, S. S... Dak. 

44 22 

i °3 34 

Campeachy. .Yucatan 

19 49 

9 ° 33 

Decatur, S. S.Tex. 

33 10 

97 30 




















































LATITUDE AND LONGITUDE, 


77 


Latitude and Longitude — Continued. 


Location. 


NORTH AND SOUTH 
AMERICA. 

Denver, S. H.Sp.. Col. 
Des Moines, C. H... Ia. 
Detroit,St.P.Ch.,Mich. 

Dover.Del. 

Dover.N. H. 

Dubuque.Ia. 

Duluth, S. S.Min. 

Eastport, Con.Ch. .Me. 
Eden ton, C.H....N. C. 
Elizabeth City, Ct. “ 

Erie, L.Penn. 

Eureka, M. Ch_Cal. 

Falls St. Anth’y..Minu. 
Fernandina, A. S.. Fla. 

Florence.Ala. 

Fort Gibson_Ind. T. 

Fort Henry.Tenn. 

Fort Laramie. .Neb.T. 
Fort Leavenworth, Ks. 

Frankfort.Ky. 

Frederick.Md. 

Fredericksburg, E. Ch., 
Va. 

Fredericton___N. B. 

Galveston, Cath’l. .Tex. 

Georgetown.__Ber. 

Georgetown, E.Ch.,S.C. 
Gloucester,U.Ch. ..Ms. 
Grand Haven, S. S., 
Mich. 

Halifax, Obs.N.S. 

Harrisburg.Penn. 

Hartford, S. H... Conn. 
Havana, Moro.. .Cuba 
Hole in the Wall, L., 
Bahamas 
Holmes’s Hole,Ch.,Ms. 

Hudson.N.Y. 

Huntsville.Ala. 

Indianapolis.Ind. 

Indianola, G. S_Tex. 

Jackson.Miss. 

Jacksonville, M. Ch., 
Fla. 

Jalapa.Mex. 

Jefferson City_Mo. 

Jersey City, Gas Ch’y. 
Kalama, M.Ch.. .W.T. 

Keokuk, S. S.Ia. 

Key West, T.Obs.,Fla. 

Kingston.Jamaica 

Kingston, C.H...C. W. 

Knoxville.Tenn. 

La Crosse, Ct.S.. .Wis. 

Lancaster.Penn. 

Lavaca, A. S.Tex. 

Leavenworth, S. S.. Ks. 
Lexington.Ky. 

Lima.,.Peru 

Little Rock.Ark. 


Latitude. 

Longitude. 

Location. 

Latitude. 

Longitude. 

N. 


w. 


NORTH AND SOUTH 


N. 


W. 


a 

0 



AMERICA. 





39 45 

104 

59 

33 

Lookport.N.Y. 

43 

II 


78 46 

4 i 35 


93 

37 

l6 

Los Angeles.Cal. 

34 

3 

5 

118 14 32 

42 19 

46 

83 

2 

2 3 

Louisville.Ky. 

38 

3 


85 3 ° 

39 10 


75 

3 ° 


Lowell, St. Ann’s Ch., 




43 13 


70 

54 


Mass. 

4 2 

38 

46 

71 19 2 

42 29 

55 

90 

39 

57 

Machias, Th.Me. 

44 

43 

I 

67 27 21 

46 48 


9 2 

8 


Macon, Arsenal_Ga. 

3 2 

50 

2 5 

83 37 36 

44 54 

15 

66 

59 

14 

Madison, Dome.. .Wis. 

43 

4 

33 

89 2 4 3 

36 3 

2 4 

76 

3 6 

3 i 

Marblehead, L. ..Mass. 

4 2 

30 

14 

7 ° 5 o 39 

36 17 

58 

70 

13 

2 3 

Martinique, S.P’t.W.I. 

14 

27 

60 55 

42 8 

43 

80 

4 

12 

Matagorda, G.S.. .Tex. 

28 

41 

2 9 

95 57 56 

40 48 

II 

124 

9 

41 

Matamoras. “ 

2 5 

5 2 

50 

97 2 7 50 

44 58 

40 

93 

IO 

30 

Matanzas.Cuba 

2 3 

3 

81 40 

30 40 

18 

8l 

2 7 

47 

Memphis, S.S. . .Tenn. 

35 

7 . 


90 7 

34 47 

13 

87 

4 i 

40 

Mexico.Mex. 

*9 

25 

45 

99 5 6 

35 47 

35 

95 

i 5 

IO 

Milwaukee.Mich. 

43 

2 

2 4 

87 54 4 

36 30 

22 

88 

3 

40 

Minneapolis,U. C.,Min. 

44 

58 

38 

93 *4 8 

42 12 

IO 

104 

47 

43 

Mississippi City, G. S., 


39 21 

14 

94 

44 


Miss. 

30 

22 

54 

89 1 57 

38 14 


84 

40 


Mobile, E. Ch.Ala. 

Monterey, Az. S... Cal. 

30 

4 1 

26 

88 2 28 

39 2 4 


77 

18 


3 6 

35 

S. 

53 

21 

121 52 59 

38 18 

6 

77 

27 

38 

Montevideo.. .Rat Is’d 

34 


56 13 

46 3 


66 

38 

15 


N. 



29 18 

17 

94 

47 

26 

Montgomery, S. H., Ala. 

32 

22 

45 

86 18 

32 22 

2 

64 

37 

6 

Montreal.C. E. 

45 

31 

73 3 2 56 

I33 22 

8 

79 

ib 

49 

Mound City.Ill. 

37 

4 

47 

89 12 

4 2 36 

46 

70 

39 

59 

Nantucket, L_Mass. 

Nantucket, S. Tower, 

4 i 

2 3 

2 4 

70 2 24 

43 5 


86 

18 


Mass. 

4 1 

l6 

57 

7 ° 5 57 

44 39 

4 

■ 63 

35 


Nashville, U_Tenn. 

36 

9 

33 

86 49 3 

40 16 


76 

50 


Nassau, L.N. P. 

25 

5 

2 

77 21 2 

4 1 45 

59 

7 2 

40 

45 

Natchez.Miss. 

31 

34 


91 24 42 

2 3 9 

82 

21 

2 3 

Nebraska, Junction of 






Forks of Platte Riv. 

4 i 

5 

5 

101 21 24 

25 5i 

5 

77 

10 

6 

New Bedford, B. Ch., 



41 27 

13 

70 

35 

59 

Mass. 

4 i 

38 

IO 

70 55 36 

42 14 


73 

46 


New Haven, Col.,Conn. 

4 i 

18 

28 

7 2 55 45 

34 36 


86 

57 


New London,P.Ch. “ 

4 i 

21 

l6 

72 5 29 

39 55 


86 

5 


New Orleans, Mint, La. 

2 9 

57 

46 

90 3 28 

28 32 
32 23 

28 

96 

90 

31 

8 

I 

New York, C.H.,N.Y. 

40 

4 2 

44 

74 2 4 




Newbern, E. Sp.. .N.C. 

35 

6 

21 

77 5 

30 19 

43 

81 

39 

14 

Newburg, A. Sp., N.Y. 

4 i 

30 

6 

74 33 

19 30 

8 

96 

54 

3 ° 

N ewburyport, L., Mass. 

4 2 

48 

30 

70 52 28 

38 36 


9 2 

8 


New Castle, E.Ch. ,Del. 

39 

39 

36 

75 33 48 

4 ° 43 

28 

74 

2 

2 4 

Newport, Sp.R. I. 

41 

2 9 

12 

71 18 49 

46 26 


122 

50 

39 

Norfolk, C. H.Ya. 

3 6 

50 

47 

76 7 22 

40 23 


91 

2 5 

Norwalk.Conn. 

4 i 

2 

50 

73 2 5 35 

24 33 

3 i 

81 

48 

3 i 

Norwich. “ 

4 i 

33 


72 7 

17 58 

76 

46 


Ocracoke, L.N.C. 

Ogdensburg, L.. .N.Y. 

35 

6 

28 

75 58 51 

44 8 


76 

28 

37 

44 

45 


75 30 

35 59 


83 

54 

Old Point Comfort,Va. 

37 

2 

76 18 6 

43 58 

5 o 

9 1 

14 

48 

Olympia.Wash.T. 

47 

3 


122 55 

40 2 

36 

76 

20 

33 

Omaha, P. Ch_Neb. 

4 i 

L 5 

43 

95 56 14 

28 37 

3 6 

96 

37 

21 

Oswego, S. S.N.Y. 

43 

28 

3 2 

76 35 5 

39 2 9 

94 

58 


Ottawa.Can. 

45 

2 3 


75 4 2 

38 6 


84 

18 


Panama, Cath’l. ..N.G. 

8 

57 

9 

79 2 7 17 

S. 




Parkersburg-W. Va. 

39 

l6 

2 

81 34 12 

12 3 


77 

6 


Pascagoula.Miss. 

30 

20 

42 

88 32 45 

N. 




Pensacola, Sq’re. .Fla. 

30 

2 4 

33 

87 12 53 

34 4 ° 


9 2 

12 


Petersburg, C. H... Va. 

37 

13 

47 

77 24 16 


G* 

























































73 


LATITUDE AND LONGITUDE 


Latitude and Longitude— Continued. 


Location. 


NORTH AND SOUTH 
AMERICA. 

Philadelphia, S. H., Pa. 
Pike’s Peak, S.S. .Col. 

Pittsburg.Penn. 

Plattsburg, Sp_N.Y. 

Plymouth, Pier .. .Ms. 

Point Hudson_W.T. 

Port au Prince... W. I. 
Port Townshend, A.S., 
Wash. T. 


Portland, C. II.Me. 

Portland, S. S. 0 . 


Porto Bello.N. G. 

Porto Cabello, Mara¬ 
caibo 

Portsmouth, L.. .N. H. 
Prairie du Chiem.Wis. 
Princeton, S.Cap., N. J. 
Providence, U. Ch., R. I. 
Provincetown, Sp., 

Mass. 

Puebla de los Angelos, 
Mex. 

Quebec, Citadel. .Can'a 
Queenstown.... “ 
Raleigh, Square. .N.C. 
Richmond, Cap.... Va. 

Rio de Janeiro, S. Loaf. 

Rochester, R. H. .N.Y. 
Rockland, E. Ch... Me. 
Sackett’s Harbor,N.Y. 

Sacramento.Cal. 

Salem, So.Mass. 

Salt Lake City, Obs., 
Utah 

Saltillo.Mex. 

San Antonio.Tex. 

San Buenaventura, 

G. S.Cal. 

San Diego, A. S_ “ 

San Francisco, C. S. 

Station.Cal. 

San Josd, Sp. “ 

San Luis Obispo.. “ 

San Pedro. “ 

Sandusky, L. 0 . 

Sandy Hook, L.. .N. J. 
Santa Barbara, M. Ch., 
Cal. 

Santa Clara, C. Ch.. ‘ ‘ 

Snnt.n. Pirn•7 TP S u 


Uil l JOi UZJj J. . kJ . . 

Santa Fd.N. Mex. 

Savannah, Sp.Ga. 


Schenectady.N. Y. 

Sherman, R. R. D., Wy. 
Shreveport. S. S.... La. 
Smithville, G.S.. .N.C. 

Springfield.Mass. 

Springfield, S.H_Ill. 

Springfield, S. S_ “ 

St Augustine.Fla. 


Latitude. 

Longitude. 

N. 

w. 

0 / 11 

0 1 II 

39 56 53 

75 9 3 

38 48 

104 59 

40 32 

80 2 

44 4 i 57 

73 26 54 

41 58 44 

70 39 12 

48 7 3 

122 44 33 

18 33 

72 16 3 

48 6 56 

122 44 58 

43 39 28 

70 15 1 

45 30 

122 27 30 

9 34 

79 40 

10 28 

68 7 

43 4 16 

70 42 34 

43 2 

9 1 8 35 

40 20 40 

74 39 55 

41 49 26 

71 24 19 

42 3 

70 11 18 

19 15 

98 2 21 

46 49 12 

7 1 12 15 

43 9 

79 8 

35 46 50 

78 38 5 

37 32 16 

77 26 4 

s. 


22 56 

43 9 

N. 


43 8 17 

77 5 i 

44 6 6 

69 6 52 

43 55 

75 57 

38 34 4 i 

121 27 44 

42 31 10 

70 53 58 

40 46 4 

m 53 47 

25 26 22 

101 4 45 

29 25 22 

98 29 15 

34 i 5 46 

119 15 56 

32 43 68 

117 9 40 

37 48 

122 23 19 

37 19 58 

121 53 39 

35 10 38 

120 43 31 

33 43 20 

n8 16 3 

41 32 30 

82 42 15 

40 27 4O 

74 9 

34 26 10 

119 42 42 

37 20 49 

121 26 56 

36 57 3 1 

122 I 29 

35 4 i 6 

106 1 22 

32 4 52 

81 5 26 

42 48 

73 55 

4 i 7 5 ° 

105 23 33 

32 3 ° 

93 45 

33 54 58 

7818 

39 47 57 

89 39 20 

42 6 

72 36 

29 48 3 ° 

, 81 35 


Location. 


NORTH AND SOUTH 
AMERICA. 

St. Augustine, P. Ch., 
Fla. 

St. Bartholomew. S. 

Point.W. I. 

St. Christopher, N. Pt., 
W. I. 

St. Croix, Obs_ “ 

St. Domingo__ “ 

St.Eustatia,Town. “ 
St. Jago de Cuba, En¬ 
trance .W. I. 

St. John.N. B. 

St. Joseph.Mo. 

St. Louis, W.U. “ 

St. Mark’s, Fort..Fla. 
St. Martin’s, Fort, W. I. 
St. Mary's, M. H.. .Ga. 

St. Paul.Minn. 

St. Thomas, Fort Ch’n, 
W. I. 

St. Vincent’s, S. Point, 
W. I. 

Staunton.Va. 

Stockton, S.S.Tex. 

Stonington, L.. .Conn. 
Sweetwater River, 

Mouth of_Neb.T. 

Sydney, S. S.N. S. 

Syracuse.N.Y. 

Tallahassee.Fla. 

Tampa Bay,E. Key “ 
Tampico, Bar.... Mex. 
Taunton, T. C. Ch., Mass. 
Tobago, N. E. P’r.W. I. 

Toronto.Can. 

Trenton, P.Ch ...N. J. 
Trinidad, Fort... W. I. 

Troy, D.Ch.N.Y. 

Tuscaloosa.Ala. 

Utica, Dut.Ch. ...N.Y. 

Valparaiso,Fort. .Chili 

Vandalia.Ill. 

Vera Cruz... *.. .Mex. 
Vicksburg, S. S.. .Miss. 

Victoria.Tex. 

Vincennes.Ind. 

Virginia City,S.S.,M.T. 

Washington. . .Capitol 

Watertown, Ars’l. .Ms. 

West Point.N. Y 

Wheeling.Va. 

Wilmington, E. Ch., 
N. C. 

Wilmington,T.H. .Del. 
Worcester, Ant.H. .Ms. 

Yankton, S.S.Dak. 

Yazoo.Miss. 

York. .Penn. 

York town.Va. 


Latitude. Longitude. 


N. 


29 53 20 

W 53 3 ° 

17 24 

17 44 3 ° 

18 29 
17 29 


19 58 
45 14 
23 3 
38 8 

30 9 
18 5 

30 43 12 
44 52 4 6 

18 21 

13 9 
38 8 51 
30 50 
41 19 36 


42 27 
46 12 

43 3 
30 28 

27 3 6 
22 15 
41 54 
11 20 


49 


43 39 

40 J 3 

10 39 

42 43 
33 12 

43 6 
S. 

33 2 
N. 

38 50 

19 11 52 

32 23 

28 46 57 
38 43 
45 20 

38 53 20 

42 21 41 

41 23 26 
40 7 

34 H 2 

39 44 27 

42 16 17 
42 45 

33 5 
39 58 
37 13 


W. 

0/11 
81 18 41 
62 56 54 

62 50 
64 40 42 
69 52 

63 

75 52 
66 3 30 
IO9 40 44 
90 12 4 
84 12 30 

63 3 

81 32 53 

95 4 54 

64 55 18 


61 14 

79 4 
102 50 
7 i 54 


15 


107 

60 

76 

84 

82 

97 

7 i 

60 

79 

74 

61 

73 

87 

75 


45 27 

12 

9 16 

36 

45 15 
5 i 5 i 
5 55 

27 

23 21 
45 50 
32 

2 16 
42 

13 


71 41 

89 2 

96 8 36 
9 ° 54 

97 1 

87 25 
[12 3 

77 36 

7 i 9 45 
73 57 1 
80 42 


13 


77 56 38 

75 33 3 
71 48 
97 30 
90 20 

76 40 

76 34 

















































LATITUDE AND LONGITUDE 


79 


V 

Hiatitu.d.e and. Longitude— Continued. 


Location. 


EUROPE, ASIA, AFRICA, 
AND THE OCEANS. 

Aleppo .. 

Alexandria, L. 

Algiers, L. 

Amsterdam. 

Antwerp. 

Archangel. 

Athens.. 

Barcelona. 

Batavia, Ohs... 

Bencoolen, Fort, Su’a. 

Berlin, Obs.... 

Bombay, F.S. 

Botany Bay, C. Roads. 

Bremen. 

Bristol. 

Brussels, Obs. 

Bussorah. 

Cadiz. 

Cairo. 

Calais. 

Calcutta. 

Candia. 

Canton . 

Cape Clear. 

Cape of G. Hope, Obs.. 
Cape St. Mary, Mad’r.. 

Ceylon, Port Pedro ... 
Christiana. 

Congo River. 

Constantinople, St. S.. 

Copenhagen. 

Corinth. 

Cronstadt. 

Dover. 

Dublin. 

Edinburgh. 

Falkland Islands, St. 
Helena, Obs. 

Fayal, S.E. Point. 

Feejee Group, Ovolau, 
Obs. 

Florence. 

Funchal, Madeira. 

Geneva. 


Latitude. 

Longitude. 

Location. 

Latituc 

N. 

E. 

EUROPE, ASIA, AFRICA, 

N. 



AND THE OCEANS. 


36 11 

37 10 

Genoa . 

44 24 

31 12 

29 53 



36 47 

3 4 

Gibraltar. 

36 7 

52 22 

4 53 

Glasgow. 

55 52 

5 i 13 

4 24 

Greenwich. 

51 28 

64 32 

40 33 



37 58 

23 44 



4 i 23 

2 II 

Hamburg. 

53 33 

S. 


Havre. 

49 29 

6 8 

106 50 


S. 

3 48 

102 19 

Hawaii or Owyhee.... 

20 23 

N. 



N. 

52 30 16 

13 23 45 

Hongkong . 

22 16 

18 56 

72 54 

Honolulu. 

21 18 

34 2 

151 13 

Hood Isl’d,Gallapagos. 

1 23 

N. 


Hood’s Island, Mar- 

S. 

53 5 

8 49 

quesas. 

9 26 


W. 


N. 

5i 27 

2 35 

Jeddo or Tokio. 

35 4 ° 


E. 

Jerusalem. 

3 i 48 

50 51 11 

4 22 

Leghorn, L. 

43 32 

3 ° 3 ° 

48 

Leipsic. 

51 20 


w. 

Leyden. 

52 9 

36 3 2 

6 18 




E. 

Lisbon. 

38 42 

30 3 

31 18 

Liverpool, Obs. 

53 24 

50 58 

1 5 i 



22 34 

88 20 

Madras. 

14 4 

35 3 i 

25 8 



23 7 

113 14 

Madrid. 

40 25 


W. 



51 26 

9 29 

Majorca, Castle. 

39 34 

S. 

E. 



33 56 3 

18 28 45 

Malaga. 

36 43 

25 39 

45 7 



N. 


Malta, Valetta. 

35 54 

9 49 

80 23 

Manila. 

14 36 

59 55 

i° 43 

Marseilles.... 

43 18 

S. 


Messina, L. 

38 12 

6 8 

12 9 

Mocha . 

13 20 

N. 


Moscow. 

55 4 ° 

41 1 

28 59 

Muscat. 

23 37 

55 41 

12 34 

Naples, L. 

40 50 

37 54 

22 52 



59 59 

29 47 

New Castle. 

54 58 

5 i 8 

1 19 

New Hebrides, Table 

S. 


w. 

Island. 

15 28 

53 23 12 

6 20 30 

Niphon, Cape Idron, 

N. 

55 57 

3 12 

Japan . 

34 36 

S. 


Odessa. 

46 28 

15 55 

5 45 

Palermo, L. 

38 8 

N. 


Paris, Obs. 

\ 4 S 50 

38 3 ° 

28 42 

Pekin. 

39 54 

S. 

E. 



17 4i 

U 8 53 

Plymouth. 

50 21 

N. 



S. 

43 4 6 

11 16 

Port Jackson. .N.S.W. 

35 5 i 


W. 

Porto Praya, Cape Yerd 

N. 

32 38 

16 55 

Islands. 

14 54 


E. 


S. 

46 11 59 

6 9 iS 

Prince of Wales Island. 

10 46 


28 


E. 

O / /,• 

8 53 
W. 

5 22 
4 16 

E. 

9 58 

6 

W. 

i 55 54 
E. 

114 14 45 
i 57 3 ° 36 
W. 

89 46 
E. 

138 57 

139 40 
37 20 
10 18 

12 22 

4 29 15 
W. 

9 9 
3 

E. 

80 15 45 
W. 

3 42 
E. 

2 23 
W. 

4 26 
E. 

14 30 

121 2 

5 22 

1 5 35 
43 12 
35 33 
58 35 
14 16 

W. 

1 37 
E. 

167 7 

138 5 ° 35 
3 ° 44 

13 22 

2 20 
116 28 

W. 

4 9 
E. 

151 18 
W. 

23 3 

E. 

142 12 






















































































8o 


LATITUDE AND LONGITUDE, 


Latitude and Longitude- Continued. 


Location. 


EUROPE, ASIA. AFRICA, 
AND THE OCEANS. 


Queenstown. 


Rome, St. Peter’s. 
Rotterdam. 


Santa Cruz.Ten’fe 

Scilly, St. Agnes, L_ 

Senegal, Fort. 


Sevastopol. 

Seville. 

Siam. 


Sierra Leone. 


Singapore. 
Smyrna... 


Southampton 
St. Helena_ 


Latitude. 

Longitude. 

Location. 

Latitude. 

Longitt 

N. 

W. 

EUROPE, ASIA, AFRICA, 

N. 

E. 

n 

n 

AND THE OCEANS. 

0 


5 i 47 ~ 

8 19 

St. Petersburg. 

59 56 " 

30 19 


E. 

Suez. 

2 9 59 

32 34 

4 1 54 

12 27 

Surat, Castle. 

21 II 

72 47 

5 54 

4 29 


s. 



w. 

Sydney.N. S. W. 

33 33 42 

151 23 

28 28 

1616 



W. 

49 54 

6 21 

Tahiti or Otaheite.... 

17 45 

149 3 o 

l6 I 

16 32 


N. 

E. 


E. 

Tangier. 

35 47 

5 54 

44 37 

33 30 

Toulon. 

43 7 

5 22 

36 59 

5 58 

Tripoli. 

34 54 

13 11 

14 55 

IOO 

Tunis, City. 

3 6 47 

10 6 

S. 

W. 

Venice. 

40 50 

14 26 

8 3 ° 

00 

H 

CO 

H 

Vienna. 

48 13 

16 23 

N. 

E. 

Warsaw, Obs. 

52 13 5 

21 2 

1 17 

103 50 


S. 


38 26 

27 7 

Wellington.. .New Z’d 

41 14 

174 44 


w. 


N. 


51 

I 30 

Yokohama. 

35 26 

i 39 39 

S. 



S. 


15 55 

5 45 

Zanzibar Island, Sp... 

6 28 

39 33 


Observatories .—Not included in previous Table. 
Longitude given in Time. 


Location. 

Latitude. 

Longitude. 

1 Location. 

1 

Latitude. 


N. 

w. 


N. 


0 / 11 m 

h. m. s. 


0 1 /> / n 

Albany, Dudley .. 

42 39 49.55 

4 54 59-52 

Madras. 

13 4 8.1 

Alleghany, Penn.. 

40 27 36 

5 20 2.9 

Marseilles. 

43 W 5 ° 

Birr Castle, Earl 





of Rosse. 

53 5 47 

31 40.9 

Mitchell’s, Cin., 0 . 

39 6 26 

Cambridge, U. S... 

42 22 52 

4 44 3°-9 

Moscow. 

55 45 i 9 - 8 



E. 

Munich, Bogenh’n 

48 8 45 

Cambridge, Eng... 

52 12 51.6 

Q 

22.75 

Palermo .. 

38 6 44 

Cape of G. Hope.. 

33 56 3 

1 13 55 

Portsmouth. 

50 48 3 


N. 


Quebec. 

46 48 30 

Copenhagen, Un’y. 

55 40 53 

50 19.8 



Crescent City, A. 


W. 

Rome, College.... 

4 1 53 52-2 

S., Cal. 

4 i 44 43 

8 16 49.1 

Salt Lake City, 


Dublin. 

53 23 13 

25 22 

Utah 

40 46 4 

Edinburgh. 

55 57 23.2 

12 43.6 

San Francisco,So., 




E. 

Cal. 

37 47 55 

Florence. 

43 46 4 i -4 

45 3-6 


S. 

Geneva. 

46 11 59.4 

24 37-7 

Santiago de Chili. 

33 26 24.8 



W. 


N. 

Georgetown, U.S.. 

38 53 39 

5 8 12.5 

St. Croix, W. I_ 

17 44 30 

Gibbes’s, Charles- 





ton, U. S. 

32 47 7 

5 19 44-7 

St. Petersburg, A.. 

59 56 29.7 




Stockholm. 

50 20 11 

Greenwich. 

51 28 38 

— 


s.' 



E. 

Sydney. 

33 51 41.1 

Hamburg. 

53 33 5 

39 54 -i 

Tifft’s, Key West. 

N. 

Leipsic. 

51 20 20.1 

40 28.5 

Fla. 

24 33 3 i 

Leyden. 

52 9 28.2 

17 57-5 

Unkrechtsberg, 01 - 



W. 

mutz. 

49 35 4 ° 

Liverpool. 

53 24 47 - 8 

12 O.II 



L. M. Rutherfurd, 



Washington. 

38 53 39 

New York. 

40 43 49 

4 55 57 

W T est Point, N.Y.. 

41 23 26 


Longitude. 


E. 

h. m. s. 

5 20 57.3 
21 29 

w. 

5 37 59 
E. 

2 30 16.96 
46 26.5 
53 24.17 
W. 

4 23.9 
4 44 49.02 
E. 

49 54-7 
W. 

7 27 35.1 

8 9 38.1 

4 42 18.9 

4 18 42.8 
E. 

2 1 i3-5 

1 12 24.8 


W. 

5 27 14.1 
E. 

1 9 o. 1 

W. 

5 8 12.03 

4 55 48 












































































DIFFERENCE IN TIME. 


8l 


DIFFERENCE IN TIME. 

Difference in Time at following Locations. 

Longitude computed both from JVeiv York and Greenwich. 

Exact Difference of Time between New York and Greenwich is 4 h. 56 m. 
1.6 sec., but in following table 2 seconds are given when the decimal in any 
reduction exceeds .5 seconds. 

F representing Fast, and S Slow. 


Location. 

New York. 

Greenwich. 


h. m. s. 

h. m. 

Acapulco. 

i 43 15 S - 

6 39 17 S. 

Albany . 

1 F. 

4 55 2 

Alexandria.. Egypt 

6 55 34 

1 59 32 F. 

Algiers. 

5 8 18 

12 16 

Amsterdam. 

5 16 5 

19 32 

Antwerp. 

5 *3 38 

17 36 

Apalachicola. 

43 54 S. 

5 39 56 S. 

Astoria. 

3 19 T 7 

8 15 19 

Atlanta. 

4 1 32 

5 37 33 

Auburn. 

9 5 ° 

5 5 52 

Augusta.Ga. 

31 34 

5 27 36 

Augusta.Me. 

16 55 F. 

4 39 6 

Austin. 

1 34 55 S. 

6 30 27 

Baltimore. 

10 26 

5 6 28 

Bangor. 

20 54 F. 

4 35 8 

Barbadoes, S.Pt_ 

57 34 

3 58 28 

Barnegat, L. 

22 S. 

4 56 24 

Bath. 

16 46 F. 

4 39 16 

Baton Rouge. 

1 9 10 S. 

6 5 12 

Beaufort.N.C. 

10 38 

5 6 39 

Beaufort.S. C. 

26 39 

5 22 40 

Belfast. 

20 x F. 

4 36 r 

Benicia. 

3 12 36 S. 

8 8 38 

Berlin. 

5 49 37 F. 

53 35 F. 

Bismarck. 

1 46 30 s. 

6 42 32 S. 

Bombay, F.S. 

9 47 38F. 

4 5i 36F. 

Boston, S. H. 

11 47.6 

4 44 M S. 

Bremen. 

5 3 1 18 

35 16 F. 

Bridgeport. 

3 *7 

4 52 44 s - 

Brooklyn, N. Yard. 

4 

4 55 58 

Brunswick_Me. 

16 12 

4 39 50 

Brunswick.Ga. 

29 56 s. 

5 25 58 

Brussels. 

5 13 3 o F . 

17 28 F 

Buenos Ayres. 

1 2 34 

3 53 28 S. 

Buffalo, L. 

19 54 S. 

5 i 5 56 

Burlington.Ia. 

1 8 24 

6 4 26 

Burlington_N. J. 

3 29 

4 59 30 

Burlington.Vt. 

1638 

5 12 40 

Bushnell.Neb. 

1 59 30 

6 55 32 

Cadiz. 

4 3° 50 F. 

25 12 

Cairo. 

7 1 J 4 

2 5 12 F. 

Cairo.Ill. 

1 43 S. 

5 56 45 s - 

Calais.Me. 

26 57 F. 

4 29 4 

Calcutta... 

10 49 22 

5 53 20F 

Callao. 

12 50 S. 

5 8 52 S. 

Cambridge.. .Mass. 

11 30F. 

4 44 3 i 

Canton. 

12 28 58 

7 32 56 F. 

Cape Girardeau.... 

1 2 10S. 

5 58 12 S. 

Cape of Good Hope. 

6 957F. 

1 13 55 F. 

Cape Horn. 

26 58 

4 29 4 S. 

Cape May. 

2 56 S. 

4 58 58 

Cape Race . 

1 23 46 F. 

3 32 16 

Carthagena. 

6 30 S. 

5 2 32 

Castine. 

21 2 F. 

4 35 


Location. 

New York. 

Greenwich. 


h. m. s. 

h. m. s. 

Cedar Keys. 

36 9S. 

5 32 11 S. 

Chagres. 

24 3 

5 20 5 

Charleston. 

23 41 

5 19 43 

Charlestown. 

11 48 F. 

4 44 13 

Cheboygan. 

41 37 s - 

5 37 38 

Chicago. 

54 30 

5 50 3 i 

Chickasaw. 

56 24 

5 52 26 

Cincinnati. 

41 57 

5 37 59 

Cleveland. 

30 40 

5 26 42 

Colorado Springs.. 

2 3 i 5 

6 59 17 

Columbia. 

28 7 

5 24 8 

Columbus. 

35 57 

5 3 1 59 

Concord. 

10 6 F. 

4 45 56 

Constantinople.... 

6 51 58 

1 55 56 F. 

Copenhagen. 

5 46 18 

50 16 

Corpus Christi. 

1 33 47 s - 

6 29 48 S. 

Council Bluffs. 

I 27 IO 

6 23 12 

Crescent City. 

3 20 44 

8 16 45 

Darien. 

29 41 

5 25 43 

Davenport. 

1 12 30 

6 2 32 

Dayton. 

40 42 

5 3 6 44 

Deadwood. 

1 58 30 

6 54 32 

Denver. 

2 3 57 

6 59 58 

Detroit. 

36 76 

5 3 2 10 

Dover.Del. 

35 58 

5 2 

Dover.N. H. 

12 26 F. 

4 43 36 

Dublin. 

4 30 4 ° 

25 22 

Dubuque. 

1 6 38S. 

6 2 40 

Duluth . 

I 12 IO 

6 8 32 

Eastport . 

28 6 F. 

4 27 56 

Edenton. 

10 24 S. 

5 6 26 

Edinburgh. 

4 43 14 F. 

12 48 

ElizabethCity,N. C. 

8 52 S. 

5 4 54 

Erie. 

24 i5 

5 20 17 

Eureka. 

3 20 37 

8 16 39 

Falls St. Anthony.. 

1 16 40 

6 12 42 

Fernandina. 

2 9 50 

5 2 5 5 i 

Fire Island, L. 

3 10F. 

4 5 2 5 i 

Florence . Ala. 

54 45 s - 

5 50 47 

Fort Gibson . 

1 2 4 59 

6 21 1 

Fort Henry. .Tenn. 

56 13 

5 52 15 

Fort Laramie . 

2 3 9 

6 59 11 

Fort Leavenworth. 

1 22 54 

6 18 56 

Frederick . 

13 10 

5 9 12 

Fredericksb’g. .Va. 

1 3 49 

5 9 5 i 

Fredericton. . .N.B. 

29 29F. 

4 26 33 

Funchal . 

3 48 22 

1 7 40 

Galveston . 

1 23 8 S. 

6 19 10 

Geneva . 

5 20 39 F. 

24 37 F. 

Geneva.N.Y. 

12 14 S. 

5 8 16S. 

Genoa . 

5 3 i 34 F. 

35 32 F. 

Georgetown. . .Ber. 

37 33 

4 18 28 S. 

Georgetowm. . .S. C. 

21 6 S. 

5 i 7 7 

Gibraltar .. 

4 34 34 F. 

21 28 






































































































82 


DIFFEKENCE IN TIME, 


Location. 


Glasgow. 

Gloucester. 

Grafton. 

Grand Haven. 

Greenwich. 

Halifax. 

Hamburg. 

Harrisburg. 

Hartford. 

Havana, Morro... 

Havre. 

Hawaii or Owyhee 

Hongkong. 

Honolulu. 

Hudson. 

Huntsville. 

Indianapolis. 

Indianola. 

Jackson. 

Jacksonville. 

Jalapa. 

Jeddo or Tokio... 

Jefferson City_ 

Jersey City. 

Jerusalem. 

Kalama. 

Keokuk. 

Key West. 

Kingston_Can. 

Kingston_Jam. 

Knoxville. 

La Crosse. 

La Guayra. 

Lancaster. 

Lavaca. 

Leavenworth. 

Leghorn. 

Lexington. 

Lima. 

Lisbon. 

Little Rock. 

Liverpool. 

Lockport. 

Los Angeles. 

Louisville. 

Lowell. 

Machias Bay. 

Macon.. 

Madison. 

Madrid. 

Malaga. 

Malta. 

Manila. 

Maracaibo. 

Marblehead, L.... 

Marseilles. 

Martinique. 

Matagorda.. 

Matamoras. 

Matanzas. 

Memphis. 

Mexico. 


Difference in Ti me—Continued. 


New York. 


Greenwich. 


h. 

m . 

s. 

h. 

m. 

S. 

4 

38 

58 F. 


17 

4 s. 


13 

22 

4 

42 

40 


24 

5S. 

5 

20 

7 


49 

IO 

5 

45 

12 

4 

56 

1.6 


— 



41 

42 F. 

4 

14 

40 

5 

35 

54 


39 

52 F. 


11 

18 y. 

5 

7 

20 S. 


5 

19 F. 

4 

50 

43 


33 

24 s. 

5 

29 

26 

4 

56 

26 F. 



24 F. 

5 

27 

34 S. 

IO 

23 

36 S. 

12 

27 

iF. 

7 

36 

59 F. 

15 

27 

30 

IO 

31 

28 


1 

12 

4 

54 

40 S. 


SI 

46 s. 

5 

47 

48 


48 

18 

5 

44 

20 

I 

30 

2 

6 

26 

4 

I 

4 

30 

6 

O 

32 


30 

35 

5 

26 

37 

I 

.31 

3 6 

6 

27 

38 

14 

l6 

2 F. 

9 

20 

F. 

I 

12 

30 S. 

6 

8 

32 s. 



8 

4 

56 

IO 

7 

25 

22 F. 

2 

29 

20 F. 

3 

15 

21 S. 

8 

II 

23 S. 

I 

9 

38 

6 

5 

40 


31 

13 

5 

27 

14 


9 

53 

5 

5 

54 


II 

2 

5 

7 

4 


39 

34 

5 

35 

36 

I 

8 

58 

6 

4 

59 


27 

54 

4 

28 

8 


19 

21 S. 

5 

15 

22 

I 

3° 

27 

6 

26 

29 

I 

23 

4° 

6 

19 

52 

5 

37 

14 F. 


4i 

12 F. 


4i 

10 S. 

5 

37 

12 S. 


12 

22 

5 

8 

24 

4 

19 

26 f’. 


3 6 

36 F. 

I 

12 

46 s. 

6 

8 

48 S. 

4 

44 

2 F . 


12 



19 

2 S. 

5 

15 

4 

2 

56 

l 6 

7 

52 

18 


45 

58 

5 

42 



IO 

45 F. 

4 

45 

l 6 


26 

12 

4 

2 9 

49 


38 

28 s. 

5 

34 

30 

I 

I 

35 

5 

57 

36 

4 

4i 

14 F. 


14 

48 

4 

38 

18 


17 

44 

5 

54 

2 


58 

F. 

13 


IO 

8 

4 

8 


9 

2 

4 

47 

S. 


12 

41 

4 

43 

21 

5 

17 

20 


21 

28 F. 


52 

22 

4 

3 

40 S. 

I 

27 

50 s. 

6 

23 

52 

I 

33 

50 

6 

29 

5i 

I 

3° 

38 

5 

26 

40 

I 

4 

26 

6 

O 

28 

I 

40 

19 

6 

36 

20 


Location. 

New York. 

Greenwich. 


/ i. m. a. 

h. m. s. 

Milwaukee. 

55 35 S. 

5 51 3 6S - 

Minneapolis. 

1 16 55 

6 12 57 

Mississippi City.. 

i 6 

5 56 8 

Mobile.. 

56 7 

5 52 9 

Montauk Point... 

18 22 F. 

4 37 40 

Monterey. 

3 11 3 ° S. 

8 7 32 

Montevideo. 

1 11 10 F. 

3 44 52 

Montgomery. 

49 10 S. 

5 45 12 

Montreal. 

1 50 F. 

4 54 12 

Montserrat. 

47 14 

4 8 48 

Moscow. 

7 18 14 

2 22 12 F. 

Mound City. 

1 26 S. 

5 56 28 S. 

Nantucket. 

15 38 F. 

4 40 24 

Naples. 

5 53 6 

57 4F. 

Nashville. 

5115 s. 

5 47 16 S. 

Nassau. 

13 23 

5 9 24 

Natchez. 

1 9 37 

6 5 39 

Nebraska. 

1 49 24 

6 45 26 

New Bedford. 

12 19 F. 

4 43 42 

New Haven. 

4 19 

4 5 i 43 

New London. 

7 40 

4 48 22 

New Orleans..... 

1 4 12 S. 

6 14 

New York. 

— 

4 56 1.6 

Newbern. 

12 18 

5 8 20 

Newburg. 

I 

4 56 2 

Newburyport. 

12 32 F. 

4 43 30 

New Castle. 

4 49 34 

6 28 

New Castle... Del. 

6 13 S. 

5 2 15 

Newport. 

10 46 F. 

4 45 i 5 

Norfolk. 

9 8S. 

5 5 9 

Norwalk. 

2 19 F. 

4 53 42 

Norwich. 

7 34 

4 48 28 

Ocracoke. 

7 54 S. 

5 3 55 

Odessa. 

6 58 58 F. 

2 2 56 F. 

Ogdensburg. 

5 58 S. 

5 2 S. 

Old Point Comfort 

9 11 

5 5 12 

Olympia. 

3 i 5 38 

8 11 40 

Omaha. 

1 27 43 

6 23 45 

Oswego. 

10 19 

5 6 20 

Ottawa. 

6 46 

5 2 48 

Paducah. 

58 22 

5 54 24 

Palermo. 

5 49 3 o F. 

53 28 F. 

Panama. 

1 21 48 S. 

5 17 49 

Paris. 

5 5 22 F. 

9 20 F. 

Parkersburg. 

30 15 s. 

5 26 37 S. 

Pekin. 

12 41 54 F. 

7 45 52 F. 

Pensacola. 

52 50 s. 

5 48 52 S. 

Petersburg. 

i 3 35 . 

5 9 37 

Philadelphia. 

4 34 6 

5 3 6- 2 

Pike’s Peak. 

2 3 5 o 

6 59 52 

Pittsburg. 

24 6 

5 20 8 

Plattsburg. 

2 14 F. 

4 53 48 

Plymouth. 

4 39 26 

16 36 

Plymouth.. .Mass. 

13 25 

4 42 37 

Port Au Prince, 



St. Domingo 

16 34 

4 39 28 

Port Townshend.. 

3 14 58 S. 

8 11 

Portland. 

15 2 F. 

4 41 

Portland.Or. 

3 13 48 S. 

8 9 50 

Porto Praya. 

3 23 50 F. 

I 32 12 

Porto Rico. 

33 26 

4 22 36 

Portsmouth. 

13 11 

4 42 51 





























































































































DIFFERENCE IN TIME 


83 


Location. 


Prairie du Chien. 

Princeton. 

Providence. 

Provincetown.... 

Quebec. 

Queenstown, L.... 

Raleigh. 

Richmond. 

Rio de Janeiro... 

Rochester. 

Rockland. 

Rome. 

Rotterdam..._ 

Sackett's Harbor. 

Sacramento. 

Salem. 

Salt Lake City... 

Saltillo. 

San Antonio. 

San Buenaventura 

San Diego . 

San Francisco, 

c. s. s. 

San Francisco, P. 

San Jose. 

Sandusky. 

Sandy Hook. 

Santa Barbara.... 

Santa Clara. 

Santa Cruz. 

Santa Cruz.Ten’fe 

Santa Fe. 

Savannah. 

Schenectady. 

Seville. 

Sherman. 

Shreveport. 

Siam. 

Sierra Leone. 

Singapore. 

Smithville. 

Smyrna. 

Southampton. 

Springfield_Ill. 

Springfield. .Mass. 

St. Augustine. 

St. Croix, Obs. 

St. Helena. 

St. Jago de Cuba.. 

St. John. 

St. Joseph_Mo. 


Difference in Time- Continued. 

New York. 


h. m. 

1 8 

2 

10 
i 5 

11 

4 42 

18 

13 

2 3 
i5 

19 

5 45 
5 13 

7 

3 9 

12 
3i 
48 
37 

I 

52 


5 . 

33 S. 
38 

24 F. 

16 

13 

46 S 
31 s. 

43 

26 F. 
22 S. 

34 
5° 

58 

46 S. 
49 

26 F. 
34 s - 

17 
55 

2 

37 


3 13 32 
3 13 50 
3 11 33 
34 47 

1 F. 
3 2 49 S. 
3 9 46 

3 12 4 
3 5 o 58 F. 
2 8 4 S. 
28 20 
22 F. 
10 

33 S. 
58 
2F. 
50 S. 
22 F. 

3 s - 
3° F. 

2 

36 s. 
39 F- 
13 

19 F. 
2 

26 S. 
48 F. 
41 S. 


4 32 
2 5 
1 18 
11 36 
4 2 
n 5i 

16 
6 44 

4 5 ° 

1 2 

5 

29 

37 
4 33 
7 
3 i 

2 22 


Greenwich. 

Location. 

New York. 

I Greenwich. 

h . 

m. 

8 . 


h . 

m . 

8 . 

h . 

m . 

8 . 

6 

4 

34 S. 

St. Louis. 


4 

47 s - 

6 


48 s. 

4 

58 

40 

St. Mark’s. 


40 

48 

5 

36 

50 

4 

45 

37 

St. Mary’s. 


30 

IO 

5 

26 

12 

4 

40 

45 

St. Paul. 

II 

24 

18 

6 

20 

20 

4 

44 

49 

St. Petersburg_ 

6 

57 

18 F. 

2 

1 

16 F. 


33 

l6 

St. Thomas, Fort.. 


36 

20 

4 

19 

41 S. 

5 

14 

32 

Staunton. 


20 

15 s. 

5 

16 

17 

5 

9 

44 

Stockholm. 

6 

8 

26 F. 

I 

12 

25 F. 

2 

52 

36 

Stonington. 


8 

26 

4 

37 

36 s. 

5 

11 

24 

Suez. 

7 

6 

18 

2 

10 

16 F. 

4 

36 

27 

Sweetwater River, 








49 

48 F. 

Mouth of.. 

2 

15 

S. 

7 

11 

aS. 


17 

56 

Sydney.N.S. 


8 

2 F. 

4 

48 


5 

3 

48 S. 

Sydney.. .N. S.W. 

15 

I 

34 

IO 

5 

32 F. 

8 

5 

5i 

Syracuse. 


8 

35 s - 

5 

4 

37 S. 

4 

43 

36 

Tahiti or Otaheite. 

14 

44 

2 f. 

9 

48 

F. 

7 

27 

35 

Tallahassee. 


42 

22 s. 

5 

38 

24 S. 

6 

44 

19 

Tampa Bay. 


34 

59 

5 

31 

I 

6 

33 

57 

Tampico Bar. 

I 

35 

25 

6 

31 

27 

7 

57 

4 

Taunton . 


II 

38 F. 

4 

44 

24 

7 

47 

39 

Toronto. 


21 

32 s. 

5 

17 

3.3 




Toulon. 

5 

17 

30 F. 


21 

28 F. 

8 

9 

33 

Trenton. 


3 

2 S. 

4 

59 

3 S. 

8 

9 

5i 

Tripoli. 

5 

48 

36 F. 


52 

44 F. 

8 

7 

35 

Troy. 


3 

54 

4 

52 

9 S. 

5 

30 

49 

Tunis. 

5 

3 6 

26 


40 

24 F. 

4 

56 

I 

Turk’s Island. 


II 

22 

4 

44 

40 S. 

7 

58 

50 

Tuscaloosa. 


54 

46 s. 

5 

50 

48 

8 

5 

48 

Utica. 


4 

50 

' 5 


52 

8 

8 

6 

Valparaiso. 


9 

18 

4 

46 

44 

I 

5 

4 

Vandalia. 

I 


6 

5 

56 

8 

7 

4 

5 

Venice. 

5 

53 

46 F. 


57 

44 F. 

5 

24 

22 

Vera Cruz. 

I 

28 

33 s - 

6 

24 

34 S. 

4 

55 

40 

Vicksburg. 

I 

7 

34 

6 

3 

36 


23 

52 

Victoria.Tex. 

I 

32 

2 

6 

28 

4 

7 

I 

34 

Vienna . 

6 

I 

34 F - 

I 

5 

32 F. 

6 

i5 


Vincennes. 


53 

38 s. 

5 

49 

40 S. 

6 

40 

F. 

Virginia City. 

2 

32 

IO 

7 

28 

12 


53 

12 S. 

Warsaw. 

6 

20 

ii F. 

I 

24 

9 f. 

6 

55 

20 F. 

Washington. Obs.. 


12 

10 s. 

5 

8 

12 S. 

5 

12 

5S. 

West Point. 



14 F. 

4 

55 

48 

I 

48 

28 F. 

Wheeling. 


26 

46 S. 

5 

22 

48 


6 

S. 

Wilmington. .Del. 


6 

II 

5 

2 

12 

5 

58 

37 

Wilmington. .N.C. 


15 

45 

5 

II 

47 

4 

5° 

24 

Worcester. 


8 

49 E. 

4 

47 

13 

5 

25 

15 

Yankton. 

I 

33 

58 s. 

6 

30 


4 

18 

43 

Yazoo . 

I 

5 

18 

6 

I 

20 



23 

Yeddo. 

14 

14 

42 F. 

9 

18 

40 F. 

5 

3 

28 

Yokohama. 

14 

14 

43 

9 

18 

41 

4 

24 

14 

York. 


IO 

38 s. 

5 

6 

40 S. 

7 

18 

43 

Yorktown. 


IO 

14 

5 

6 

l6 


To Compute Difference of Time Detween NJew York and. 
Greenwich and any Location not given in Table. 

Rule. —Reduce longitude of location to time, and if it is W. of as¬ 
sumed meridian it is Slow ; if E., it is Fast. 

If difference for New York is required, and it exceeds 4 h. 56 m. 
2 sec., subtract this time, and remainder will give difference of time, S.; 
and if it (4 h. 56 m. 2 sec.) does not exceed it, subtract difference from it, 
and remainder will give difference of time, F. 

























































































84 


TIDES 


TIDES. 

Tide-Table fox* Coast of TJnited States, 


Showing Tune of High-water at Full and New Moon , termed Establish¬ 
ment o f the Port , being Mean Interval between Time of Moon's Transit 
and Time of High-water. ( U. S. Coast and Geodetic Survey.) 


Locations and Time. 

Spring. 

Neap. 

Locations and Time. 

Spring. 

Neap. 

COAST FROM EASTPORT 





CHESAPEAKE BAY AND 





TO NEW YORK. 

h. 

m. 

Feet. 

Feet. 

RIVERS. 


h. 

m. 

Feet. 

Feet. 

Eastport. 

...Me. 

II 

30 

15 


Old Pt. Comfort§ 

..Ya. 

8 

w 

3 

2 

Campo Bello*. 

U 

II 


25 


Cape Henry*... 

44 

7 

51 

6 


Portland. 

U 

II 

25 

9.9 

7.6 

Point Lookout.. 

. Md. 

12 

58 

1.9 

.7 

Cape Ann*.... 

i 4 

II 

30 

II 


Annapolis. 

44 

w 

4 

I 

.8 

Portsmouth... 

.N.H. 

II 

23 

9.9 

7.2 

Bodkin Light... 

44 

18 

8 

1 3 

.8 

Newburyport.. 

.Mass. 

II 

22 

9 - 1 

6.6 

Baltimore. 

44 

18 

59 

1. e, 

•9 

Salem. 

U 

II 

13 

10.6 

7.6 

James R. (CityPt 

)>Va 

14 

37 

3 

2-5 

Cape Cod*. 

U 

II 

3 ° 

6 


Richmond. 

4 4 

16 58 

3-4 

2-3 

Boston Light .. 

u 

II 

12 

IO.9 

8.1 







Bostonf. 

u 

II 

27 

10.3 

8-5 

COASTS OF N. AND S. 





Nantucket. 

u 

12 

24 

3-6 

2.6 

CAROLINA, GEOR-GIA, 





Edgartown .... 

u 

12 

16 

2-5 

1.6 

AND FLORIDA. 






Holmes’s Hole. 

4 i 

II 

43 

1.8 

i -3 

Hatteras Inlet.. 

N.C. 

7 

4 

2.2 

1.8 

Tarpaulin Cove 

u 

8 

4 

2.8. 

1.8 

Cape Hatteras .. 

44 

9 

I 

5 


Wood’s Hole, n. 

side. 

7 

5 ° 

4-7 

3 - 1 

Beaufort. 

44 

7 

26 

3-3 

2.2 

N. Bedford (Dump-) 




2 8 

Smithv’le (C.Fear) “ 

7 

J 9 

5-5 

3-8 

ing Rock) 


7 

57 

4-0 


Charleston!! (C. 

H. ) 





New York!.... 

. N.Y. 

8 

13 

5-4 

34 

Wharf.S.C. ( 

7 

20 

O 

4.1 

Albany*. 

.4 

3 

3 ° 

I 


Fort Pulaski.... 


7 

20 

8 

5-9 







Savannah . 

4 4 

8 

13 

7.6 

5-5 

LONG ISLAND SOUND. 





St. Augustine... 

Fla. 

8 

21 

4.9 

3-6 

Newport. 

. R. I. 

7 

45 

4.6 

3 -i 

Cape Florida ... 

4 4 

8 84 

1.8 

1.2 

Point Judith... 

44 

7 

32 

3-7 

2.6 

Key West. 

44 

9 

22 

1.6 

I 

Montauk Point. 

. N.Y. 

8 

20 

2.4 

1.8 

Tampa Bay .... 

44 

II 

21 

1.8 

I 

Watch Hill .... 

.. R.I. 

9 


3 - 1 

2.4 

Cedar Keys. 

44 

13 

15 

3-2 

1.6 

Providence*... 

1. 

8 

25 

5 








Stonington .... 

... Ct. 

9 

7 

3-2 

2.2 

WESTERN COAST. 





Little Gull Isl’d 

. N.Y. 

9 38 

2.9 

2-3 

San Diego. 

Cal. 

9 38 

5 

2-3 

New London... 

... Ct. 

Q 

28 

Q. I 

2. I 

San Pedro 

44 





New Haven.... 

44 

II 

l6 

6.2 

5-2 

Cuyler’s Harbor 

4 4 

y 

9 

oy 

25 

T / 

2.8 

Bridgeport .... 

4 4 

II 

II 

8 

4-7 

San Luis Obispo. 

44 

10 

8 

4.8 

2.4 

Oyster Bay.... 

. N.Y. 

II 

7 

9.2 

5-4 

Monterey. 

44 

10 

22 

4-3 

2-5 

Sand’s Point... 

41 

II 

1 3 

8.9 

6.4 

South Farallone 

44 

10 

37 

4.4 

2.8 

New Rochelle.. 

44 

II 

22 

8.6 

6.6 

San Francisco.. 

44 

12 

6 

4-3 

2.8 

Throg’s Neck.. 

44 

II 

20 

9.2 

6.1 

Mare Island .... 

44 

13 

40 

5-2 

4.1 

Hell Gate*. 

44 

9 

35 

6 


Benicia. 

44 

14 

IO 

5 -i 

3-7 







Ravenswood.... 

44 

12 

36 

7-3 

4-9 

COAST OF NEW JERSEY. 





Bodega. 

44 

11 

17 

4-7 

2.7 

Cold Spring Inlet.N.J. 

7 

32 

5-4 

3-6 

Humboldt Bay.. 

44 

12 

2 

5-5 

3-5 

Sandy Hook... 

.. N. J. 

7 

2 9 

5-6 

4 

Astoria. 

.Or. 

12 

42 

7-4 

4.6 

Amboy. 

44 

8 

15 

5 


Nee-ah Harbor, Wash. 

12 

33 

7-4 

4.8 

Cape May Landing “ 

8 

*9 

6 

4-3 

Port Townshend 

44 

3 49 

5-5 

4 

Egg Harbor*... 

44 

9 

34 

5 














MISCELLANEOUS. 





DELAWARE BAY AND 





Bay of Fundy*. 

N.S. 

12 


60 


RIVER. 






Blue Hill Bay*. 

44 

11 


12 


Delaware Breakwater 

8 


4-5 

3 

St. John’s*. 

4 4 

12 


3 ° 


Higbee’s (Cape May).. 

8 

33 

6.2 

3-9 

Kingston*. 

Jam. 

2 

3 ° 

2 


Egg Isl’d Light 

. .N.J. 

9 

4 

7 

5 -i 

Halifax*.N. S. 



7-5 

• 5 

New Castle .... 

.. Del. 

II 

53 

6.9 

6.6 

Pensacola*. 

.Fla. 



i -5 

•4 

Philadelphia... 

.Penn. 

13 

44 

6.8 

5 -i 

Galveston*. 

Tex. 



1.6 

.8 


* Refers to rise and fall of tide alone. I % § | see p. 83. 

Note. —Mean interval ba3 been increased 12 h. 26 min. (half a mean lunar day) for some ports in Del¬ 
aware River and Chesapeake Bay, to give succession of times from the mouth ; hence 12 h. 26 min. is 
to lie subtracted from the Establishments which are greater than that, to give the interval required. 

































































TIDES 


35 


Bench. Marks referred, to in preceding Table. 

t Boston —Top of wall or quay, at entrance to dry-dock in Charlestown navy- 
ya*rd, 14.76 feet above mean low-water. 

i New \ ork. Lower edge of a straight line, cut in a stone wall, at head of wooden 
wharf on Governor’s Island, 14.56 feet above mean low-water. 

§ Old Point Comfort, \ a. —A line cut in wall of light-house, one foot from ground, 
on southwest side, n feet above mean low-water. 

II Charleston, S. C. — Outer and lower edge of embrasure of gun No. 5, at Castle 
Pinckney, 10.13 feet above mean low-water. 


Establishment of - the Port for several Locations 


Port. [ Time. 


, L 

Amsterdam.3 

Antwerp. 4 

Beachy Head.. Eng. 11 


Belfast. IO 

Bordeaux. 6 

Bremen. 6 

Brest Harbor. 3 

Bristol. I 7 

Bristol Quay. 6 

Cadiz. x 

Calais..... xx 

Calf of Man. n 


Cape St. Vincent.... 2 


m. 

25 

50 

43 

50 

47 

21 

27 

40 

49 

5 

3° 


Europe, etc. 


| Port. 

Time. 

Port. 

Chatham. 

h. m. 

1 2 

7 49 
4 

10 46 

11 12 

11 12 

11 30 

1 14 

8 

10 11 

6 29 

3 57 

2 30 

Liverpool . 

Cherbourg. 

Clear Cape. 

London Bridge. 

Newcastle. 

Cowes. 

Dover Pier. 

Dublin Bar. 

Portsmouth D.-yard, 
Eng. 

Quebec . 

Funchal. 

Gravesend.Eng. 

Greenock. 

Ramsgate Pier. 

Rye Bay.Eng. 

Sheerness. 

| Holyhead. 

Hull.Eng. 

Land’s End. 

Lisbon. 

Sierra Leone. 

Southampton. .Eng. 
Thames R.,mo’tli “ 
Woolwich.“ 


in 

Time. 

h. m. 
11 16 
2 7 

1 22 

11 41 
8 

10 27 

11 20 

57 
8 15 

11 40 

12 

2 15 


Rise and Fall of Tides in Grnlf of ^Mexico. 


Locations. 

Mean. 

Spring. 

Neap. 

Locations. 

Mean. 

Spring. 

Neap. 


Feet. 

Feet. 

Feet. 



Feet. 

Feet. 

Feet. 

St. George’s Island_Fla. 

I. I 

1.8 

.6 

Isle Derniere. 

...La. 

1.4 

1.2 

•7 

Fort Morgan (Mobile) 




Entrance to Lake 

Cal-) 




Bay).Ala. ) 

I 

i -5 

•4 

casieu. 

.La.) 

i -5 

i-f 

.6 

Cat Island.Miss. 

i -3 

1.9 

.6 

Aransas Pass. 

U 

I. I 

1.8 

.6 

Southwest Pass.La. 

I. I 

1.4 

•5 

Brazos Santiago... 

U 

•9 

1.2 

•5 


Tides of Grnlf of IVEexico. 


On Coast of Florida, from Cape Florida to St. George’s Island, near Cape San Bias, 
the tides are of the ordinary kind, but with a large daily inequality. From St. 
George’s Island, Apalachicola entrance, to Derniere Isle, the tides are usually of the 
single-day class, ebbing and flowing but once in 24 (lunar) hours. At Calcasieu en¬ 
trance, double tides reappear, and except for some days about the period of Moon’s 
greatest declination, tides are double at Galveston, Texas. At Aransas and Brazos 
Santiago the single-day tides are as perfectly well marked as at St. George’s, Pensa¬ 
cola., Fort Morgan, Cat Island, and the mouths of the Mississippi. For some 3 to 5 
cfciys, however, about the time when the Moon’s declination is nothing, there are 
generally two tides at all these places in 24 hours, the rise and fall being quite small. 

Highest high and lowest low waters occur when greatest declination of Moon 
happens at full or change. Least tides when Moon’s declination is nothing at first 
or last quarter. 

Tides of Bacific Coast. 

On Pacific coast there is, as a general rule, one large and one small tide during 
each day, heights of two successive high-waters occurring, one A.M., and other 
P. M. of same 24 hours, and intervals from next preceding transit of Moon are very 
different. These inequalities depend upon Moon’s declination. When Moon’s de¬ 
clination is nothing, they disappear, and when it is greatest, either North or South, 
they are greatest. The inequalities for low water are not same as for high, though 
they disappear, and have greatest value at nearly same time. 

When Moon’s declination is North, highest of two high tides of the 24 hours oc¬ 
curs at San Francisco, about 11.5 hours after Moon’s southing (transit); and when 
declination is South, lowest of the two high tides occurs about this interval. 

Lowest of two low-waters of the day is the one which follows next highest high- 
water. 


FT 
































































86 


STEAMING DISTANCES 


STEAMING DISTANCES. 

Distances between various Ports of United. States 

and. Canada. 


By Cake, River, and Canal. 


Locations. 

Lake 

and 

River. 

Canal. 

Total. 

Locations. 

Lake 

and 

River. 

Canal. 

Total. 


Miles. 

Miles. 

Miles. 


Miles. 

Miles. 

Miles. 

Duluth to Buffalo... 

1024 

I 

1025 

Chicago to NewYork, 




Chicago to Buffalo .. 

925 

— 

9 2 5 

via Oswego. 

ii 95 

232 

1427 

Duluth to Oswego... 

IX 33 

27 

1160 

Chicago to Montreal. 

1190 

71 

1261 

Chicago to Oswego.. 

x °34 

26 

1060 

Buffalo to Colborne, 




Duluth to New York, 




via Welland Canal. 

— 

26.77 

26.77 

via Buffalo. 

1166 

353 

1519 

Buffalo to New York. 

142 

352 

494 

via Oswego. 

1294 

233 

1527 

Welland Canal to 




Duluth to Montreal. 

1289 

72 

1361 

Montreal. 

3 ° 4-5 

7°-5 

375 

Chicago to New 




Montreal to Kingston 

126.25, 

120 

246.25 

York, via Buffalo. 

1067 

352 

x 4 r 9 

Ottawa to Kingston. 

— 

126.25 

126.25 


Distances between various Ports and New York 

and London. 

Not included in preceding Table. 


Ports. 

Miles. 

Miles. 

Ports. 

Miles. 

Miles. 

Ports. 

Miles. 

Miles. 

Alexandria... 

N.Y. 

4893 

Lond. 

3 i°2 

Cape Race. 

N. Y. 

1 004 

Lond. 

2 249 

New Orleans. 

N.Y. 

1790 

Lond. 

4 73 ° 

Amsterdam.. 
Barbadoes... 

3291 

x8 55 

262 
3 812 

Cowes. 

Funchal. 

3092 
2 760 

200 

1 3°3 

Norfolk. 

Pensacola... 

308 

1623 

3 447 

4 654 

Batavia. 

Bermudas ... 

8972 

682 

II 492 

3!42 

Galway. 

Gibraltar. 

2 720 

3 260 

721 

1 325 

Philadelphia. 
Quebec . 

262 

1360 

3 4°4 
3080 

Bombay. 

Boston. 

Bremen. 

Bristol. 

8522 

356 

3428 

2979 

10703 

3 ° 3 ° 
408 

5 ° x 

Glasgow. 

Halifax. 

Havana. 

Hobart Town.. 

2 9 1 3 

59° 

1161 
9187 

765 

2 706 

4197 

11368 

Queenstown. 
Rio Janeiro.. 
St. Johns.... 
Southampton 

2780 

497 ° 

1064 

3 io 3 

551 
5 200 
2 214 

211 

Buenos Ayres 

6010 

6 280 

Kingston, Jam. 

1456 

4 3°5 

Swan River.. 

8480 

10 661 

Cadiz. 

Calcutta. 

3 X 25 

935 ° 

1 i *5 
ii 53 1 

Lima. 

Madras. 

10050 
8 707 

IO I49 

10 888 

Tortugas_ 

Washington. 

1151 

461 

4182 

3 612 


Distances between various Ports of England, 
Canada, United States, etc. 

, Not included in preceding Table. 


Ports. 


Halifax to 

Liverpool. 

St. Thomas. 

St. Johns, N. F. .. 

Quebec to Glasgow. 

Liverpool to 

Boston. 

Quebec. 

Philadelphia. 

Callao. 

Fastnet. 

Cape Race. 

Aspinwall. 

Port Said. 

Melbourne. 

Rio Janeiro. 

San Francisco... 
via Panama.... 
maTehuantopec 


Miles. 


2563 

1 563 

520 

2563 

2 955 
2855 
3147 

n 379 
283 
1992 
4650 

3 290 
13290 

5 125 
13 800 

7 378 

6 400 


Ports. 


Miles. 


Liverpool to 

Havana . 

Portland. 

Baltimore..... 

N. Orleans to Havana 
Cape Race to 

Fastnet. 

Halifax. 

Boston. 

St. Johns, N. F., to 

Quebec. 

Boston. 

Greenock. 

Bermudas to Nassau. 
Panama to 
San Juan del Sud . 
Gulf of Fonseca... 

Acapulco. 

Manzanilla. 


4100 

2770 

3400 

570 

1711 

457 

835 

891 

890 

1848 

804 

57 ° 

739 

1416 

1724 


Ports. 


Panama to 

San Diego. 

Monterey. 

San Francisco 
San Francisco to 
San Juan del Sud. 

Acapulco.. 

Manzanilla....... 

San Diego. 

Monterey. 

Humboldt. 

Columbia R. Bar.. 

Vancouver. 

Portland. 

Port Townshend .. 

Victoria.... 

Yokohama. 

Honolulu. 

Honolulu to Callao.. 


Miles. 


2897 

3 I 9 8 

3240 

2685 

1841 

*543 

474 

io 5 

200 

53 ° 

638 

650 

732 

7 i 5 

475 ° 

2080 

5 X 45 






































































































STEAMING DISTANCES. 


87 


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FRACTIONS. 


89 


FRACTIONS. 

A Fraction, or broken number, is one or more parts of a Unit. 

Illustration. —12 inches are 1 foot. Here, 1 foot is unit, and 12 inches its parts; 
3 inches therefore, are one fourth of a foot, for 3 is fourth or quarter of 12. 

A Vulgar Fraction is a fraction expressed by two numbers placed one 
above the other, with a line between them; as, 50 cents is the 5 of a dollar. 

Upper number is termed Numerator , the lower Denominator. Terms of a frac¬ 
tion express numerator and denominator; as, 6 and 9 are terms of |. 

A Proper fraction has numerator equal to, or less than denominator; as, etc. 

An Improper fraction is reverse of a proper one; as, etc. 

A Mixed fraction is a compound of a whole number and a fraction; as, 5 1 -, etc. 

A Compound fraction is fraction of a fraction; as, ^ of etc. 

A Complex fraction is one that has a fraction for its numerator or denominator, 
1 5 . „ 31 

or both; as, _ 2 _, or J*_, or _, or etc. 

6 4 3 6 

<f 

Note.—A Fraction denotes division, and its value is equal to quotient, obtained by 
dividing numerator by denominator; thus, J^ 2 - is equal to 3, and - 2 ^- is equal to 4I. 


Reduction of Fractions. 

To Compute Common Measure 01 * greatest 1ST umber 
tliat Avill clivide Two or more jST- umbers without a 
Remainder. 

Rule. —Divide greater number by less; then divide divisor by remainder; and so 
on, dividing always last divisor by last remainder, until there is no remainder, and 
last divisor is greatest common measure required. 

Example i. —What is greatest common 936) 1908 (2 
measure of 1908 and 936 ? 1872 

36) 936 (26 
7 2 

216. Hence 36. 

2.—How many squares can there be obtained in an area of 90 by 160 feet? 

Here 10 is greatest common measure. 

Hence, = 16, and AO _ 9. therefore 16 X 9 = 144. 

To Compute least Common. NJultiple of Two or more 

Numbers. 

Rule. —Divide given numbers by any number that will divide the greatest num¬ 
ber of them without a remainder, and set quotients with undivided numbers in a 
line beneath. 

Divide second line in same manner, and so on, until there are no two numbers 
that can be divided; then the continued product of divisors and last quotients will 
give common multiple required. 

Example. — What is least 5 ) 40.50.25 
common multiple of 40, 50, 5 ) 8 . 10 . 5 
and 25 ? 2) 8 . 2 . 1 

4 . 1 . 1. Then 5X5X2X4XiXi = 200. 

To Reduce a Fraction to its Lowest Term. 

p ULE .—Divide terms by any number or series of numbers that will divide them 
without a remainder, or by their greatest common measure. 

Example.— Reduce of a foot to its lowest terms. 

= = = or 9 ins. 

H* 






9 o 


FRACTIONS. 


To Reduce a Mixed. Fraction. to its Equivalent, an Im¬ 
proper Fraction. 

Rule. — Multiply whole number by denominator of fraction and to product add 
numerator; then set that sum above denominator. 

Example i.— Reduce 2df to a fraction. 23 X 2 __ 112 —ZS!, 
u 0 0 6 3 

2.—Reduce inches to its value in feet. 123 -f- 6 = 20 § = x foot 8|- ins. 


To Reduce a Complex Fraction to a Simple one. 


Rule.— Reduce the two parts both to a simple fraction, multiply numerator of re¬ 
duced fraction by denominator of reduced denominator, and denominator of numer¬ 
ator fraction by numerator of denominator fraction. 


Example.— Simplify complex fraction 


2 f 

4 ' 



8 

'3 

24 

“S' 


8 X 5 — 4 ^__ 5 _ 

3 X 24 = 72 ~ 9 ' 


To Reduce a "WTiole iN'nin'ber to an Equivalent Fraction 
Raving a given Denominator. 

Rule. —Multiply whole number by given denominator, and set product over said 
denominator. 

Example. —Reduce 8 to a fraction, denominator of which shall be 9. 

8 X 9 = 72; then result required. 


To Reduce a Compound Fraction to an Equivalent 

Simple one. 

Rule. —Multiply all numerators together for a numerator, and all denominators 
together for a denominator. 

Note.— When there are terms that are common, they may be cancelled. 

Example.— Reduce A of A of |- to a simple fraction. 

Y x 4" x 1 " — '2 6 4 — b 0r > h x ± x § = b hy cancellin 9 2 ’s and 3 ’s. 


To Reduce Fractions of different Denominations to 
Equivalents Raving a Common Denominator. 

Rule.— Multiply each numerator by all denominators except its own for new nu¬ 
merators; and multiply all denominators together for a common denominator. 

Note. — In this, as in all other operations, whole numbers, mixed or compound 
fractions, must first be reduced to form of simple fractions. 

2. When many of denominators are same, or are multiples of each other, ascertain 
their least common multiple, and then multiply the terms of each fraction by quo¬ 
tient of least common multiple divided by its denominator. 

Example. — Reduce A -|, and to a 1X3X4 = 12 
common denominator. 2X2X4 — 16 

3X2X3=18 

2X3X4 = 24 


— 13. JL6 _ 18 
24 — 24 — 2 4 5 

0r T2 > TS and tV 


^Addition. 


Rule.— If fractions have a common denominator, add all numerators together, 
and place sum over denominator. 


Note.— If fractions have not a common denominator, they must be reduced to 
one. Also, compound and complex must be reduced to simple fractions. 

Example i. — Add ^ and -J together. -)-1- = |- = 1. 


2.—Add A of f of T 6 ^ to 2-J- of f. 


Then, 


1 v* 3 w 6 _ 18 

‘2 x q X Tfi — so- 

18 1 5 1 _ 4 0 8 0 


o 1 of 3 _ 17 v 3 _ 5 1 

2 8 OI T — X 4 — 1T2' 


IF + fi = + 2 5 6 0 — I xlo’ reduced to equivalent fractions having 


a common denominator and thence to its lowest terms. 





FRACTIONS, 


91 


Siob traction.. 

Role.— Prepare fractions same as for other operations, when necessary; then 
subtract one numerator from the other, and set remainder over common denom¬ 
inator. 

Example.— What is difference 6X9 = 54 ) 

between | and I ? 3 X 8 = 24 | = yg — f ¥ = |-|- = T y. 

9 8 8X9 = 72) 


M/ultiplication. 

Rule.— Prepare fractions as previously required; multiply all numerators to¬ 
gether for a new numerator, and all denominators together for a new denominator. 

Example i.— What is product of and ||? = ^ = 

2.—What is product of 6 and ^ of 5 ? -*j-x§of5=:^x^ = -^p = 20 . 

Division. 

Rule.— Prepare fractions as before; then divide numerator by the numerator, 
and denominator by the denominator, if they will exactly divide; but if not, invert 
the terms of divisor, and multiply dividend by it, as in multiplication. 

Example i. —Divide by -f. 5. _ 6. _ x 2.. 

_ OividP 5 w 2 5 _i_ 2 — 5 v 15.— 15 v 5 7 5 _ 2 5 . .1 

2 ’ Divide ¥ t>y T -q . T g _ ¥ x -g- — -9- X 2 — - f¥ — -g- — 4 g. 


Application of Reduction of Fractions. 

To Compute Value of a Fraction, in Parts of a Whole 

Number. 

r ule . —Multiply whole number by numerator, and divide by denominator; then, 
if anything remains, multiply it by the parts in next inferior denomination, and 
divide by denominator, as before, and so on as far as necessary; so shall the quo¬ 
tients placed in order be value of fraction required. 

Example i.—W hat is value of \ of of 9? 

i of f = f, and § X * = \ 8 - = 3 - 

2.—Reduce of a pound to an avoirdupois ounce. 4) 3 (° 

4 16 ounces in a lb. 

4) 48 (12 ounces. 

To Reduce a Fraction, from one Denomination to 

another. 

Rule.— Multiply number of required denomination contained in given denomina¬ 
tion by numerator if reduction is to be to a less na?ne, but by denominator if to a 
greater. 

Example i.—R educe 1 of a dollar to fraction of a cent. 

\ X IOO = 1 ^ a = -*^. 

2. —Reduce A of an avoirdupois pound to fraction of an ounce. 

1 T f. 1 6 .8 n 2 

6 X 10 — 6 — 3 — ^3* 

3. —Reduce A of ^ of a mile to the fraction of a foot. 

2„f3_ 6 s, - 31680 — 2640 

S 01 T — 12 x 52«o — j-g— — j . 

For Rule of Three in Vulgar Fractions, see Decimals, page 94. 






9 2 


DECIMALS. 


DECIMALS. 

A Decimal is a fraction, having for its denominator a unit with 
as many ciphers annexed as the numerator has places; it is usually ex¬ 
pressed by writing the numerator only, with a point at the left of it. Thus, 

A is -4; TAG is ; 8 S; TGGGG is -°°75; and y^oGo is -00125. When there is 
a deficiency of figures in the numerator, prefix ciphers to make up as many 
places as there are ciphers in denominator. 

Mixed numbers consist of a whole number and a fraction; as, 3.25, w ? hich is the 


same as 3 -^ 0 5 0 , 


or 


325 

lbo- 


Ciphers on right hand make no alteration in their value; for .4, .40, .400 are deci¬ 
mals of same value, each being jL., or J-. 


.Addition.. 

Rule. — Set numbers under each other according to value of their places, as in 
whole numbers, m which position the decimal points will stand directly under each 
other; then begin at right hand, add up all the columns of numbers as in integers, 
and place the point directly below all the other points. 

Example. —Add together 25.125 and 293.7325. 25.125 

293-7325 

318.8575 sum. 


Subtraction. 

Rule. —Set numbers under each other as in addition; then subtract as in whole 
numbers, and point off decimals as iu last rule. 

Example. —Subtract 15.15 from 89.1759. 89.1759 

1515 

74.0259 remainder. 


^Multiplication. 

Rule. —Set the factors, and multiply them together same as if they were whole 
numbers; then point off in product just as many places of decimals as there are 
decimals in both factors. But if there are not so many figures in product, supply 
deficiency by prefixing ciphers. 

Example.— Multiply 1.56 by .75. 1.56 

•75 

780 

1092 

1.1700 product. 


33 y Contraction. 

To Contract tlie Operation so as to retain only- as many 
Decimal places in Product as may be required. 

Rule. —Set unit’s place of multiplier under figure of multiplicand, the place of 
which is same as is to be retained for the last in product, and dispose of the rest of 
figures in contrary order to which they are usually placed. 

In multiplying, reject all figures that are more to right hand than each multiply¬ 
ing figure, and set down the products, so that their right-hand figures may fall in a 
column directly below each other, and increase first figure 
in every line with what would have arisen from figures 
omitted; thus, add 1 for every result from 5 to 14, 2 from 
15 to 24, 3 from 25 to 34,4 from 35 to 44, etc., and the sum 
of all the lines will be the product as required. 

Example. —Multiply 13.57493 by 46.2051, and retain only 
four places of decimals in the product. 


Note.— When exact result is required, increase last figure with what would have arisen from all the 
figures omitted. 


I 3-574 93 
1 502.64 

54 299 72 

8 144 96 -f- 2 for 18 
271 50-f- 2 “ 18 
6 79 + 4 “ 35 
_14+1 “ 5 


627.23 11 













DECIMALS. 


93 


Division. 

Rule. —Divide as in whole numbers, and point off in quotient as many places for 
decimals as decimal places in dividend exceed those in divisor; but if there are not 
so many places, supply deficiency by prefixing ciphers. 

Example. Divide 53 by 6.75. 6.75) 53.00000 (=7.851-!-. 

Here 5 ciphers are annexed to dividend to extend division. 

By Contraction. 

Rule.—T ake only as many figures of divisor as will be equal to number of figures, 
both integers and decimals, to be in quotient, and ascertain how many times they 
may be contained in first figures of dividend, as usual. 

Let each remainder be a new dividend; and for every such dividend leave out 
one figure more on right-hand side of divisor, carrying for figures cut off as in Con¬ 
traction of Multiplication. 

Note.— When there are not so many figures in divisor as there are required to bo in quotient, con¬ 
tinue first operation until number of figures in divisor are equal to those remaining to be found in quo¬ 
tient, after which begin the contraction. 


Example.— Divide 2508.92806 

02.410315) 2508.928 06 (27.1498 

13.849 

912 

by 92.4x035, so as to have only 

1848 207 -|- i 

9241 

S32+4 

four places of decimals in quo- 

660 721 

4 608 

80 

tient. 

646 872 -f- 2 

3 696 

74 + 2 


x 3 849 

912 

6 


Reduction of Decimals. 


To Reduce a Vulgar Fraction to its Equivalent Decimal. 

Rule. —Divide numerator by denominator, annexing ciphers to numerator to ex¬ 
tent that may be necessary. 

Example. —Reduce to a decimal. 5) 4.0 

“Tb 

To Compute Value of a Decimal in Terms of an Inferior 

Denomination. 

Rule. —Multiply decimal by number of parts in next lower denomination, and 
cut off as many places for a remainder, to right hand, as there are places in given 
decimal. 

Multiply that remainder by the parts in next lower denomination, again cutting 
off for a remainder, and so on through all the parts of integer. 

Example 1.—What is value of .875 dollars? .875 

100 

Cents, 87.500 
10 


Mills, 5.000 = 87 cents 5 mills. 

2. —What is volume of .140 cube feet in inches? 

. 140 

1728 cube inches in a cube foot. 

241 920 cube ins. 

3. —What is value of .00129 °f a f° ot? -01548 ins. 


To Reduce a Decimal to an Equivalent Decimal of a 
Higher Denomination. 

Rule.— Divide by number of parts in next higher denomination, continuing op¬ 
eration as far as required. 


Example i. —Reduce 1 inch to decimal of a foot. 


12 


1.000 00 


•083 33 + foot. 


2.—Reduce 14" 12'" to decimal of a minute. 14" 12"' 

60 


60 852/" 

60 14. 2" 

. 236 66'+ minute. 
















94 DECIMALS.-DUODECIMALS.-MEAN PROPORTION. 


When there are several numbers , to be reduced all to decimal of highest. 
Rule.— Reduce them all to lowest denomination, and proceed as for one denomi¬ 
nation. Feet. Ins. Be. 

Example. — Reduce 5 feet 10 inches and 3 5 10 3 

barleycorns to decimal of a yard. 23. 


70 

3 


3 2 I 3- 
12 71. 

3 _ 5 - 9 l66 

1.9722-}- yards. 


Rule of Three. 

Rule. — Prepare the terms by reducing vulgar fractions to decimals, compound 
numbers to decimals of the highest denomination, first and third terms to same 
denomination; then proceed as in whole numbers. 

Example. —If .5 of a ton of iron cost .75 of a dollar, .5 : .75 :: .625 

what will .625 of a ton cost? .625 

• 5 ) -46875 

•9375j dollar. 


DUODECIMALS. 


In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, 
inches, and twelfths of an inch. 

Rule. —Set dimensions to be multiplied together one under the other, feet under 
feet, inches under inches, etc. 

Multiply each term of multiplicand, beginning at lowest, by feet in multiplier, and 
set result of each immediately under its corresponding term, carrying 1 for every 
12 from one term to the other. In like manner, multiply all multiplicand by inches 
of multiplier, and then by twelfth parts, setting result of each term one place farther 
to right hand for every multiplier. And sum of pi’oducts will give result. 

Example.— How many square inches are Feet. ins. Twelfths, 
there in a board 35 feet 4.5 inches long and 12 35 4 6 

feet^T inches wide ? 12 3 4 

424 6 o 

8 10 1 6 

11 960 

434 3 11 o o 


"Value of Duodecimals in Square 


1 Foot 
1 Inch 


Feet and. Indies. 
Sq..Ft. Sq. Ins. 

Ta of I twelfth = or .083 333, etc. 

006 944, etc. 


TS of T2 of 


— 2TT7 3 6 


Sq. Ft. Sq. Ins. 

.=: 1 or 144. 

— -X- “ T? 

.— Ta I2- 

I Twelfth. yyy ** I. 

Illustration. — What number of square inches are there in a floor 100 feet 
6 inches long and 25 feet 6 inches and 6 twelfths broad? 

2566 feet 11 ins. 3 twelfths — 2566 feet 135 ins. 


MEAN PROPORTION. 

Mean Proportion is proportion to two given numbers or terms. 

Rule.— Multiply two numbers or terms together, and extract square root of their 
product. 

Example_ What is mean proportionate velocity to 16 and 81 ? 

16 X 81 — 1296, and f 1296 = 36 mean velocity. 


















RULE OF THREE.-COMPOUND PROPORTION. 


RULE OF THREE. 

Rule of Three. — It is so termed because three terms or numbers are 
given to ascertain a fourth. 

It is either Direct or Inverse. 

It is Direct when more requires more, or less requires less; thus, if 3 bar¬ 
rels of flour cost $18, what will 10 barrels cost? 

In this case Proportion is Direct , and stating must be, 

As 3 : 10 :: 18 • 60. 

It is Inverse when more requires less, or less requires more; thus, if 6 men bnild 
a certain quantity of wall in 10 days, in how many days will 8 men build like quan¬ 
tity ? Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men 
perform same work ? 

Here the Proportion is Inverse , and stating must be, 

As 8 : 6 10 : 7.5, and 2 : 3 7 : 10.5. 

The fourth term is always ascertained by multiplying 2d and 3d terms together, 
and dividing their product by 1st term. 

Of the three given numbers necessary for the stating, two of them contain the 
supposition, and the third a demand. 

Rule.— State question by setting down in a straight line the three necessary 
numbers in following manner : 

Let third term be that of supposition , of same denomination as the result, or 4th 
term is to be, making demanding number 2d term, and the other number 1st term 
when question is in Direct Proportion, but contrariwise if in Inverse Proportion; 
that is, let demanding number be 1st term. 

Multiply 2d and 3d terms together, and divide by 1st, and product will give re¬ 
sult, or 4th term sought, of same denomination as 2d term. 

Note. — I f first and third terms are of different denominations, reduce them to same. If, after divis¬ 
ion, there is any remainder, reduce it to next lower denomination, divide by divisor as before, and 
quotient will be of this last denomination. 

Sometimes two or more statings are necessary , which may always he known by 
nature of question. 

Example r.— If 20 tons of iron cost $225, what will Tons. Tons. Dolls. 

500 tons cost ? 20 : 5 °° • • 22 5 

500 

2 1 o) II 2501 o 

5 625 dollars. 

2 .—A wall that is to be built to height of 36 feet, was raised 9 feet by 16 men in 
6 days; how many men could finish it in 4 days at same rate of working? 

Days. Days. Men. Men. 

4 : 6 16 : 24 

Then, if 9 feet requires 24 men, what will 27 men require? 

9 : 27 " 24 : 72 men. 


COMPOUND PROPORTION. 

Compound Proportion is rule by means of which such questions as 
would require two or more statings in simple proportion (Rule of Three) 
can be resolved in one. 

As rule, however, is but little used, and not easily acquired, it is deemed prefer¬ 
able to omit it here, and to show the operation by two or more statings in Simple 
Proportion. 

Illustration i.— How many men can dig a trench 135 feet long in 8 days, when 
16 men can dig 54 feet in 6 days? 


Feet. Feet. Men. Men. 

First .As 54 : 135 " 16 : 40 

Days. Days. Men. Men. 
Second .As 8 : 6 40 : 30 







96 COMPOUND PROPORTION.-INVOLUTION.—EVOLUTION. 


2.—If a man travel 130 miles in 3 days of 12 hours each, how many days of 10 
hours each would he require to travel 360 miles? 


Miles. Miles. Days. Days. 

First .As 130 : 360 3 : 8.307-}- 

Hours. Hours. Days. Days. 
Second .As 10 : 12 :: 8.307 : 9.9684 


3.—If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, 
in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 
6 deep? 120 days. 

By Caixcellation. 

Rule. —On right of a vertical line put the number of same denomination as that 
of required answer. 

Examine each simple proportion separately, and if its terms demand a greater 
result than 3 d term , put larger number on right and lesser on left of line ; but if its 
terms demand a less result than 3 d term , put smaller number on right and larger 
on left of line. 

Then Cancel the numbers divisible by a common divisor, and evolve the 4th term 
or result required. 

Take Illustration 1, page 95: 3d term, or term of supposition of same denomination 
as required result, 16 men. 


Statement. 135 feet require more men than 54 feet , 
16 and 8 days less men than 6 days. 

54 135 

8 6 2X5X3 = 30 men. 


Result by Cancellation. 

2 

% h m 5 

£ ^3 


Illustration 3 .— 3 d term, 15 days. 


Statement. 


60 

8 

30 

6 

3 


15 

12 

12 

300 

8 

6 


60 men require less days than 12 men, 
8 hours more days than 12 hours, 300 feet 
more days than 30 feet, 8 feet more days 
than 6 feet, and 6 feet more days than 
3 feet. 


3 X 4 X 10 = 120 days. 


Result by Cancellation. 


% 

V 

3 


n 

n 3 
n 4 
10 


0 


INVOLUTION. 

Involution is multiplying any number into itself a certain number of 
times. Products obtained are termed Poioers. The number is termed the 
Root , or first power. 

When a number is multiplied by itself once, product is square of that 
number; twice, cube ; three times, biquadrate ; etc. Thus, of the number 5 . 

5 is the Root, or 1st power. 

5X5= 25 “ Square , or zd power, and is expressed <f. 

5X5X5 = 125 “ Cube, or 3d power, and is expressed 53. 

5X5X5X5 = 625 “ Biquadrate, or 4th power, and is.expressed 54 

The lesser figure set superior to number denotes the power, and is termed the 
Index or Exponent. 

Illustration i. —What is cube of 9? 

2. —What is cube of f ? 

3. —What is 4th power of 1.5? 


729. 

2 7 
6T- 
5.0625. 


EVOLUTION. 

Evolution is ascertaining Root of any number. 

Sign y placed before any number indicates that square root of that number is re¬ 
quired or shown. 

Same character expresses any other root by placing the index above it. 

Thus, y 2 5 == 5; 4 + 2 = V 36- 

^27 = 3, and ^64= 4. 

Roots which only approximate are termed Surd Roots. 










EVOLUTION. 


97 


To Extract Sqxiare Root. 

Rule.—P oint off given number from units’ place, into periods of two figures each. 

Ascertain greatest square m left-hand period, and place its root in quotient; sub¬ 
tract square number from this period, and to remainder bring down next period 
for a dividend. 

Double this root for a divisor; ascertain how many times it is contained in divi¬ 
dend, exclusive of right-hand figure, which, when multiplied by number to be put 
to right hand of this divisor, product will be equal to, or next less than dividend; 
place result in quotient, and also at right hand of divisor. 

Multiply divisor by last quotient figure, and subtract product from dividend; 
bring down next period, and proceed as before. 

Note. —Mixed decimals must be poiuted off both ways from units. 


Example i.— What is square root of 2? 

11 2.000000 (i- 4 i-(-. 

1 1 


24 IIOO 

41 

96 

281 

1 4 °° 

I 

1 281 


282 I 11900 


2. What is square root of 144? 

ij 144 (12 
i| 1 

221044 
I 44 


Squiare Roots of Fractions. 

Rule.— Reduce fractions to their lowest terms, and that fraction to a decimal, 
and proceed as in whole numbers and decimals. 

N ote .—When terms of fractions are squares, take root of each and set one above the other; as 
A is square root of 

Example.— What is square root of ? .866025 4. 


To Compute 4:fh. or 8tli Root of a TSTnm'ber, etc. 
Rule. —For the 4th root extract square root twice, and for 8th root thrice, etc. 


To Extract Cube Root. 

Rule.—F rom table of roots (page 272) take nearest cube to given number, and 
term it the assumed cube. 

Then, as given number added to twice assumed cube, is to assumed cube added 
to twice given number, so is root of assumed cube to required root, nearly; and by 
using in like manner the root thus found as an assumed cube, and proceeding in 
like manner, another root will be found still nearer; and in like manner as far as 
may be deemed necessary. 

Example.— What is cube root of 10 517.9? 

Nearest cube, page 272; 10648, root 22. 10648. 10517.9 

2 2 

21296 21035.8 
xo 517.9 10648. 

31 813.9 • 3 1 683.8 :: 22 : 21.9-f-. 


To Ascertain or to Compute tlxe Scpuare or Cube Roots of 
Roots, Whole Numbers, and. of Integers and Decimals, 
see TaTole of Squares and Cubes, and Rules, pp. 2/2, 300. 


To Extract any Root whatever. 

Let P represent number. I Let A represent assumed power, r its root. 

n “ index of the power. | R “ required root of P. 

Then, as sum of n -|- 1 X A and n — 1 X P is to sum of n ~f-1 X P aud n — 1 X A, 
so is assumed root r to required root R. 

Illustration. —What is cube root of 1500? 

Nearest cube, page 272, is 1331, root n. 

P = 1500, n — 3, A = 1331, r — 11; 
then, nfiXA = 5324, n-fiXP = 6000 
n — 1 x P = 3000, n — 1 X A = 2662 

8324 8662:: 11: 11.446-f-. 

I 












•POSITION. 


98 EVOLUTION.—PROPERTIES OF NUMBERS.- 

To Compute the Root of an Even. Tower greater than 
any given in Table of Square and. Cube Roots. 
Rule.—E xtract square or cube root of it, which will reduce it to half the given 
power; then square or cube root of that power; and so on until required root is ob¬ 
tained. 

Example 1.—Suppose a 12th power is given; the square root of that reduces it to 
a 6th power, and the square root of 6th power to a cube. 

2.—What is biquadrate, or 4th root, of 2 560000? 

y /2 560 000 —1600, and y /1600 = 40. 

Note. —For other rules for extraction of roots see pp. 301-4. 


PROPERTIES OF NUMBERS. 

1. A Prime Number is that which can only be measured (divided without a re¬ 
mainder) by 1 or unity. 

2. A Composite Number is that which can be measured by some number greater 
than unity. 

3. A Perfect Number is that which is equal to the sum of all its divisors or ali¬ 
quot parts; as 6 = •§. 

4. If sum of the digits constituting any number be divisible by 3 or 9, the whole 
is divisible by them. 

5. A square number cannot terminate with an odd number of ciphers. 

6. No square number can terminate with two equal digits, except two ciphers or 
two fours. 

7. No number, the last digit of which is 2, 3, 7, or 8, is a square number. 


Powers of tire first NTirie 1STrim tiers. 


1st. 

2d. 

3 d- 

4 th. 

5 th. 

6 th. 

7 th. 

8 th. 

9 th. 

I 

I 

I 

I 

I 

I 

I 

I 

1 

2 

4 

8 

16 

32 

64 

128 

256 

512 

3 

9 

27 

81 

243 

729 

2 187 

6 561 

19683 

4 

l6 

64 

256 

I 024 

4096 

16384 

65536 

262 144 

5 

25 

125 

625 

3125 

15625 

78 125 

39 ° 625 

1953125 

6 

36 

216 

1296 

7776 

46656 

279936 

1 679 616 

10077 696 

7 

49 

343 

2401 

16 807 

117649 

823 543 

5 764 801 

40353607 

8 

64 

512 

4096 

32768 

262 144 

2097 152 

16 777 216 

134217728 

9 

81 

729 

6561 

59 °49 

531 44 i 

4 782 969 

43 046 721 

387420489 


POSITION. 

Position is of two kinds, Single and Double, and it is determined by- 
number of Suppositions. 

Single ^Position. 

Rule.—T ake any number, and proceed with it as if it were the correct one; then, 
as result is to given sum, so is supposed number to number required. 

Example i. —A commander of a vessel, after sending away in boats i, L, and i 
of his crew, had left 300; what number had he in command? 

Suppose he had.600. 

4 of 600 is 200 

o 

■1- of 600 is 100 

of 600 is 150 450 

150 : 300 :: 600 : 1200 men. 








































































POSITION.-FELLOWSHIP. 


99 


2. — A person asked his age, replied, if £ of my age be multiplied by 2, and that 
product added to half the years I have lived, the sum will he 75. How old was he ? 

37.5 years. 

DoWble ^Position. 

Rule.—A ssume any two numbers, and proceed with each according to conditions 
of question; multiply results or errors by contrary supposition; that is, first posi¬ 
tion by last error, and last position by first error. 

If errors are too great, mark them +; and if too little, —. 

Then, if errors are alike, divide difference of products by difference of errors; but 
if they are unlike , divide sum of the products by sum of errors. 

Example 1.— A asked B how much his boat cost; he replied, that if it cost him 6 
times as much as it did, and $30 more, it would have cost him $300. What was 
price of the boat ? 


Suppose it cost him 


60.30 

6 times. 6 times. 


360 

and 30 more 

39 ° 

3 °° 

9 °+ 

30 2d position. 


180 

and 30 more 
210 

300 

90— 

60 xst position. 


go 2700 5400 

9 ° 54 °° 

180) 8100 (45 dollars. 

2.—What is length of a fish when the head is 9 inches long, tail as long as its head 
and half its body, and body as long as both head and tail ? 6 feet. 


FELLOWSHIP. 

Fellowship is a method of ascertaining gains or losses of individuals 
engaged in joint operations. 

Single IT'ello’wsliip. 

Rule.— As the whole stock is to the whole gain or loss, so is each share to the 
gain or loss on that share. 

Example.— Two men drew a prize in a lottery of $9500. A paid $3, and B $2 for 
the ticket; how much is each share? 

5 : 9500 :: 3 : 5700, A’s share. 

5 : 9500 :: 2 ; 3800, B’s share. 

Double Fellowsliip, 

Or Fellowship with Time. 

Rule.— Multiply each share by time of its interest; then, as sum of products is to 
product of each interest, so is whole gain or loss to each share of gain or loss. 

Example.— A cutter’s company take a prize of $10 000, which is to be divided ac¬ 
cording to their rate of pay and time of service on board. The officers have been 
on board 6 months, and the crew 3 months; pay of lieutenants is $100, ensigns $50, 
and crew $10 per month; and there are 2 lieutenants, 4 ensigns, and 50 men; what 


is each one’s share ? 

2 lieutenants. $100 = 200 X 6 = 1200 

4 ensigns. 50 — 200 x 6 = 1200 

50 men. 10 = 500 X 3 = 1500 

39 °° 


3900 : 1200 ;; 10000 : 3076.92-4- 2 = 1538.46 dolls. 
3900 : i2oo ;; 10000 : 3076.92-4- 4= 769.23 “ 
3900 : 1500 10000 : 3846.16-7-50= 76.92 11 


Lieutenants 
Ensigns 
Men .. 
















100 


PERMUTATION, 


PERMUTATION. 

Permutation is a rule for ascertaining how many different ways any 
given number of numbers of things may be varied in their position. 

Permutation of the three letters abc, taken all together, are 6 ; taken two 
and two , are 6; and taken singly , are 3 . 

Rule. —Multiply all the terms continually together, and last product will give 
result. 

Example i.—H ow many variations will the nine digits admit of? 

1X2X3X4X5X6X7X8X9 = 362 880. 

2.—How many years would there be required to elapse before 10 persons could 
be seated in a varied position collectively, each day at dinner, including one day in 
every 4 years for a leap year ? 9935 years , 42 days. 

When only part of the Numbers or Elements are taken at once. Rule.— 
Take a series of numbers, beginning with number of things given, decreasing by 1, 
until number of terms equals number of things or quantities to be taken at a time, 
and product of all the terms will give sum required. 

Example i.—H ow many changes can be made with 2 events in 5? 

5 —1 — 4, and 4X5 = 2 terms. Hence, 5 X 4 = 20 changes. 

2 . —How t many changes of 2 will 3 playing cards admit of? 

3 — 1 = 2 , and 2X3 = 2 terms. Hence, 2X3 = 6 changes. 

3. —How many changes can be rung with 4 bells (taken 4 and 4 together) out of 6 ? 

4 —1 = 3, and 3X4X5X61=4 terms or changes. 

Hence, 3X4X5X6 = 360 changes. 


When several of the Elements are alike. Rule. —Ascertain the permutations 
of all the numbers or things, and of all that can be made of each separate kind or 
division; divide number of permutations of whole by product of the several partial 
permutations, and quotient will give number of permutations. 

Example. —How many permutations can be made out of the letters of the word 
persevere (9 letters, having 4 e’s and 2 r’s)? 

1X2X3X4X5X6X7X8X9 = 362880; 

1 X 2 X 3 X 4 = 24 for the e’s; 1X2 = 2 for the r’s, and 24 X 2 = 48. 

Hence, 362 880 -r- 48 = 7560. 

Or, Add logarithms of all the terms together, and number for the sum will give 
result. 

Example i.— How many permutations can be made with three letters or figures? 
Log. 1 =. 00 

2 = • 30103 

3 — - 4-77 121 3 

.7781513 = log. of number 6. 

2.—How many variations will 15 numbers in 16 places admit of? 

Add logarithms of numbers 1 to 16 and take logarithm of their sum— 
viz., 13.320661 97 = 20922 789 888000. 

Number of positions of the blocks in the “ 15 puzzle ” is as above for their 16 permutations. 

IPeriTiurtatioxis, 

Whereby any questions of Permutation may be solved by Inspection , number of 

terms not exceeding 20. 1 


I 

1 

5 

120 

9 

362 880 

13 

6 227 020 800 

2 

2 

6 

720 

10 

3 628 800 

14 

87 178 291 200 

3 

6 

7 

5040 

11 

39 916 800 

15 

1 307 674 368 000 

4 

24 

8 

40 32O 

12 

479001600 

16 

20922 789 888000 


355 687 428 096 000 
6 4°2 373 7°5 728000 
121 645 100 408 832 000 
2 432 902 oo 3 176 640000 
















ARITHMETICAL PROGRESSION. 


IOI 


ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers increasing or de¬ 
creasing by a constant number or difference; as, i, 3, 5, 7, 12, 9, 6, 3. The 
numbers which form the series are designated Terms; the first and last 
are termed Extremes , and the others Means. 

When any three of following elements are given, the remaining two can be ascer¬ 
tained —viz., First term, Last term, Number of terms, Common Difference , and Sum 
of all the terms. 

To Compute First Term. 

When Last term , Number of terms, and Sum of series are given. Rule. — From 
quotient of twice sum of series, divided by number of terms, subtract last term. 

I — d S d n — 1 ,- 

Or,- - ;-; and y (l -f-. 5 d)- — 2dS±.sd = a. a represent- 

71 — I 71 2 

ing 1 st, l last, n number of, and S sum of all terms , and d common difference. 


Illustration. —A man travelled 390 miles in 12 days, travelling 60 miles last day. 
How far did be travel first day? 

* 2 = 65, and 65 — 60 = 5 first term. 

12 

„ k > tti >1 I : v'T O v/ j f t*?' * // j • • • f -. 1 . f v • J 

To Compxxte Fast Term. 

When First term, Common Difference , and Number of terms are given. Rule. — 
Multiply the number of terms less 1, by common difference, and to product add first 
term. 

Example.— A man travelled for 12 days, at the rate of 5 miles first day, 10 second, 
and so on; bow far did be travel the last day ? 

12 — 1 X 5 — 55 > and 55 + 5 = 60 miles. 

When First term , Number of terms, and Sum of series are given. Rule. — Divide 
twice sum of series by number of terms, and from quotient subtract first term. 

Or, — — a \ V2dS-\-(a — .5d) 2 ±.5 cZ; and— — — = l. 

n n 2 

Illustration.— A man travelled 360 miles in 12 days, commencing with 5 miles 
first day; how far did be travel last day ? 

QQQ V 2 

- - ~ 65, and 65 — 5 = 60 miles. 

12 

To Compute Number of Terms. 

When Common Difference and Extremes, or First and Last term, are given. 
Rule.—D ivide difference of extremes by common difference, and add 1 to quotient. 

Example. —A man travelled 3 miles first day, 5 second, 7 third, and so on, till be 
went 57 miles in one day; bow many days bad be travelled at close of last day? 

57 — 3-^2 = 27, and 27 —(- 1 28 days. 


When Sum of series and Extremes are given. Rule.— Divide twice sum of series 
by sum of first and last terms. 

I — a , fzS . (2 a — d \ 2 , d — 2 a 

0r , —+ ■; V -7 + (vr ±^r ; 


and 


2 l -j— d ^ 
2 d ) 


2 S 2 Z + d 
2 d 


n. 


Illustration. —A man travelled 840 miles, walking 3 miles first day and 57 last 
day; how many days was be travelling? 

840 X 2 1680 

^ — — 28 days. 


3 + 57 


60 

I* 

















102 


ARITHMETICAL PROGRESSION. 


To Compute Common Difference. 

When Number of terms and Extremes are given. Rule. —Divide difference of 


extremes by i less than number of terms. 
2 S — 2 an 


Or, 


n (n- 


lf-ax l — a 

2 IS — l — a 


, 2 nl — 2 S , 

and —- -- = d. 

n (n — i) 


Illustration. —Extremes are 3 and 15, and number of terms 7; what is common 
difference? 


15 —3-H7— 


12 1 12 t-p 

—, and — = 2 com. dif. 
6 6 


To Compute Snm of tlie Series or of all Terms. 

When Extremes and Number of terms are given. Rule. —Multiply number of 
terms by half sum of extremes. 

Iax {l —a) l-\-a 


Or, 2 a-\-d (n — 1) X .5 

and 2 l-(dxn- 


_ 2 d 

■1) X .5 « = S. 


Illustration.—H ow many times does hammer of a clock strike in 12 hours? 

12 X i2-j-i = 156, and 156-j-2 — 78 times. 

To Compute any Number of ^Arithmetical JVfeans or 
Terms between two Extremes. 

Rule. — Subtract less extreme from greater, and divide difference by 1 more 
than number of means or terms required to be ascertained, and then proceed as 
in rule. 


To Compute Two Arithmetical Means or Terms between 

two given Extremes. 

Rule. —Subtract less extreme from greater, and divide difference by 3, quotient 
will be common difference, which being added to less extreme, or taken from great¬ 
er, will give means. 

Example i. —Compute two arithmetical means between 4 and 16. 

16 — 4 -7- 3 = 4 com. dif. 

4 + 4 = 8 one mean. 

16 — 4 = 12 second mean. 

2.—Compute four arithmetical means between 5 and 30. 


30 — 5 = 25, and 25-7- x — 5 — com. dif. 

5 -{-5 = 10 = 1 st mean. I 15 + 5 = 20 = 3 d mean. 

10+ 5 = 15 = 2d “ I 20 + 5 = 25 = 4 th “ 

IMiscellarxeous Illustrations. 

1. A steamer having been purchased upon following terms — viz.: $5000 upon 
transfer of bill of sale and balance in monthly instalments, commencing at $4500 
for first month, and decreasing $500 in each month, until whole sum is paid. 

1st. How many months must elapse before final payment? 

2d. What was amount of purchase-money, or sum of Series ? 

Here are first and last terms — viz., 500 and 5000, and common difference, 500. 
Hence , To compute number of terms and amount of purchase, 

5000 — 500 -r- 500 = 9, and 9 + x = 10 = number of terms or months, and 10 X 


5000 + 500 


10 X 2750 = $27 500, amount of purchase. 


2. If 100 stones are placed in a right line, one yard apart; how many yards must 
a person walk, to take them up one at a time and put them into a basket, one yard 
from first stone? 

First term 2, last term 200, and number of terms 100. 

200 + 2 


Hence, 100 X 


: 10100 yards. 


2 















GEOMETRICAL PROGRESSION. 


103 


3. If in the sinking of curb of a well, $3 is to be given for first foot in depth, $5 
for second, $7 for third, and increasing in like manner to a depth of 20 feet, what 
would it cost? 

First term 3, common difference 2, and number of terms 20. 

Hence, 20 — 1 X 2 -j- 3 = 41, last term. 

20 

Then, 3 -(-41 X — = 440, sum of all terms, or cost of curb. 

4. If a contractor engaged to sink a curb to depth of 20 feet for $400, and the 
contract was annulled when he had reached a depth of 8 feet; how much had he 
earned ? 

400-r- 20 = number of terms. But inasmuch as 400 may be divided into 20 terms 
in arithmetical proportion in many different ways, according to value of 1st term, 
it becomes necessary to assume the value of the first foot as value of 1st term. 

Assuming it at I5, the required proportion will be, 1st term 5, number of terms 20, 
sum of ser ies 400. 

400 — 5 X 20 X 2 600 ,, . 

Hence, ----— — -— = 111, common difference. 

’ 20 X (20 — 1) 380 T 9 ’ * 

Then, 5 -|- 1 y^ X 7 = i6y^ = 1st term -{-product of common difference and 8th 
term less 1, which added to 5 ^ 21-L, and X 4 = half number of terms for which 
cost is sought — 84^ dollars , sum earned. 


GEOMETRICAL PROGRESSION. 


Geometrical Progression is any series of numbers continually in¬ 
creasing; by a constant multiplier, or decreasing by a constant divisor, as 
1, 2, 4, 8,16, etc., and 15, 7.5, 3.75, etc. 

The constant multiplier or divisor is the Ratio. 

When any three of following elements are given , remaining two can be computed , 
viz.: First term, Last term, Number of Terms, Ratio , and Sum of all Terms. 


To Compute First Term. 

I 

When Radio, Last Term , and Number of Terms are given. Rule. — Divide last 
term by ratio raised to a power denoted by number of terms less 1. 


Or, -— and rl — S (r — 1) = a. 

’ r n — 1 

S sum of all terms, and r ratio. 


a representing 1st term, l last, n number of, 


Illustration. —Last term is 4374, number of terms 8, and ratio 3; what is first 
term ? 

Ayn_ __ 4374 ji rs i i erm 

38-1 2187 

To Compute Last Term. 

When First Term and Ratio are Equal. Rule.—W rite a few of leading terms 
of series and place their indices over them, beginning with a unit. Add together 
the most convenient and least number of indices to make the index to term required. 

Multiply terms of the series of these indices together, and product will give term 
required. 


Or, Multiply first term by ratio raised to a power, denoted by number of terms 
less 1. 

Example 1.—First term is 2, ratio 2, and number of terms 13; what is last term ? 

Indices, 1234 5 

Terms, 2, 4, 8, 16, 32. 

Then, 5 5 —3 = 1 3 = sum of indices, and 32 X 32 X 3 = 8192 = last term. 

Or, 2 X 2 1 — 8192. Also by inspection of table, page 105,13th term = 8192. 








104 


GEOMETRICAL PROGRESSION, 


2.—The price of 12 horses being 4 cents for first, 16 for second, and 64 for third, 
and so on; what is price of last horse ? 

Indices, 1234 
Terms, 4, 16, 64, 256. 

Then, 4-|-4-1-4 = 12= sum of indices, and 256 X 256 X 256 = 2563 = $167 77216. 


When First Term and Ratio are Different. Rule. —Write a few of leading terms 
of series, and place their indices over them, beginning with a cipher. Add together 
the most convenient indices to make an index less by 1 than term sought. 

Multiply terms of these series belonging to these indices together, and take 
product for a dividend. 

Or, Raise first term to a powrnr, index of which is 1 less than number of terms 
multiplied; take result for a divisor; proceed with their division, and quotient will 
give term required. 

Example i. — First term is 1, ratio 2, and number of terms 23; what is the last 
term? 

Indices, 01234 5 
Terms, 1, 2, 4, 8, 16, 32. 

Then, 5 + 5 + 5 + 5 + 2 = 22 = sum of indices , and 32 X 32 X 32 X 32 X 4 = 
4 194 304, and 4 194 304 -i- the 5th power (6 — 1) of 1 = 1 — 4194 304. 

Or, 1 x 2 2 3 1 = 4194 304. By inspection of table, page 105, 23d term = 4 194 304. 


2.—If 1 cent had been put out at interest in 1630, wiiat w r ould it have amounted 
to in 1834, if it had doubled its value every 12 years? 

1834 —1630_204, wiiich -4-12 = 17, and 17 -(- 1 = 18 = number of terms. 

Indices, 01234 7 

Terms, 1, 2, 4, 8, 16, 128. 

Then, 7 -j— 4 3 —J— 2 —f— 1 = 17, and 128 x 16X8X4X2X1 = 131072, and 131072 

-r-1, the 4th pow-er (5 — 1) of 1 = $ 1 31072. 

When First Term , Ratio , and Sum of the series are given. Rule.—F rom sum of 
series subtract quotient of first term subtracted from sum of series divided by 

ratio. ^ „ T 

Or, a X 1 1 — l. 

Example.—F irst term is 2, ratio 3, and sum of series 2186; what is last term? 
n , 2186—2 

2180-= 2186 — 728 = 1458, last term. 


To Oompu.te jNT nm'ber of Terms. 

When Ratio , First, and Last Terms are given. Rule.—D ivide logarithm of quo¬ 
tient of product of ratio and last term, divided by first term, by logarithm of ratio. 


Or, 


log. (a-fSr-i)-log. a 


log. I — log. a 


log. r 


and 


log. (S — a) — log. (S — l) 
log. I — log. (r l — r — 1 S) 


log. r 


x = n. 


Example. — Ratio is 2, and first and last terms are 1 and 131072 * what is num¬ 
ber of terms? 

log. 2 X ^ 072 = log. 262 144 = 5.41854, and 5.418 54 -7- log. of 2 = 5-418 54 = x8. 

1 -S 01 ^ 

To Compute Sum of Series. 

When First Term., Ratio, and Number of Terms are given. Rule.—R aise ratio to 
a power index of which is equal to number of terms, from which subtract 1 • then 
divide remainder by ratio less 1, and multiply quotient by first term. 


rl — a l n ^/l — a n ^/a 
Or,--; —-— --— and 


l — ] 


• = S. 


1 r' 















GEOMETRICAL PROGRESSION. 


105 


Illustration i. —First term is 2, ratio 2, and number of terms 13; what is sum 
of series ? 

2 1 = 8192 — i = 8191, and 8191 -r- (2 — 1) = 8191, and 8191 x 2 = 16 382. 

2 - —If a man were to buy 12 horses, giving 2 cents for first horse, 6 cents for 


second, and so on, what would they cost him ? 


$5314.40. 


To Compute Ratio. 

When First Term, Last Term, and Numbers of Terms are given. Rule.— Divide 
last term by first, and quotient will be equal to ratio raised to power denoted by 1 
less than number of terms; then extract root of this quotient. 

S — a 
Or, 


S — L 


■■ r. 


Illustration.— First term is 2, last term 4374, and number of terms 8; what is 
ratio ? 

4374 8—1. - - 

- = 2187, and y/ 2187 3, ratio. 


HVIiscel Ian eons Illustrations. 

1. What is 9th term in geometrical progression 3, 9, 27, 81, etc.? and what is 
sum of terms ? 

1st term — 3, number of terms 9, and ratio 3. 

Hence, by rule to compute last term, 1st term and ratio being equal— 

Indices, 1234 
Terms, 3, 9, 27, 81. 

Then, 24-3 + 4 = 9 = sum °f indices, and 9X27X81=119 683 = last term. 

By rule to compute sum of terms— 

•2 9 — 1 19682 


X 3 


: 9841 X 3 = 29 523, sum of terms. 


3 — 1 2 

2. First term is 1, ratio 2, and last term 131072; what is sum of series? 


131 072 X 2 — 1 = 262 143, and 262 143 - 7-2 — 1 = 262 143. 

3. What are the proportional terms between 2 and 2048? 


4 + 2 = 6, and 6 —1 = 5, and 




Hence, 2 : 8 : 32 : 128 : 512 : 2048. 

4. Sum of series is 6560, ratio 3, and number of terms 8; what is first term? 

9 — I 2 

6560 X -= 6560 X >- 7 - = 2, first term. 


3 “ — 1 


6560 


Greoxnetrical 3 3 i*ogi*essiorxs, 

Whereby any questions of Geometrical Progression and of Double Ratio may be 
solved by Inspection, number of terms not exceeding 56. 


I. 

1 

15 

16384 

29 

268 435 456 

43 

4398046511104 

2 

2 

l6 

32768 

3 ° 

536 870912 

44 

8796093022208 

3 

4 

17 

65536 

3 i 

1 073 741 824 

45 

17 592 186044416 

4 

8 

l8 

131 ° 7 2 

32 

2 i 47 483648 

46 

35 184 372088 832 

5 

16 

*9 

262 144 

33 

4294967 296 

47 

70368 744177664 

6 

32 

20 

524 288 

34 

8589934 592 

48 

140737488355328 

7 

64 

21 

1 048 576 

35 

17 179 869 184 

49 

281 474976 710656 

8 

128 

22 

2097 152 

36 

34 359 738 368 

5 ° 

562949 953421 312 

9 

256 

2 3 

4 i 94 304 

37 

68719476736 

5 i 

1125 899 906 842 624 

10 

512 

24 

8 388 608 

38 

i 37 438 953 472 

52 

2 251 799 813 685 248 

11 

1024 

25 

16 777 216 

39 

274877906 944 

53 

4503 599 627 370496 

12 

2048 

26 

33554432 

40 

549755813888 

54 

9007 199 254 740992 

13 

4096 

27 

67108 864 

4 i 

1 099 511 627 776 

55 

18 014 398 509 481 984 

H 

8192 

28 

134 217 728 

42 

2 199023255 552 

56 

36 028 797 018 963 968 


Illustrations.— 12th power of 2 = 4096, and 7th root of 128 = 2. 























106 


ALLIGATION. 


ALLIGATION. 

Alligation is a method of finding mean rate or quality of different ma¬ 
terials when mixed together. 

To Compute Mean TPri.ee of a Affixture. 

When Prices and Quantities are known. Rule. — Multiply each quantity by its 
rate, divide sum of products by sum of quantities, and quotient will give rate of the 
composition. 

Example.— If 10 lbs. of copper at 20 cents per lb., 1 lb. of tin at 5 cents, and 1 lb. 
of lead at 4 cents, be mixed together, what is value of composition? 

10 X 20 = 200 
1X5= 5 

_fX 4 =_4 

12 ) 209 (17.416 cents. 

To Compute Quantity of each. -Article. 

When Prices and Mean Price are given. Rule. —Write prices of ingredients, one 
under the other in order of their values, beginning with least, and set mean price 
at left. Connect with a line each price that is less than mean rate with one or more 
that is greater. 

Write difference between mixture rate and that of each of simples opposite price 
with which it is connected; then sum of differences against any price will express 
quantity to be taken of that price. 

Example.— How much gunpowder, at 72, 54, and 48 cents per pound, will compose 
a mixture worth 60 cents a pound ? 

(48 \ 12, at 48 cents. 

60 < 54X7 12, at 54 cents. 

(72/ 12 -)- 6 = 18, at 72 cents. 

Here, 72 — 60 = 12 at 48, 72 — 60 = 12 at 54, 60 — 48 = 12, and 60— 54 — 6 = 

12 -f- 6 = 18 at 72. 

Then 12 X 48 + 12 X 54 + 18 X 72 = 2520, and 2520-7- 12 -J-12 -f-12 + 6 = 60 cents. 

Note. — Should it be required to mix a definite quantity of any one article, the quantities of each, 
determined by above rule, must be increased or decreased in proportion they bear to defined quantity. 

Thus, had it been required to mix 18 pounds at 48 cents, result would be 18 at 48, 
18 at 54, and 27 at 72 cents per pound. 

When the whole Composition is limited. Rule.— As sum of relative quantities, 
as ascertained by above rule, is to whole quantity required, so is each quantity so 
ascertained to required quantity of each. 

Example. —Required 100 pounds of above mixture. 

Then, i2-j-i2-(- 18 = 42. Then, 42 t 100 12 .- 28.571 pounds. 

42 : 100 :: 12 : 28.571 pounds. 

42 : 100 18 : 42.857 pound's. 

When Price of Several Articles and Quantity of one of them is given. Rule. —As¬ 
certain proportionate quantities of ingredients by previous rule. 

Then, as number opposite ingredients, quantity of which is given, is to given 
quantity; so is number opposite to each ingredient to quantity required of that in¬ 
gredient. 

Example. — Having 35 lbs. of tobacco, worth 60 cents per pound, how much of 
other qualities, worth 65, 70, and 75 cents per pound, must be mixed with it, so as to 
sell mixture at 68 cents-'per pound? 

By previous rule, it is ascertained there must be 7 lbs. at 60, 2 at 65, 3 at 70, and 
8 at 75 cents; but as there are 35 lbs. at 60 cents to be taken, other quantities and 
kinds must be increased in like manner. 

Hence, 7 : 35 : : 2 : 10 = 10 at 65 cents. 

7 : 35 : : 3 : I 5 — J 5 u 7 ° cents. 

7 : 35 : : 8 : 40 = 40 “ 75 cents. 





SIMPLE INTEREST. 


107 


SIMPLE INTEREST. 


To Compute Interest on any G-iven Snm for a Period, 
of One or more Years. 

Rule. —Multiply given sum or principal by rate per cent, and number of years; 
point off two figures to right of product, and result will give interest in dollars and 
cents for 1 year. 

Example. —What is interest upon $ 1050 for 5 years at 7 per cent. ? 

1050 X 7 X 5 = 36 75°) and 3 6 7 - 50 — $ 367.50. 

When Time is less than One Year. Rule.— Proceed as before, multiplying by 
number of months or days, and dividing by following units—viz., 12 for months, 
and 365 or 366, as the case may be, for days. 


Example.— What is interest upon $1050 for 5 months and 30 days at 7 per cent.? 
5 months and 30 days = 183 days. 1050 X 7 — 3685, and 36.85 = $ 36.85. 


The operation of computing interest may he performed thus: 

Assuming interest upon any sum at 6 per cent. = 1 per cent, for 2 months. 
Interest at 5 per cent, is -Uh less than at 6 per cent. 

Interest at 7 per cent, is ^th greater than at 6 per cent. 


Taking preceding example—2 months = 1 per cent.= 10.50 


2 “ —1 " xo.50 

1 “ =i “ 5-25 

30 days = 1 month = 5.25 


u 


3i-5o 

Add ^ for 7 per cent. = 5.25 


$36.75 


Note. —Difference between this amount and preceding arises from 183 days being taken in one case, 
and half a year, or 182.5 days, in the other. 

In every computation of interest there are four elements—viz., Principal, Time, 
Rate, and Interest or Amount, any three of which being given, remaining one can 
be ascertained. 


To Compute [Principal. 

When Time, Rate per Cent., and Interest are given. Rule. —Divide given interest 
by interest of $1, etc., for given rate and time. 

Example. —What sum of money at 6 per cent, will in 14 months produce $ 14? 

14 - 4 - .07 =2 200 dollars. 

To Compute Rate per Cent. 

When Principal, Interest, and Time are given. Rule. —Divide given interest by 
interest of given sum, for time, at 1 per cent. 

Example. — If $32.66 was discounted from a note of $400 for 14 months, what 
was that per cent. ? 

Interest on 400 for 14 months at 1 per cent. = 4.66. 

Then 32.66 - 4 - 4.66 = 7 per cent. 

To Compute Time. 

When Principal, Rate per Cent., and Interest are given. Rule. —Divide given in¬ 
terest by interest of sum, at rate per cent, for one year. 

Example. —In what time will $ 108 produce $ n.34, at 7 per cent. ? 

Interest on 108 for one year is 7.56. 

11.34-4-7.56 = 1.5 years. 

Illustration i. —If an amount of $ 2175 is returned for a period of 15 months, 
rate of interest having been 7 per cent., what was principal invested? $2000. 

2.—If $ 1000 in 18 months will produce $ 1090, what is rate ? 6 per cent. 





io8 


COMPOUND INTEREST. 


COMPOUND INTEREST. 

If any Principal be multiplied by number (in following table) opposite 
years, and under rate per cent., sum will be amount of that principal at com¬ 
pound interest for time and rate taken. 

Example. —What is amount of $500 for 10 years at 6 per cent. ? 

Tabular number.... 1.79084, and 1.79084 X 500 = 895.42 dollars. 


TO 

** 

CJ 

3 

4 

5 

6 

to 

C 3 

3 

4 

5 

6 


Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 


Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 

I 

1.03 

1.04 

1.05 

1.06 

13 

1.46853 

1.665 07 

1.885 64 

2. I32 92 

2 

1.0609 

1.081 6 

1.102 5 

1.123 6 

X 4 

i- 5 x 529 

1.73167 

i -979 93 

2.2609 

3 

1.092 73 

1.124 86 

1.157 62 

I. I9I OI 

1.5 

I - 557 97 

1.800Q5 

2.078 92 

2 - 39 6 55 

4 

1-125 51 

1.169 86 

1-215 5 

1.262 47 

l6 

1.604 71 

1.872 98 

2.182 87 

2-540 35 

5 

1.15927 

1.216 68 

1.27698 

1.338 22 

17 

1.652 85 

i -947 99 

2 . 292 OI 

2.692 77 

6 

I-I 94 °5 

1.265 3 2 

i -34 

1.418 51 

18 

1.702 44 

2.025 81 

2.406 61 

2-85433 

7 

1.229 87 

I- 3 I 593 

1.407 1 

i- 5°3 63 

x 9 

1-753 5 

2.106 84 

2.52695 

3-025 59 

8 

1.266 77 

1-36857 

1-477 45 

i -593 84 

20 

1.806 11 

2-19113 

2.653 29 

3.207 13 

9 

1.30477 

1-42331 

i- 55 i 32 

1.689 47 

21 

1.860 29 

2.278 76 

2.78596 

3-399 56 

IO 

1.34392 

1.480 24 

1.628 89 

1.790 84 

22 

1.916 1 

2.36992 

2.925 26 

3-60353 

II 

1.384 24 

1-539 45 

i- 7 IQ 33 

1.898 29 

23 

x -973 6 

2.464 21 

3-07152 

3.81974 

12 

1.42576 

1.601 03 

I-795 85 

2.012 19 

24 

2.032 79 

2-5633 

3.22509 

4.04873 


For any other Rate or Period. —Multiply logarithm of rate 1 by period, and 
number for logarithm will give tabular amount as above. 

Illustration.— What is tabular number for 4 per cent, for 10 years? 

Log. of 1.04 = .017 033 3, which x 10 = .170 333, and number for log. = 1.48024. 

Time in Years in Avliicli a Sum of Money Avill be 
d.on'bled. at Several Races of Interest. 


Rate. 

Time. 

Rate. 

Time. 

Rate. 

Time. 

Rate. 

Time. 

Per cent. 

1 

69.68 

Per cent. 

4 

17.67 

Per cent. 

7 

10.34 

L’er cent. 

10 

7.27 

2 

35 

5 

14. 21 

8 

9.01 

20 

3-8 

3 

23-44 

6 

n.88 

9 

8.04 

3 ° 

2.64 


Yalne of $ 1 , etc.. Computed Semi-annually for a IPeriod. 

of IS Years. 


Years. 

3 

Per Cent. 

4 

Per Cent. 

5 

Per Cent. 

6 

Per Cent. 

Years. 

3 

Per Cent. 

4 

Per Cent. 

5 

Per Cent. 

6 

Per Cent 

•5 

1-015 

1.02 

1.025 

1.03 

6-5 

1-2134 

1.2936 

I -3785 

1.4684 

I 

1.0302 

I.O404 

1.0506 

1.0609 

7 

1-2317 

I- 3 I 95 

I- 4 I 3 

1 5102 

i -5 

1-0457 

1.0612 

1.0769 

1.0927 

7-5 

1.2502 

1-3459 

1.4483 

i- 55 8 

2 

1.0614 

1.0824 

1.1038 

1-1255 

8 

1.269 

1.3728 

1.4845 

1.6047 

2-5 

1.0773 

I. IO4I 

1.1314 

x - x 593 

8-5 

1.288 

1.4002 

1.5216 

1.6528 

3 

1.0934 

1.1262 

i -1597 

i-i 94 i 

9 

1-3073 

1.4282 

1-5597 

1.7024 

3-5 

1.1098 

1.1487 

1.1887 

1.2299 

9-5 

1.3269 

1.4568 

1-5987 

1-7535 

4 

1.1265 

I - I 7 I 7 

1.2184 

1.2668 

IO 

1.3469 

1.486 

1.6386 

1.8061 

4-5 

i - x 434 

1.1951 

1.2489 

1.3048 

10.5 

1.3671 

1-5157 

1 6796 

1.8603 

5 

1.1604 

I.219 

1.2801 

1-3439 

II 

1.3876 

1 546 

1.7216 

1.9161 

5-5 

1.178 

1.2434 

i- 3 I2 i 

1.3842 

n -5 

1.4084 

1-5769 

1.7606 

I-9736 

6 

1.1956 

1.2689 

x -3449 

1.4258 

12 

1.4295 

1.6084 

1.8087 

2.0356 


Illustration. — What is amount of $500 at semi-annual interest of 5 per cent, 
compounded for 10 years ? 

Tabular number 1.6386. Then, 500 X 1.62889 = $814.44.5. 


To Compute Interest on any- Griven Som. 


For a Period of Years. P (1 -j- r) n A ; 


and 


log. A — log. P 


(i + D 


n = p ; 


n /~A 

V p" 


n. 


P representing principal , r rate per cent, per annum, n 


log. (i + r) 

number of years, and A amount of principal and interest. 
























































DISCOUNT OR REBATE.-EQUATION OF PAYMENTS. IO9 

Illustration. —Assume as preceding, $500 at 5 per cent, for 10 years. 

500 X 1.05 10 = 500 X 1.628 89 = $814.44.5, amount. ~^'^ 5 I0 = 500, principal. 


10 /814.44.5 
V 500 


: .05, rate. 


log. 814.44.5 —log. 500 
log. (1 + .05) 


= 10, number of years. 


For any Period. — Assume elements of preceding case, interest payable semi¬ 
annually. 10X2 = 2o, number of payments; — = .025, rate. 

2 

Then, 500 X i-o25 20 = 500 X 1-638 62 = $ 819.31. 

When term of payments and rate are not given in table. 

[jog- 1) X n pj ±= log. A. 

Illustration. —Assume $1000 for 30 years, at 7 per cent, half-yearly. 

.07 - 

log. •-|- i := * OI 4 94 ° 3 j an< ^ * OI 4 94° 3 X 30 X 1000 $ 2806.78. 


DISCOUNT OR REBATE. 

Discount or Rebate is a deduction upon money paid before it is due. 

To Compute Tt.eDa.te upon any Sam. 

Rule. —Multiply amount by rate per cent, and by time, and divide product by 
sum of product of rate per cent, and time, added to ioo. 

Example i. —What is discount upon $12075 for 3 years, 5 months, and 15 days, 
at 6 per cent. ? 

3 years 5 months and 15 days = 3.4574 years. 

12075X6X3-4^74 230488.63 

- 1 -;-— -- = 2074. 53 - - $ 2074. 53. 

100+(0 X 3 - 4574 ) 120.7444 

2.—What is present value of a note for $963.75, payable in 7 months, at 6 per 
cent. ? 

6 rate. 7 months = Ag- of 1 year = 6 X 7 -f-12 = 3.5, and 3.5 —(— 100 = 103.5 = 1-035. 

963-75^- 1.035 = $931.16. 

To Compute tlie Sum for a given Time and. Rate, to yield 

a Certain Sum. 

Rule. —Divide given sum by proceeds of $ 1 for given time and rate. 

Example. —For what sum should a note be drawn at 90 days, that when dis¬ 
counted at 6 per cent, it will net $200? 

Discount on $ 1 for 90 -(- 3 days at 6 per cent. = $ .0155. 

Hence, $1—.0155 —.9845 , proceeds, and $200-7-. 9845 = $203.14.9. 


EQUATION OP PAYMENTS. 

Rule. —Multiply each sum by its time of payment in days, and divide sum of 
products by sum of payments. 

Example.—A owes B $300 in 15 days, $60 in 12 days, and $350 in 20 days; when 
is the whole due ? 

300 X 15 = 4 500 
60 X 12 = 720 
350 X 20 = 7 000 
710 ) 12220 (17 -f- days. 

K 











IIO 


ANNUITIES. 


ANNUITIES. 

To Compute Amount of .A_iiiiTiity. 

When Time and Ratio of Interest are Given. Rule. — Raise the ratio to a power 
denoted by time, from which subtract i; divide remainder by ratio less i, and quo¬ 
tient, multiplied by annuity, will give amount. 

Note. — $ i added to given rate per cent, is ratio, and preceding table in Compound Interest is a 
table of ratios. 

Example. — What is amount of an annual pension of $100, interest 5 percent., 
which has remained unpaid for four years? 

1.05 ratio; then 1.054—1 = 1.21550625 — i = . 21550625, and .215 50625-r-(1.05 
— 1) .05 = 4.310125, which x 100 = $431.01. 25. 

To Compute Present "Wortlx of aix -A.ninxity. 

When Time and Rate of Interest are Given. Rule. —Ascertain amount of it for 
whole time; divide by ratio, involved to time, and result will give worth. 

Example. — What is present worth of a pension or salary of $500, to continue 10 
years at 6 per cent, compound interest? 

$500, by last rule, is worth $6590.3975, which, divided by 1.06 10 (by table, page 
108, is 1.79084) = $ 3680.05. 

Or, Multiply tabular amount in following table by given annuity, and product 
will give present worth. 

Illustration i.—A s above; 10 years at 6 per cent. = 7. 36008, and 7.36008 X 500 
= 3.68.004 dollars. 

2. What is present worth of $150 due in one year at 6 per cent, interest per annum ? 

•94339 X 150 = 1141.50.85. 


Present "Wortlx of aix Axiixuiity of Sjfcl, at -A, 5, axxd 6 
Per Ceixt. Compound Interest for ^Periods Tinder 25 
Y ears. 


Years. 

4 Per Cent. 

5 Per Cent. 

6 Per Cent. 

Years. 

4 Per Cent. 

5 Per Cent. 

6 Per Cent. 

I 

.961 54 

•952 38 

■943 39 

13 

9.98562 

9-393 57 

8.852 68 

2 

1.886 09 

1.85941 

I-833 39 

14 

10.56307 

9.898 64 

9.29498 

3 

2-775 1 

2.723 25 

2.673 OI 

15 

11.11843 

10.37966 

9.712 25 

4 

3.6299 

3-545 95 

3-465I 

l6 

11.651 28 

10.837 78 

10.105 89 

5 

4-452 03 

4.32948 

4.212 36 

17 

12.166 26 

11.274 O7 

10.477 26 

6 

5.242 15 

5-07569 

4.91732 

18 

12.659 26 

11.689 58 

10.827 6 

7 

6.002 03 

5-786 37 

5.58238 

T -9 

I 3- x 33 88 

12.085 32 

xi.158 II 

8 

6.731 76 

6.463 21 

6.209 79 

20 

13.59029 

12.462 21 

II.46Q Q2 

9 

7-436 4 

7.107 82 

6.801 69 

21 

I4.O2Q 12 

12.821 15 

II.76407 

IO 

8. no 85 

7.721 73 

7.36008 

22 

14.45112 

i3- l6 3 

12.041 58 

II 

8.760 44 

8.306 41 

7.886 87 

23 

14.856 82 

13.48807 

12.303 38 

12 

9-385 05 

8.863 25 

8.383 84 

24 

15.24695 

13.79864 

12.550 35 


For a Rate of Interest and Term of Years not given in either Table. 

— [1 — -—r—ul = A. Notation as preceding, 
r L (! + r)»J 

Illustration. —Take $ 1 at 4 per cent, for 24 years. 

Log. 1.04 = .017 033, which x 24 = .408 799. log. .408 799 = 2.5633 = ratio raised 
to power of 24. 

Then ’ ( I_X I — 39° 122 = $15.24.695. 

To Corxipxxte Yearly Amount tliat will Tiiqxxidate a Debt 
ixx a Gfivexx HN'xxrxi'ber of Years at Coixipoxiiid Iixterest. 

P r (1 4 - r) n 

(i -f-r) TO _1 = A- Illustration. — What is amount of an annual payment that 

will liquidate a debt of $100 in 6 years at 5 per cent, compound interest? 





















ANNUITIES. 


Ill 


(i -f .05) 6 per table, page 108, 
— 1.34. 


100X .05 (1 + .05) 6 5X1.34 6.7 


= $ 19.76. 


(i-f.05) 6 — 1 1 - 34 —1 -34 

When Annuities do not commence till a certain period of time , they are said to be 
in Reversion. 


To Compute Present 'Wortli of an Annuity in Reversion. 

Rule.— Take two amounts under rate in above table—viz., that opposite sum of 
two given times and that of time of reversion; multiply their difference by an¬ 
nuity, and product will give present worth. 


Example.— What is present worth of the reversion of a lease of $40 per annum, 
to continue for 6 years, but not to commence until end of 2 years, at rate of 6 per 
cent. ? 

6 -(- 2 = 8 years.6.209 79 

2 “ . 1-833 39 

4.3764° X 40 = 1175.05.6. 


Amount of Annuity of SBl, etc.. Compound Interest, 
from 1 to SO Years. 


02 

*4 

C3 

4 

5 

6 

7 

CO 

ft* 

c3 

4 

5 

6 

7 

£ 

Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 


Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 

I 

I. 

I. 

I. 

I. 

II 

13.48635 

14.206 7Q 

14.97164 

15-7836 

2 

2.04 

2.05 

2.06 

2.0 7 

12 

15.025 8 

I 5 - 9 I 7 13 

16.869 94 

17.88845 

3 

3. i2r 6 

3-152 5 

3.1836 

3.2149 

13 

16.626 84 

17.712 98 

18.882 14 

20.140 64 

4 

4.24646 

4.310 12 

4.37462 

4-439 94 

14 

18.291 91 

19.59863 

21.015 °7 

22.550 49 

5 

5.41632 

5.52563 

5.63709 

5-75074 

15 

20.023 59 

21.57856 

23.27597 

25.12902 

6 

6.63297 

6.801 91 

6.975 32 

7-153 29 

16 

21.82453 

23.657 49 

25.672 53 

27.888 05 

7 

7.898 29 

8.142 01 

8-39384 

8.654 02 

17 

23.697 5i 

25.84037 

28.212 88 

30.840 22 

8 

9.21423 

9.549 11 

9.897 47 

10.259 8 

l8 

25.64541 

28.13238 

30.90565 

33-999 °3 

9 

10.58279 

11.026 56 

11.491 32 

11.97799 

*9 

27.671 23 

30-539 

33.759 99 

37.37896 

IO 

12.006 11 

12.57789 

13.18079 

13.816 45 

20 

29.778 08 

33.06595 

36-78559 

40.99549 


Illustration.— What is amount of $ 1000 for 20 years at 5 per cent.? 

5 per cent, for 20 years = 33.065 95; hence, 1000 X 33-06595 = $ 33.06.595. 


To Compute -A.mou.nt of an Annuity for any Reriod 

and. Rate. 

Rule. —From table for Compound Interest, page 108, take value for rate per cent, 
for 1 year, and raise it to a power determined by time in years, from which subtract 
1, divide remainder by rate, and quotient multiplied by annuity will give amount 
required. 

Example.— What will an annuity of $ 50, payable yearly, amount to in 4 years, at 
5 per cent. ? 

By table, page 108, 1.054 = 1.2155. 

1.2155 —1-=-(1.05 — 1) = 4.31, and 4.31 x 50 = 1215.50. 


For Half-yearly and Quarterly Payments. 
Multiply annuity for given time by amount in following table: 


Rate per cent. 

Half-yearly. 

Quarterly. 

Rate percent. 

Half-yearly. 

Quarterly. 

3 

1.007 445 

1.011 181 

5-5 

1.013 567 

1.020 395 

3-5 

1.008 675 

1.013031 

6 

1.014 781 

1.022 227 

4 

I.OO9 902 

1.014 877 

6.5 

1-015 993 

1.024055 

4-5 

I.on 126 

1.016 729 

7 

I.OI7 204 

1.025 88 

5 

1.012 348 

1.018 559 

7-5 

1.018 414 

1.027 704 


Illustration i. —Annuity as determined in previous case = $215.50. 

Hence, 215.50 X 1.012348 from above table = $218.16 for half yearly payments. 

2. A person 30 years of age has an annuity for 10 years, present worth of it being 
$1000, provided he may live for 10 years. What is annuity worth, assuming that 
60 persons out of every 3550, between the ages of 30 and 40, die annually ? 

3550 — 600 (-60 X 10) = 2950 would therefore be living. 

And, 3550 : 2950 :: 1000 = $830.98. 





































I 12 


PERPETUITIES.-COMBINATION. 


PERPETUITIES. 

Perpetuities are such Annuities as continue forever. 

To Compute Value of a Perpetual .A.nnnity. 

Rule.— Divide annuity by rate per cent., and multiply quotient by unit in pre¬ 
ceding table. 

Example. —What is present worth of an annuit y for $ ioo, payable semi-annually, 
at 5 percent.? 

100-^.05 = 2, and 2 X 1.012348, from preceding table = 2.024.70. 

To Compute Value of a Perpetuity in Reversion. 
Rule. —Subtract present worth of annuity for time of reversion from worth of 
annuity, to commence immediately. 

Example. —What is present worth of an estate of $50 per annum, at 5 per cent., 
to commence in 4 years? 

50-i- .65.. 1000 

$50, for 4 years, at 5 per cent. = 3.545 95 (from table, page no) x 50= 177.2975 

822.7025 

which in 4 years, at 5 per cent, compound interest, would produce $1000. 


COMBINATION. 

Combination is a rule for ascertaining how often a less number of num¬ 
bers or things can be chosen varied from a greater, or how many different 
collections may be formed without regard to order of each collection. 

Combinations of any number of things signify the different collections 
which may be formed of their quantities, without regard to the order of their 
arrangement. 

Thus, 3 letters, a, b, c, taken all together, form but one combination, abc. 
Taken two and two, they form 3 combinations, as ah, ac, be. 

Note. —Class of tlie combination is determined by number of elements or things to be taken ; if two 
are taken, the combination is of 2d class, and so on. 

Rule.—M ultiply together natural series 1, 2, 3, etc., up to the number to be taken 
at a time. Take a series of as many terms, decreasing by 1, from number out of 
which combination is to be made, ascertain their continued product, and divide 
this last product by former. 

Example i.—H ow many single combinations, as ab, ac, may be made of 2 letters 

out of 3 ? 1 x 2 2 _ 6 _ 

3X2 6 2 ^ 

2. —How many combinations may be made of 7 letters out of 12? 

iX 2 X 3X4X5X6X7 5040 3 99 i68 ° 

-- , cilia - ~ 7Q2. 

12X11X10X9X8X7X6 3991680 5040 

3. —How many different hands of cards may be held, as at whist, combinations 

13 out of 52? 635013559600. 

When two Numbers or Tilings are Combined. 

Rule.—M ultiply together natural series 1, 2, 3, etc., to one less term than number 
of combinations; ascertain their continued product, and proceed as before. 

Example.—T here are 3 cards in a box, out of which two are to be drawn in a re¬ 
quired order. How many combinations are there? 

Here there are 2 terms; hence, 2 —1 = 1, and —-— =z — =6 —1 = 6 

3X2 6 

To Compute jNTumber of Ways in which any Number of 

Distinct Objects can be Divided, among any Number. 

Rule.—M ultiply together numbers equal to number given, as often as objects 
are to be divided among them. 

Example.—I n how many different ways can 10 different cards be divided among 
3 persons? 3X3X3X3X3X3X3X3X3X3 or 3 I0 = 59049. 










COMBINATION.-CIRCULAR MEASURE. 


113 


Combinations with. Repetitions. 

In this case the repetition of a term is considered a new combination. Thus, 
1, 2, admits of but one combination, if not repeated; if repeated, however, it admits 
of three combinations, as 1, 1; 1, 2; 2, 2. 

Rule.— To number of terms of series add number of class of combination, less 1 ; 
multiply sum by successive decreasing terms of series, down to last term of series; 
then divide this product by number of permutations of the terms, denoted by class 
of combination. 

Example.— How many different combinations of numbers of 6 figures can he 
made out of n? 

— 1) = 16 = sum of number of terms , and number of class, less 1. 

16 X 15 X 14 X 13 X 12 X 11 — 5 765 760= product of sum, and successive terms to 
last term. 

iX2X3X4X5X6 = 720 permutations of class of combination. 

Thep, 5765 7 - = 8oo8. 

720 


'Variations with Repetitions. 

Every different arrangement of individual number or things, including repeti¬ 
tions, is termed a Variation. 

Class of Variation is denoted by number of individual things taken at a time. 

Rule. —Raise number denoting the individual things to a power, the exponent 
of which is number expressing class of variation. 


Example i.— How many variations with 4 repetitions can he made out of 5 fig¬ 
ures? 5 4 = 625. 


2.—How many different combinations of 4 places of figures can be made out of 
the 9 digits? 

. , . 12 X 11 X 10X9 11880 

9 -f (4 — 1) — 12, and . w w = -777- = 495- 


1 X 2 X 3 X 4 


24 


Combination without Repetitions. 

Rule.— From number of terms of series subtract number of class of combination, 
less 1; multiply this remainder by successive increasing terms of series, up to last 
term of series; then divide this product by number of permutations of the terms, 
denoted by class of combination. 


Example i.—How many combinations can be made of 4 letters out of 10, exclud¬ 
ing any repetition of them in any second combination ? 

10 — (4 — 1) = 7 = number of terms — number of class, less 1. 

7X8X9X10 = 5 ° 4 ° =prod. of remainder 7, and successive terms up to last term. 
1X2X3X4 = 24 —permutations of class of combination. 


Then, = 210. 

24 

2.—How many combinations of the 5th class, without repetitions, can be made 
of 12 different articles? 


12 — (5 — 1) = 8, and 


X 9 X 10 X 11 X 12 8 5040 


1X2X3X4X5 


= 792 - 


CIRCULAR MEASURE. 


Unit of Circular Measure is an angle which is subtended at centre of a circle 
by an arc equal to radius of that circle, being equal to 


180° 

3.1416 


57.296°. 


Circular measure of an angle is equal to a fraction which has for its numerator 
the arc subtended by that angle at centre of any circle, and for its denominator the 
radius of that circle. 


K* 










CIRCULAR MEASURE.—PROBABILITY. 


I 14 


To Compute Circular Measure of air -Angle. 

Rule.— Multiply measure of angle in degrees by 3.1416, and divide by 1S0. 

Example. —What is circular measure of 24 0 10' 8 "? 

24 0 10' 8" X 3-1416 87008X3-1416 

-—- — — — -” .042 IO. 

180 180 X 60 X 60 

To Compute Measure of an -Angle, its Circular Measure 

toeing Griven. 

Rule. —Multiply circular measure of angle by 180, and divide by 3.1416. 


PROBABILITY. 

Probability of any event is the ratio of the favorable cases, to all the 
cases which are similarly circumstanced with regard to the occurrence. If 
an event have 3 chances for occurring and 2 for failing, sum of chances 
being 5, the fraction § will represent probability of its occurring and is taken 
as measure of it. Thus, from a receptacle containing 1 white and 2 black 
balls, the probability of drawing a white ball, by abstraction of 1, is \; prob¬ 
ability of throwing ace with a die is : in other words, the odds are 2 to 1 
against first, and 5 to 1 against second. 

If m -f- n — whole number of chances, m representing number which arc favorable, 

and n unfavorable. Therefore ——— —probability of event. 

m -p n 

Probabilities of two or more single events being known, probability of their oc¬ 
curring in succession may be determined by multiplying together the probabilities 
of their events, considered singly. 

Thus, probability of one event in two is expressed by of its occurring twice in 
succession, i x or of thrice in succession, X ^ X or -J-, etc. 

Illustration i.— If a cent is thrown twice into the air, the probability of its fall¬ 
ing with its head up, twice in succession, is as 1 to 4. Thus, it may fall: 

1. Head up twice in succession. \ 

2. Head up 1st time and wreath 2d time, f 1 1 

3. Wreath up 1st time and head 2d time, f Hence, ^ , — .25 — = 4 times. 

4. Wreath up twice in succession. ) J 5 

These are the only results possible, and being all similarly circumstanced as to 
probability, the probability of each case is as 1 to 4, or odds are as 3 to 1. 

Probability of either head or w'reatli being up twice in succession is as 1 to 1, or 
chances are even, because 1st and 4th cases favor such a result;' probability of head 
once and w'reath once in any order is as 1 to 2, because 2d and 3d cases favor such a 
result; and probability of head or wreath once is as 3 to 4, or odds are as 3 to 1, be¬ 
cause 1st, 2d, and 3d, or 2d, 3d, and 4th cases favor such a result. 

Note.— i to 2 is an equal chance, for x out of 2 chances = itoi, being an equal chance ; again, 1 to 
5 is 4 to 1, for 1 out of 5 chances is x to 4. 

2. —If there are 4 white balls and 6 black in a bag, what is the chance of a person 
drawing out 2 black at two successive trials? 

This is a combination without repetition. Hence, 6 — (2 — 1) = 5, 

and -- X 6 = — = — which x 2 for successive trials 2= — or —. 

1X221 215 

3. —Suppose with two bags, one containing 5 white balls and 2 black, and the other 
7 white and 3 black. 

Number of cases possible in one drawing from each bag is (5-)-2) x (7 + 3) =7 
X 10 = 70, because every ball in one bag may be drawn alike to one in the other. 








PROBABILITY. 


I 15 


Number of cases which favor drawing of a white ball from both bags is 5 X 7 = 35, 
for every one of the 5 white balls in one bag may be drawn in combination with every 
one of the 7 in the other. For a like cause, number of cases which favor drawing of 
a white ball from 1st bag and a black one from 2d is 5 X 3 = 15; a black ball from 1st 
bag and a white ball from 2d is 7 X 2 = 14; and a black ball from both is 3 X 2 = 6. 

Probability, therefore, of drawing is as 


5 X 7 35 


5X3 15 


3 to 11, 


1 to 1, a ivhite ball from both baas. 

7° 7° 2 ' 70 70 14 

a white ball from 1st, and a black from 2d. — I 4 — _L — x to 4, a black 


70 
6 3 


7 ° 35 


7 ° 5 

3 to 32, a black ball from 


o y 2 

ball from 1st, and a white from 2d. -- 

70 

5 X 3 2 X 7 20 

both. -—-= — = 29 to 41, a white ball from one, and a black from other , 

for both 2d and 3d cases favor this result ; hence, — 4 - — = —. - X 7 — 3 — - X 7 

5 14 7 ° 7 ° 

_ 64 __ 32 __ ^ i eas t one 'ujhite ball , for the 1st, 2d, and 3d cases favor this 

7° 35 ’ ’ ’ J 

result; hence, — -f- — 4 - — = — • 

2 14 5 35 

Again, if number of white and black balls in each bag are same, say 5 white and 
2 black,5 + 2X5 + 2 = 49, then probability of drawing is as 

r y r 2C C X 2 IO 

--- = — = 25 to 24, a white ball from both. ■—— = — — 10 to 30, a white ball 

49 49 49 49 

2X5 10 

from 1 st. and a black from 2d. -= ■—= 10 to 39, a black ball from 1st, and a 

49 49 


2X2 


: 4 to 45, a black ball from both. 


white from 2d. 

49 49 

4. —When two dice are thrown, probability that sum of numbers on upper sides 
is any given number, say 7, is as follows: 

As every one of the six numbers on one die may come up alike to, or in combi¬ 
nation with the other, number of throws is 6 X 6 = 36. 

f 1 and 6 ) 

Number 7 may be a combination of < 2 “ 5 [ ; and as these numbers may be 

(3 4) 

upon either die, there are 3X2 — 6 throws in favor of the combination of 7; hence 

6 1 

probability of throwing 7 is — = —, or as 1 to 5. 

5. —Probability of a player’s partner at Whist holding a given card is as follows: 
Number of cards held by the other 3 players is 3 X 13 = 39; probability, there¬ 
fore that it is held by partner is —, but it may be one of the 13 cards which he 

39 

I 13 I 

holds; hence probability is — X 13 = — = —, or as 1 to 2. 

’ 39 39 3 

6. _Probability of a player’s partner at Whist holding two given cards is as follows: 


OQ ^ 

Number of combinations of 39 things, taken 2 and 2 together, is ——— 


1X2 


; 741; 


therefore, probability that these 2 cards are in partner’s hand is 39 x 38 


1X2 


39 X 19 


— — 1 to 740; but they may be any 2 cards in partner’s hand; therefore, since 

741 w 

!3 y J 2 

number of combinations of 13 cards, taken 2 and 2 together, is =, —, 78, 

, . 78 2 . 

probability required is — - = —, or as 2 to 17. 

741 19 


1X2 


Similarly, probability that he holds any 3 given cards is as ——, or as 22 to 681. 


















116 


PROBABILITY. 


Probabilities at a game of Whist upon following points are: 

9. to 7, that one hand has two honors, and two hands one; 

9 to 55, that two hands have each two honors ; 

3 to 29, that each hand holds an honor; 

3 to 13, that one hand has three honors , and one hand one ; 

1 to 63, that four honors are held by one hand. 

7.—If 3 half-dollars are thrown into the air, probability of any of the possible com¬ 
binations of their falling is determined as follows : 


(f + i) , = (7) 3 +f(7H^(l) 

Hence, .125 = 1 to 7 in favor of 3 heads. 

7 (7)= - 375=3 10 5 “ 

(~) 3 =- 3 75 = 3 t0 5 “ 

(~~^j — .125 = i to 7 “ 


3 , 3 X 2 X 


1 X 2 X 3 \ 2 / 




3x2 
1X2 
3 X 2 X i 
1X2X3 


2 heads and 1 tail. 
1 head and 2 tails. 


3 tails. 


And in like manner, if 5 were thrown up, probability of any of their possible 
combinations would be determined as follows : 




5 1 5X4X3 

I X 2 \ 2 / '1X2X3 
5X4X3X2X1 / I \5 


(f \ 5 _I 5 x 4 x 3 x 2 ( 1\5 
\ 2 / ' 1X2X3X4I2/ 


IX2X3X4X5 


i(v) ; 


Hence, .031 25 — 1 to 31 in favor of 5 heads ; 

Y = • 156 25 = 5 to 27 
5 X 4 /i \5 

1 — 1 = .3125 = 10 to 22 

^^=.3125 = IO tO 22 

{ff-- 1 5 6 25 = 5 to 27 
1X2X3X4X512) — .Ojl 25 _ 


1X2 
5 X 4 X 3 
1X2X3 
5 X 4 X 3 X 2 

1 X 2 X 3 X 4 
5X4X3X2X1 


1 to 


4 heads and 1 tail; 

3 heads and 2 tails ; 
2 heads and 3 tails ; 
1 head and 4 tails; 

5 tails. 


All Wagers are founded upon the principle of product of the event, 
and contingent gain, being equal to amount at stake. 

Illustration i.— Suppose 3 horses, A, B, and C, are entered for a race, and X 
wagers 12 to 5 against A, n to 6 against B, and 10 to 7 against C. 

If A wins, X wins 6 + 7 — 12 = 1 

“B “ X “ 5 -|- 7 —11 = 1. 

“C “ X “ 5 + 6-10=1. 

Hence, X wins 1, whichever horse wins, from having taken field against each 
horse at odds named. 


Odds given in fa¬ 
vor of 


,5.67 18 

anCl i7~'i7'i7~i7 = I ' o6 = 1-06 t0 1 in f avor °f iaJcer of odds. 


A are 5 to 12 ) 


( A 

in favor of A. 

B “ 6 “ 11 - 

; corresponding probabil¬ 
ity is 

\ 17 
XT 

“ b; 

O 

M 

O 


l Tf 

“ C, 













PROBABILITY. 


ii; 


2.—Odds given upon first seven favorite horses for Oaks Stakes of 1828 were so 
great, that probability in favor of taker of the odds when reduced was as follows: 


1st, 5 to 2 ; 2d, 5 to 2; 3d, 4 to 1; 4th, 7 to 1; 5th, 14 to 1; 6th, 14 to 1; 7th, 15 to 1 

4 X 3 X 161= 192 
1 X 7 X 16=1112 
3 X 7 X 3 = 63 


7 ^7 ^5 8 15 15 16 


4 + A + A = 4,^ + _3_ 
7 i5 16 7 '3 16 


' 7 X 3 X 16 336 

= 367 -f- 336 = 1.092 = 1.092 to 1, in favor of taker of odds, yet neither of the horses 
upon which these odds were given won. 


3. —If odds are 3 to 1 against a horse in a race, and 6 to 1 against another horse 
in a second race, probability of 1st horse winning is |, and of other A. Therefore 
probability of both races being won is ^g-, and odds against it 27 to i,or 1000 to 37.037. 
Odds upon such an event were given in 1828 at 1000 to 60, or 16.67 t0 *• 

4. —Two persons play for a certain stake, to be won by winner of three games or 
results. One having won one and the other two, they decide to divide the sum, 
proportionate to their interest. How much of it should each one receive? 


Operation. —If winner of two games should win game to be played, he would be 
entitled to the whole sum; if he lost, he would be entitled to half of it. Now as 

one event is as probable as the other,-1-= ■—, half of which —, or share 

122 4 

of winner of two games. 


When events are wholly independent, so that occurrence of one does not 
affect that of the other, probability that both will occur is product of proba¬ 
bilities that each will occur. 


Note. —It is indifferent whether events are to occur together or consecutively. 

Illustration i.— Assume three boxes, each containing white and black balls as 
follows: 

6 whito, 5 black; 7 white, 2 black; 8 white, 10 black. What is chance of drawing 
from them a white, black, and a white ball? 


628 6 —I— 2 ~4“ 8 

Probabilities are —, — , and — , product of which = — ' ——— = 17.625 to 1. 

11 9 18 297 

2.—A gives an answer correctly 3 times out of 4, B 4 times out of 5, and C 6 out 
of 7. What is probability of an event which A and B declare correct and C denies? 
Operation.— Compound probability that A and B answer correctly and C denies 

(all 3 of tvhich are in favor of event) is — X — X — = — = — . 

4 5 7 i 4 ° 35 


Compound probability that A and B deny and C is correct (all 3 of which are 
... 1 1 6 6 3 

against event) is — X — X — — — = —. 

4 5 7 140 70 


Then correct, divided by sum _ 3 _ [ z , 3 \_ -8714 _ M 2 

of correct and incorrect, “ 35 \Is + 7°/ “ .857 14+ .428 57 I 


Odds loetw-een TLesnlts or Chances, and between any 
jST niTiloer and Whole Number, at various Odds against 
each, also Value of each Chance in parts of IOO. 


Odds against 
each. 

Value of j 
Chance. 

Odds against 
each. 

Value of 
Chance. 

Odds against 
each. 

Value of 
Chance. 

Odds against 
each. 

Value of 
Chance. 

Even 

50 

2 to 1 

33-33 

6.5 to 1 

13-33 

15 to 1 

6.25 

11 to 10 

47.62 

2. 5 “ 1 

28.57 

7 “ 1 

12.5 

18 “ 1 

5.26 

6 “ 5 

45-45 

3 “ 1 

25 

7-5 “ 1 

11.76 

to 

0 

M 

4.76 

5 “ 4 

44.44 

3-5 “ 1 

22.22 

8 “ 1 

II. II 

25 1 

3 - 8 4 

5-5 “ 4 

42.1 

4 “ 1 

20 

8.5 “ 1 

10.52 

30 “ 1 

3.22 

6 “4 

40 

4-5 “ 1 

18.18 

9 “ 1 

IO 

40 “ 1 

2.44 

6-5 “ 4 

38.1 

5 “ 1 

16.66 

9-5 1 

9-52 

50 “ 1 

1.96 

7 “ 4 

36-3 6 

5-5 “ 1 

15 - 3 8 

10 “ 1 

9.09 

60 “ 1 

1.64 

7-5 “ 4 

34 - 7 8 

6 “ 1 

14.28 

12 “ 1 

7-7 

100 “ 1 

•99 

Operation. — Divide 100, or unit, as case may be, by sum of odds, and 

multiply 


quotient by lesser chance or odds. 


Illustration.— 6 to 4. 6 -}- 4 = 10, and 100 - 4 -10 X 4 — 40, value of chance. 























I 1 8 WEIGHTS OF IRON, STEEL, COPPER, ETC. 

WEIGHTS OF IRON, STEEL, COPPER, ETC. 


'W'ro'u.glit Iron, Steel, Copper, and. Brass [Plates. 
soft rolled. ( American Gauge.) 


No. of 
Gauge. 

Thickness. 

Iron. 

Per Sq 
S teel. 

uare Foot. 
Copper. 

Brass. 


Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

oooo 

.46 or T 7 g- full 

18.4575 

18.7036 

20.838 

19.688 

ooo 

.409 64 

16.4368 

16.6559 

18.556 7 

I 7-532 6 

oo 

.364 8 or f light 

14.6376 

14.8328 

16.525 4 

I 5 - 6 I 3 4 

o 

.324 86 or i “ 

I 3 - 035 I 

13.2088 

14.716 2 

13.904 

X 

.2893 

11.6082 

II.7629 , 

13-1053 

12.382 

2 

.257 63 or £ full 

10.3374 

IO.4752 

11.670 6 

11.026 6 

3 

.229 42 

9-2055 

9-3283 

10.392 7 

9.819 2 

4 

.204 31 or -g- full 

8.1979 

8-3073 

9-255 2 

8-744 5 

5 

.181 94 or A- light 

7.3004 

7-3977 

8.241 9 

7.787 

6 

.162 02 

6.5011 

6.5878 

7-339 5 

6-934 5 

7 

.144 28 

5.7892 

5.8664 

6-535 9 

6.175 2 

8 

.128 49 or full 

5-1557 

5.2244 

5.820 6 

5-499 4 

9 

•114 43 

4-5915 

4.6527 

5-1837 

4.897 6 

IO 

.101 89 or jL full 

4.0884 

4.1428 

4.615 6 

4.3609 

ii 

.090 742 

3.641 

3.6896 

4.110 6 

3.883 8 

12 

.080 808 

3.2424 

3.2856 

3.660 6 

3-458 6 

I 3 

.071 961 

2.8874 

2.9259 

3-259 8 

3-079 9 

14 

.064 084 

2.5714 

2.6057 

2.903 

2.742 8 

I 5 

.057 068 

2.2899 

2.3204 

2.585 2 

2.4425 

16 

.050 82 or -Ty full 

2.0392 

2.0664 

2.302 1 

2.175 1 

17 

.045 257 

1.8159 

1.8402 

2.050 1 

1-937 

18 

.040 303 

1.6172 

1.6387 

1.825 7 

1-725 

19 

•035 89 

1.44 

1-4593 

1.625 8 

i- 536 i 

20 

.031 961 

1.2824 

1.2995 

1.447 8 

1.3679 

21 

.028 462 

1.142 

I-I 573 

1.2893 

1.218 2 

22 

•025 347 

1.017 

1.0306 

1.148 2 

1.084 9 

23 

.022 571 

•9057 

.9177 

1.022 5 

.966 04 

24 

.021 1 

.8065 

•8173 

•9io 53 

.860 28 

25 

.0179 

.7182 

.7278 

.810 87 

.766 12 

26 

.015 94 

.6396 

.6481 

.722 08 

.682 23 

27 

.014 195 

.5696 

•5772 

.643 03 

•607 55 

28 

.012 641 

.5072 

.514 

•572 64 

• 54 i 03 

29 

•on 257 

.4517 

•4577 

•509 94 

.481 8 

30 

.010 025 

.4023 

.4076 

•454 13 

.429 07 

31 

.008 928 

•3582 

•363 

•404 44 

.382 12 

32 

.007 95 

• 3 i 9 

.3232 

•360 14 

.340 26 

33 

.007 08 

.2841 

.2879 

•320 72 

•3°3 02 

34 

.006 304 

,2529 

•2563 

•285 57 

.269 81 

35 

.005 614 

.2253 

.2283 

•254 31 

.240 28 

36 

.005 

.2006 

•2033 

.2265 

.214 

37 

.004 453 

.1787 

.181 

.201 72 

.19059 

38 

.003 965 

•i 59 i 

.1612 

.179 61 

.1697 

39 

.003 531 

.1417 

.1436 

•159 95 

•151 13 

40 

.003 144 

.1261 

.1278 

.142 42 

•134 56 

Specific Gravities. 

7.704 

7.806 

8.698 

8.218 

Weights of a Cube Foot.. 

481.75 

487-75 

543-6 

5 I 3-6 

u 

“ Inch.. 

.278 7 

.2823 

.3146 

.297 2 



















WEIGHTS OF IRON, STEEL, COPPER, ETC. I 1 9 
"Wrought Iron, Steel, Copper, and Brass IMates. 


{Birmingham Gauge.) 


No. of 
Gauge. 

Thickness. 

Iron. 

Per Square Foot. 

Steel. Copper. 

Brass. 


Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

0000 

•454 or T 7 ¥ full 

18.2167 

18.4596 

20.5662 

19.4312 

000 

•425 

I 7 - 053 I 

17.2805 

19.2525 

18.19 

00 

.38 or f- full 

15-2475 

15.4508 

17.214 

16.264 

0 

•34 orl “ 

13.6425 

13.8244 

15402 

14-552 

1 

•3 

12.0375 

12.198 

13-59 

12.84 

2 

.284 

11 -3955 

n -5474 

12.8652 

12.1552 

3 

.259 or 4 full 

10.3924 

10.5309 

11.7327 

11.0852 

4 

.238 

9-5497 

9.6771 

10.7814 

10.1864 

5 

.22 

8.8275 

8.9452 

9.966 

9.416 

6 

.203 or - 5 - full 

8.1454 

8.254 

9-1959 

8.6884 

7 

.18 or t 8 ¥ light 

7.2225 

7.3188 

8.154 

7.704 

8 

.165 or A- “ 

6.6206 

6.7089 

7-4745 

7.062 

9 

.148 or 4 full 

5-9385 

6.0177 

6.7044 

6.3344 

10 

• 134 

5-3767 

5.4484 

6.0702 

5-7352 

11 

.12 or 1 light 

4.815 

4.8792 

5.436 

5 -I 36 

12 

.109 

4-3736 

4 - 43 I 9 

4-9377 

4.6652 

13 

•095 01 to light 

3.8119 

3.8627 

4-3035 

4.066 

14 

.083 

3-3304 

3-3748 

3-7599 

3-5524 

IS 

.072 

2.889 

2.9275 

3.2616 

3.0816 

16 

.065 

2.6081 

2.6429 

2.9445 

2.782 

17 

.058 

2.3272 

2.3583 

2.6274 

2.4824 

18 

•049 °r wo % ht 

1.9661 

1.9923 

2.2197 

2.0972 

19 

.042 

1.6852 

1.7077 

1.9026 

1.7976 

20 

•035 

1.4044 

1-4231 

I -5855 

1.498 

21 

.032 

1.284 

1.3011 

1.4496 

1.3696 

22 

.028 

1-1235 

1-1385 

1.2684 

1.1984 

23 

•025 or ^ 

1.0031 

1.0165 

1-1325 

1.07 

24 

.022 

.8827 

•8945 

.9966 

.9416 

25 

.02 or f-Q 

.8025 

.8132 

.906 

.856 

26 

.018 

.7222 

• 73 i 9 

.8154 

.7704 

27 

.016 

.642 

.6506 

.7248 

.6848 

28 

.014 

•5617 

.5692 

•6342 

•5992 

29 

.013 

.5216 

.5286 

.5889 

•5564 

3° 

.012 

.4815 

.4879 

•5436 

•5136 

3i 

.01 or you 

.4012 

.4066 

•453 

.428 

3 2 

.009 

.3611 

•3659 

•4077 

•3852 

33 

.008 

.321 

■3253 

.3624 

•3424 

34 

.007 

.2809 

.2846 

• 3 i 7 i 

.2996 

35 

•005 or woo 

.2006 

•2033 

.2265 

.214 

36 

.004 or -gTo 

.1605 

.1626 

.1812 

.1712 


Thickness of Sheet Silver, Gold, etc. 

By Birmingham Gauge for these Metals. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

1 

.004 

7 

.015 

13 

•036 

19 

.064 

25 

•095 

3 1 

•133 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

•143 

3 

.008 

9 

.019 

15 

-047 

21 

.072 

27 

• IT 3 

33 

-145 

4 

.01 

10 

.024 

16 

.051 

22 

•074 

28 

.12 

34 

.148 

5 

.013 

11 

.029 

17 

•057 

23 

•077 

29 

.124 

35 

.158 

6 

.013 

12 

•034 

18 

.061 

24 

.082 

30 

.126 

36 

.167 











































120 


WEIGHTS OF IRON, STEEL, COPPER, ETC, 


W'roxiglxt Iron, Steel, Copper, and Brass NWire. 
American Gauge, f. full, 1 . light. 


No. of 
Gauge. 

Diameter. 

Iron. 

Per Linejs 
S teel. 

iL Foot. 
Copper. 

Brass. 


Inch. 

Lbs. 

Lbs. 

Lbs. - 

Lbs. 

oooo 

.46 or T \ f. 

.560 74 

.566 03 

.640 513 

.605 176 

ooo 

.40964 

•444 683 

•448 879 

.507 946 

•479 908 

oo 

.364 8 or 1 1. 

.352 659 

•355 986 

.402 83 

.380 666 

o 

.324 86 or s f. 

.279 665 

.282 303 

• 3 i 9 45 i 

.301 816 

i 

.2893 

.221 789 

.223 891 

•253 342 

•239 353 

2 

•257 63 or i f. 

.175 888 

.177 548 

.200 911 

.189 818 

3 

.229 42 

.139 48 

.140 796 

•159 323 

.150522 

4 

.20431 or £ f. 

.110 616 

.111 66 

• 126 353 

.119376 

5 

.181 94 or T 3 ^ 1. 

.087 72 

.088 548 

.100 2 

.094 666 

6 

.162 02 

.069 565 

.070 221 

.079 462 

•075 075 

7 

.144 28 

.055 165 

•055 685 

.063 013 

•059 545 

8 

.128 49 or 1 f. 

•043 75 i 

.044 164 

.049 976 

.047 219 

9 

•ii 4 43 

.034 699 

.035 026 

.039 636 

•037 437 

IO 

.101 89 or jL f. 

.027 512 

.027 772 

.031 426 

.029 687 

ii 

.090 742 

.021 82 

.022 026 

.024 924 

•023 549 

12 

.080 808 

.017304 

.017 468 

.019 766 

.018 676 

13 

.071 961 

.013 722 

.013 851 

.015 674 

.014 809 

14 

.064 084 

.010 886 

.010 989 

.012435 

.011 746 

15 

.057 068 

.008 631 

.008 712 

.009 859 

.009315 

l6 

.050 82 or ^ f. 

.006 845 

.006 909 

.007 819 

.007 587 

17 

•045 257 

.005 427 

.005 478 

.006 199 

.005 857 

18 

.040 303 

.004 304 

.004 344 

.004 916 

.004 645 

19 

•035 89 

.003 413 

•003 445 

.003 899 

.003 684 

20 

.031 961 

.002 708 

.002 734 

.003 094 

.002 92 

21 

.028 462 

.002 147 

.002 167 

.002 452 

.002 317 

22 

•025 347 

.001 703 

.001 719 

.001 945 

.001 838 

23 

.022 571 

•001 35 

.001 363 

.001 542 

.001 457 

24 

.020 1 or J- f. 

.001 071 

.001 081 

.001 223 

.001 155 

25 

.0179 

.000 849 1 

.000 857 1 

.000 969 9 

.000 916 3 

26 

• OI 5 94 

.000 673 4 

.000 679 7 

.000 769 2 

.000 726 7 

27 

.014 195 

.000 534 

.000 539 1 

.000 609 9 

.000 576 3 

28 

.012 641 

.000 423 5 

.000 427 s 

.000 483 7 

.000457 

29 

.011 257 

.000 335 8 

.000 338 9 

.000 383 5 

.000 362 4 

30 

• OI ° o2 5 0 r T k f - 

.000 266 3 

.000 268 8 

.000 304 2 

.000 287 4 

31 

.008 928 

.000 211 3 

.000 213 2 

.000 241 3 

.000 228 

32 

.007 95 

.000 167 5 

.000 169 1 

.000 191 3 

.000 180 8 

33 

•007 08 

.000 132 8 

.000 134 1 

.000 151 7 

.000 143 4 

34 

.006 304 

.000 105 3 

.000 106 3 

.000 120 4 

.000 113 7 

35 

.005 614 

.000 083 66 

.000 084 45 

.000 095 6 

.000 090 15 

36 

•°°5 or ¥ fo. 

.000 066 25 

.000 066 87 

.000 075 7 

.000 071 5 

37 

•004 453 

.000 052 55 

.000 053 04 

.000 060 03 

.000 056 71 

38 

.003 965 

.000 041 66 

.000 042 05 

.000 047 58 

.000 044 96 

39 

•003 53 i 

.000 033 05 

.000 033 36 

.000 037 75 

.000 035 66 

40 

.003 144 

.000 026 2 

.000 026 44 

.000 029 92 

.000 028 27 

Specific Gravities. 

• • 7-774 

7.847 

8.88 

8.386 

Weights of a Cube Foot 

. • 485-87 

490.45 

554-988 

524.16 

U 

“ Inch 

.. .2812 

.2838 

.3212 

•3033 


Specific Gravities to determine the computations of these weights were made by 
author for Messrs. J. R. Browne & Sharpe, Providence, R. I. 



















WEIGHTS OF IRON, STEEL, COPPER, ETC 


12 I 


■Wronglit Iron, Steel, Copper, and. Brass 'Wire. 
Birmingham Wire Gauge, f. full, 1. light. 


No. of 
Gauge. 

Thickness. 

Iron. | 

Per Line 
Steel. 

al Foot. 
Copper. 

Brass. 


Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

oooo 

•454 or Jg- f. 

.546 207 

• 55 i 36 

.623 913 

.589 286 

ooo 

•425 

.478 656 

.483 172 

•546 752 

• 5 - 16 407 

oo 

.38 or | f. 

.382 66 

.386 27 

•437 099 

.412 84 

o 

•34 or 1 f. 

•30634 

.30923 

•349 921 

•330 5 

i 

•3 

•2385 

.240 75 

.272 43 

•257 3 r 

2 

.284 

•213 738 

•215 755 

.244 146 

.230 596 

3 

•259 or 1 f. 

.177 765 

.179442 

S203 054 

.191 785 

4 

.238 

.150 107 

•151 523 

.171 461 

.161 945 

5 

.22 

.128 26 

.129 47 

.146 507 

•r 3 8 376 

6 

.203 or l f. 

.109 204 

.110234 

.124 74 

.117 817 

7 

H 

00 

0 

t-t 

,®! w 

.085 86 

.086 667 

.098 075 

.092 632 

8 

.165 or } 1. 

.072 146 

.072 827 

.082 41 

.077 836 

9 

.148 or i f. 

.058 046 

•058 593 

.066 303 

.062 624 

IO 

•134 

•047 583 

.048 032 

•054 353 

.051 336 

ii 

.12 or A 1. 

.038 16 

.038 52 

•043 589 

.041 17 

12 

.109 

.031 485 

.031 782 

.035 964 

.033 968 

13 

•095 or -L 1. 

.023 916 

• 1?4 * 4 2 

.027 319 

.025 802 

14 

.083 

.018 256 

.018 428 

.020 853 

.019 696 

15 

.072 

.013 728 

.013 867 

.015 692 

.014 821 

16 

.065 

.011 196 

.011 302 

.012 789 

.012 079 

17 

.058 

.008 915 

.008 999 

.010 183 

.009 618 

18 

•°49 or ^ 1. 

.006 363 

.006 423 

.007 268 

.006 864 

19 

.042 

.004 675 

.004 719 

•005 34 

.005 043 

20 

•035 

.003 246 

.003 277 

.003 708 

.003 502 

21 

.032 

.002 714 

.002 739 

.003 1 

.002 928 

22 

.028 

.002 078 

.002 097 

.002 373 

.002 241 

23 

.025 or 

.001 656 

.001 672 

.001 892 

.001 787 

24 

.022 

.001 283 

.001 295 

.001 465 

.001 384 

25 

.02 or gL 

.001 06 

.001 070 

.001 211 

.001 144 

26 

.018 

.000 858 6 

.000 866 7 

.000 980 7 

.000 <326 

27 

.016 

.000 678 4 

.000 684 8 

.000 774 9 

.000 731 9 

28 

.014 

.0005194 

.000 524 3 

.0005933 

.000 560 4 

29 

•013 

.000 447 9 

.000 452 1 

.000 511 6 

.000 483 2 

30 

.012 

.000 381 6 

.000 385 2 

.000 435 9 

.000411 7 

3 i 

.01 or jig- 

.000 265 

.000 267 5 

.000 302 7 

.000 285 9 

32 

.009 

.000 214 7 

.000 216 7 

.000 245 2 

.000 231 6 

33 

.008 

.000 169 6 

.000 171 2 

.000 193 7 

.000 183 

34 

.007 

.000 129 9 

.000 131 1 

•ooo 148 3 

.000 140 1 

35 

; -005 or 

.000 066 25 

j .000 066 88 

.000 075 68 

.000 071 48 

36 

; .004 or -g-g-Q- 

.000 042 4 

1 .000 042 8 

.000 048 43 

.000 045 74 


Tliielmess of Plates. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

1 

•3125 

9 

.156 25 

17 

.056 25 

25 

.02344 

2 

.281 25 

10 

.140 625 

18 

•05 

26 

.021 875 

3 

•25 

11 

.125 

19 

•043 75 

27 

.020 312 

4 

•234 375 

12 

.112 5 

20 

•037 5 

28 

.018 75 

5 

.218 75 

13 

.1 

21 

•034 375 

29 

.017 19 

6 

.203 125 

14 

.087 5 

22 

•031 25 

30 

.015 625 

7 

•1875 

15 

•075 

23 

.028 125 

3 i 

.014 06 

8 

•171 875 

16 

.062 5 

24 

.025 

32 

.012 5 


L 



































122 


WIRE GAUGES 


WIRE GAUGES. (English.) 

Warrington (Hylands Brothers). 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

[ No. 

Inch. 

7/0 

X 

0 

.326 

6 

.191 

11 

.I17 

I 7 

•053 

6/0 


1 

•3 

7 

.174 

12 

.1 

18 

.047 

5 /o 

%> 

2 

.274 

8 

•159 

13 

.09 

19 

.041 

4/0 

% 

3 

•25 

9 

.146 

M 

.079 

20 

.036 

3/0 

% 

4 

.229 

10 

•133 

15 

.069 

21 

•031S 

2/0 

11 / 

m 

5 

.209 

10.5 

.125 

16 

.0625 

22 

.028 




Sir Joseph Whitworth & Co. 

\s. 




No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

1 

.001 

14 

.014 

34 

•034 

85 

.085 

240 

.24 

2 

.002 

15 

.015 

36 

.036 

90 

.09 

260 

.26 

3 

.003 

16 

.016 

38 

.038 

95 

.09 

280 

.28 

4 

.004 

17 

.017 

40 

.04 

100 

.1 

300 

•3 

5 

.005 

18 

.018 

45 

•045 

no 

.11 

325 

•325 

6 

.006 

19 

.019 

50 

•05 

120 

.12 

350 

•35 

7 

.007 

2 Q 

.02 

55 

•055 

135 

•135 

375 

•375 

8 

.008 

22 

.022 

60 

.06 

150 

•15 

400 

•4 

9 

.009 

24 

.024 

65 

.065 

165 

.165 

425 

•425 

10 

.01 

26 

.026 

70 

.07 

180 

.18 

450 

•45 

11 

.Oil 

28 

.028 

75 

•075 

200 

.2 

475 

•475 

12 

.012 

30 

•03 

80 

.08 

220 

.22 

500 

•5 

13 

.013 

32 

.032 








Sir Joseph Whitworth, in 1857, introduced a Standard Wire-Gauge, rang¬ 
ing from half an inch to a thousandth, and comprising 62 measurements. 
It commences with least thickness, and increases by thousandths of an inch 
up to half an inch. Smallest thickness, of an inch, is No. 1; No. 2 

is and so on, increasing up to No. 20 by intervals of yq^y ’ from 

No. 20 to No. 40 by yofo * anc ^ from No. 40 to No. 100 by yoto- The 
thicknesses are designated or marked by their respective numbers in thou¬ 
sandths of an inch. 

This gauge is entering into general use in England. 


New Standard AN ire Grange of Great Britain, 

1884 . 


No. 

Inch. 

No. 

Inch. | 

No. 

Inch. 

No. 

Inch. 

7/0 

•5 

8 

.160 

22 

.028 

36 

.OO76 

6/0 

.464 

9 

.144 

23 

.024 

37 

.0068 

5/0 

•432 

10 

.128 

24 

.022 

38 

.006 

4/0 

•4 

11 

.Il6 

25 

.02 

39 

.OO52 

3/0 

•372 

12 

.104 

26 

.Ol8 

40 

.OO48 

2/0 

•348 

13 

.092 

27 

.0164 

4 * 

.OO44 

O 

•324 

14 

.08 

28 

.OI48 

42 

.OO4 

I 

•3 

15 

.072 

29 

.OI36 

43 

.OO36 

2 

.276 

16 

.064 

30 

.OI24 

44 

.OO32 

3 

.252 

17 

.056 

31 

•OIl6 

45 

.0028 

4 

.232 

18 

.048 

32 

.OIOS 

46 

.OO24 

5 

.212 

r 9 

.04 

33 

.OI 

47 

.002 

6 

.192 

20 

.036 

34 

.OO92 

48 

.OOl6 

7 

.176 

21 

.032 

35 

.0084 

49 

.0012 


No. 50, .001 inch. 

































































WIRE GAUGES.-GAS PIPES AND WIRE COED. 123 


French. (Jauges de Fils de Fer ). 

French wire-gauges, alike to the English, have been subjected to variation. 
Following table contains diameters of the numbers of the Limoges gauge. 


Wire-Gauge ( Jauge de Limoges ). 


Number. 

Millimetre. 

Inch. 

Number. 

Millimetre. 

Inch. 

Number. 

Millimetre. 

Inch. 

O 

•39 

.0154 

9 

x *35 

•0532 

18 

3-4 

•134 

I 

•45 

.0177 

IO 

1.46 

•057s 

19 

3-95 

.156 

2 

•56 

.0221 

II 

1.68 

.0661 

20 

4-5 

• I 77 

3 

.67 

.0264 

12 

1.8 

.0706 

21 

5 -i 

.201 

4 

•79 

.0311 

13 

1.91 

.0752 

22 

5-65 

.222 

5 

•9 

•0354 

14 

2.02 

•0795 

23 

6.2 

.244 

6 

I.OI 

.0398 

15 

2.14 

.0843 

24 

6.8 

.268 

7 

1.12 

.0441 

l6 

2.25 

.0886 




8 

I.24 

.0488 

17 

2.84 

.112 





For (Galvanized Iron Wire. 


Number. 

Millimetre. 

Inch. 

Number. 

Millimetre. 

Inch. 

Number. 

Millimetre. 

Inch. 

I 

.6 

.0236 

9 

1.4 

•0551 

J 7 

3 - 

. 

M 

W 

OO 

i 

2 

•7 

.0276 

IO 

x -5 

.0591 

18 

3-4 

•134 

3 

.8 

•0315 

II 

1.6 

.063 

19 

3-9 

•154 

4 

•9 

•0354 

12 

1.8 

.0709 

20 

4.4 

•173 

5 

I. 

•0394 

13 

2. 

.0787 

21 

4.9 

• x 93 

6 

1.1 

•0433 

14 

2.2 

.0866 

22 

5-4 

.213 

7 

1.2 

•0473 

15 

2.4 

•0945 

23 

5-9 

.232 

8 

i-3 

.0512 

l6 

2.7 

.106 





For Wire and Bars. 


Mark. 

Millimetre. 

Mark. 

Millimetre.' 

Mark. 

Millimetre. 

Mark. 

Millimetre. 

Mark. 

Millimetre 

p 

5 

7 

12 

x 3 

20 

19 

39 

25 

70 

I 

6 

8 

13 

14 

22 

20 

44 

26 

76 

2 

7 

9 

14 

i5 

24 

21 

49 

27 

82 

3 

8 

IO 

15 

l6 

27 

22 

54 

28 

88 

4 

9 

II 

l6 

17 

30 

23 

59 

29 

94 

5 

IO 

12 

18 

18 

34 

24 

64 

30 

IOO 

6 

II 










Thickness of Gras Pipes. 


Diameter. 

Thickness. 

Diameter. 

Thickness. 

Diameter. 

Thickness 

i -5 to 3 

•25 

8 to 10 

•5 

14 to 15 

•75 

4 “6 

•375 

12 “ 13 

.625 

16 “ 48 

•875 


Copper "Wire Cord. 

Circumference and Safe Load. 


Inch. 


Circumference.25 

Safe load in Lbs. 34 


Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Ins. 

Ius. 

•375 

•5 

.625 

•75 

I 

1.125 

1.25 

50 

75 

112 

168 

224 

336 

448 


Zinc—sheets. 


Thickness and Weight per Square Foot. 


Inch. 

,0311 = IO OZ 
.0457 = 12 OZ 


Inch. 

0534 = 14 oz 
0611 = 16 oz 


Inch. 

0686 = 18 oz 
0761 = 20 OZ, 

















































































124 WEIGHT AND STRENGTH OF WIRE, IRON, ETC. 


WEIGHT AND STRENGTH OF WIRE, IRON, ETC. 
"Weight and. Strength of Warrington Iron Wire. 


Manufactured by Rylands Brothers. (England.) 


Weight per ioo Lineal Feet . 



Diame¬ 

ter. 


Breaking 

W eight. 



| 

Breaking 

Weight. 

No. 

Weight 

An¬ 

nealed. 

Bright. 

No. 

Diameter. 

Weight. 

An¬ 

nealed. 

Bright. 

Gauge. 

Inch 

Lbs. 

Lbs. 

Lbs. 

Gauge. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

7/0 

X 

64.46 

3490 

5233 

9 ‘ 

.146 

5-5 

298 

447 

6/0 

% 

56.66 

3066 

4603 

10 

•133 

4-43 

247 

370 

5 /o 

Xe 

49-36 

2673 

4OOO 

10.5 

.125 

4-03 

2l8 

327 

4/0 

/U 2 

42.53 

2303 

3457 

n 

.117 

3-53 

I 9 I 

288 

3/0 

% 

36.26 

1963 

2945 

12 

.1 

2.66 

145 

217 

2/0 

3^2 

30.46 

1653 

2473 

13 

.09 

2.1 

113 

169 

O 

.326 

27.36 

i486 

2226 

14 

.079 

1.6 

87 

130 

I 

•3 

23-3 

1257 

1885 

15 

.069 

1.23 

66 

99 

2 

.274 

19.36 

IO46 

1572 

16 

.0625 

.96 

53 

77 

3 

•25 

16.13 

873 

1309 

17 

•053 

•73 

39 

59 

4 

.229 

13-53 

732 

1098 

18 

.047 

•56 

3 i 

46 

5 

.209 

II.26 

6 lO 

9i3 

*9 

.041 

•43 

23 

35 

6 

.191 

9.4 

509 

763 

20 

.036 

•33 

18 

27 

7 

.174 

7.8 

422 

633 

21 

•031 25 

.26. 

14 

21 

8 

•159 1 

6-53 

353 

5i9 

22 

.028 

.2 

11 

16 


To Compute Length of IOO Pounds of "Wire of a Given 

Diameter. 


Rule. —Divide following numbers by square of diameter, in parts of an 
inch, and quotient is length in feet. 


37.68 for wrought iron. 
37.45 for steel. 


33.42 for copper. 
34.41 for brass. 
13.64 for platinum. 


28 for silver. 
15.3 for gold. 


Window Glass. 

Thickness and "Weight per Square Foot. 


No. 

Thickness. 

Weight. 

No. 

Thickness. 

Weight. 

No. 

Thickness. 

Weight. 


Inch. 

Oz. 


Inch. 

Oz. 


Inch. 

Oz. 

12 

•059 

12 

17 

.083 

17 

2'6 

.125 

26 

13 

.063 

13 

19 

.O9I 

19 

32 

•154 

32 

15 

.O7I 

IS 

21 

. I 

21 

36 

. 167 

36 

l6 

•°77 

16 

24 

.III 

24 

42 

.2 

42 


Tex*ne Plates. 

Term Plates —Are of iron covered with an amalgam of lead. 


Thickness and. W r eight of Galvanized Sheet Iron. 


Sheet 2 Feet in Width by from 6 to 9 Feet in Length (M. Lejferts). 


•eSniif) 

Weight 

per 

Sq. Foot. 

Wire 

Gauge. 

Weight 

per 

Sq. Foot. 

Wire 

Gauge. 

Weight 

per 

Sq Foot. 

Wire 

Gauge. 

Weight 

per 

Sq. Foot. 

Wire 

Gauge. 

Weight 

per 

Sq. Foot. 

Wire 

Gauge. 

Weight 

per 

Sq. Foot. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

29 

12 

26 

15 

23 

20 

20 

27 

17 

36 

14 

53 

28 

13 

25 

16 

22 

22 

19 

30 

16 

42 

13 

61 

27 

14 

24 

18 

21 

24 

18 

35 

15 

46. 

12 

70 
































































WEIGHTS OP METALS. 


125 


Wrought Iron. 


"Weight of Sqnare Rolled. Iron, 
From .125 Inch to 10 Inches, one foot in length. 


Side. 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Ins. 

Lbs. 1 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

•053 

2.125 

I 5-263 

4.125 

57-517 

6.25 

I32.O4 

•25 

.211 

•25 

17.II2 

•25 

61.055 

•5 

142.816 

•375 

•475 

•375 

19.066 

•375 

64.7 

•75 

I 54 -OI 2 

•5 

•845 

•5 

21.12 

•5 

68.448 

7 

165.632 

.625 

1.32 

.625 

23.292 

.625 

72.305 

•25 

177.672 

•75 

1.901 

•75 

25-56 

•75 

76.264 

•5 

190.136 

•875 

2.588 

•875 

27-939 

•875 

80.333 

•75 

203.024 

1 

3-38 

3 

30.416 

5 

84.48 

8 

216.336 

.125 

4.278 

.125 

33 -oi 

.125 

88.784 

•25 

230.068 

•25 

5.28 

•75 

35-704 

•25 

93- i6 8 

•5 

244.22 

•375 

6-39 

•375 

38-503 

•375 

97-657 

•75 

258,8 

•5 

7.604 

•5 

41.408 

•5 

102.24 

9 

273.792 

.625 

8.926 

.625 

44.418 

.625 

106.953 

•25 

289.22 

•75 

10.352 

•75 

47-534 

•75 

111.756 

•5 

305 056 

•875 

11.883 

.875 

50.756 

•875 

116.671 

•75 

321-33 

2 

I 3-52 1 

4 

54.084 1 

6 

121.664 

10. 

327.92 


Illustration. —What is weight of a bar 1.5 inches, by 12 inches in length? 

In column 1st, find 1.5; opposite to it is 7.604 lbs., which is 7 lbs. and .604 of a lb. 
If lesser denomination of ounces is required, result is obtained as follows: 


Multiply remainder by 16 , point off the decimals, and the figures remain¬ 
ing on left of the point will give number of ounces. 

Thus, .604 of a lb. = .604 X 16 = .9.664 = 7 lbs . 9.664 ounces . 

To Compuite 'Weight for less than a Foot in Length. 
Operation. —What is weight of a bar 6.25 inches square and 10.5 inches long? 

In column 7th, opposite to 6.25 is 132.04, which is weight for a foot in length. 

6.25 X 12 inches =132.04 6 ins. =.5 =66.02 

3 “ =.25 =33.01 

1.5 “ = .125 = 16.505 

115.535 lbs . 


"Weight of fvngle Iron, 

From 1.25 to 4.5 Inches, one foot in length. 
Thickness measured in Middle of each Side. 


L Equal Sides. 


Sides. 

Thick¬ 

ness. 

Weight 

Ins. 

Inch. 

Lbs. 

I.25XI.25 

•1875 

i -5 

i -5 X1.5 

•1875 

2 

I- 75 XI -75 

•25 

3 

2 X 2 

•25 

3-5 

2.25X2.25 

•3125 

4-5 

2-5 X 2.5 

•3125 

5 

3 X3 

•375 

7 

3-5 X3.5 

•4375 

9 

4 X4 

•5 

12.5 

4-5 X 4.5 

.-5 

14 

4-5 X 4-5 

•5625 

16 


L Unequal Sides. 


Sides. 

Thick¬ 

ness. 

Weight 

Ins. 

Inch. 

Lbs. 

3 X2.5 

•375 

6.25 

3 - 5 X 3 

•4375 

7-75 

3 - 5 X 3 

•4375 

9.6 

4 X 3 

•5 

11 

4 X3.5 

•5 

n -5 

4 X 3-5 

•5 

n -75 

4 - 5 X 3 

•5 

n -75 

5 X3 

•5 

12.65 

5 X 3 

•5625 

13-7 

5-5 X 3-5 

•5 

14-5 

5 - 5 X 3-5 

•5625 

15.6 


L Unequal Sides. 


Sides. 

Thick¬ 

ness. 

Weight. 

Ins. 

Inch. 

Lbs. 

6 X3.5 

•625 

18 

6 X 4.5 

•625 

20 

T 



2 X 2.375* 

•375 

5-5 

2.5X2.87 5 

•375 

6-5 

3 - 5 X 3.5 

•4375 

10.5 

4 X3.5 j 

•4375 

•75 

13 

4 X3.5 

•75 

13-5 


* This column gives depth of web added to the thickness of base or flange. 

L* 









































126 


WEIGHTS OF METALS 


"Weiglnt of Round Rolled Iron, 

From .125 Inch to 12 Inches in Diameter , one foot in length. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

.041 

2 

10.616 

4-375 

5 °- 8 J 5 

7-5 

149.328 

.25 

.165 

.125 

n.988 

•5 

53-76 

•75 

159-456 

•3125 

•259 

•25 

13-44 

.625 

56.788 

8 

169.856 

•375 

•373 

•375 

14-975 

•75 

59-9 

•25 

180.696 

•4375 

.508 

•5 

16.588 

5 

66.35 

•5 

191.808 

•5 

.663 

.625 

18.293 

.125 

69.731 

•75 

203.26 

•5625 

.84 

•75 

20.076 

•25 

73 -I 72 

9 

215.04 

.625 

1.043 

•875 

21.944 

•375 

76.7 

•25 

227.152 

.6875 

1.254 

3 

23.888 

•5 

80.304 

•5 

239.6 

•75 

1-493 

.125 

25.926 

.625 

84.001 

•75 

252.376 

.875 

2.032 

•25 

28.04 

•75 

87.776 

IO 

265.4 

I 

2.654 

•375 

30.24 

6 

95-552 

•25 

278.924 

.125 

3-359 

•5 

3 2 - 5 12 

•25 

103.704 

•5 

292.688 

•25 

4.147 

.625 

34.886 

•375 

107.86 

•75 

306.8 

•375 

5.019 

•75 

37-332 

•5 

112.16 

II 

321.216 

•5 

5-972 

•875 

39.864 

•625 

116.484 

•25 

336.004 

.625 

7.01 

4 

42.464 

•75 

120.96 

•5 

351-104 

•75 

8.128 

.125 

45-174 

7 

130.048 

•75 

366.536 

•875 

9-333 

•25 

47-952 

•25 

139-544 

12 

382.208 


Weight of Flat Rolled. Iron, 

From .5X.125 Inch to 5.5X4.5 Inches .'*' one foot in length. 


Thickness. 

W eight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 


.5 


.875 


1.25 


1.5 

.125 

.211 

•75 

2.217 

•5 

2.112 

•75 

3.802 

•25 

.422 

•875 

2-583 

.625 

2.64 

•875 

4-435 

•375 

•634 


1 

•75 

3.168 

I 

5.069 

•5 

•845 


JL 

•875 

3.696 

1.125 

5-703 


B25 

.125 

.422 

I 

4.224 

1.25 

6-337 



•25 

•845 

1.125 

4-752 

1-375 

6.97 

.125 

•25 

.264 

.528 

•375 

•5 

1.267 

1.69 


1.375 

1.625 

•375 

.792 

.625 

2.112 

.125 

•58 

.125 

.686 

•5 

1.056 

•75 

2.534 

•25 

1.161 

•25 

1-372 

.625 

1.32 

.875 

2.956 

•375 

1.742 

•375 

2.059 


75 


1.125 

•5 

2.325 

•5 

2.746 





.625 

2.9O4 

•625 

3-432 

.125 

.316 

.125 

•475 

•75 

3-484 

•75 

4 -ii 9 

•25 

•6 33 

•25 

•95 

•875 

4.065 

•875 

4.805 

•375 

•95 

•375 

1.425 

I 

4.646 

I 

5-492 

•5 

1.265 

•5 

1.901 

1-125 

5.227 

1-125 

6.178 

•625 

1.584 

•625 

2-375 

1.25 

5.808 

1.5 

6.864 

•75 

1.9 

•75 

2.85 

i -375 

6.389 

I -375 

7 - 55 i 


.875 

•875 

I 

3-326 

3.802 


1.5 

i -5 

8.237 

.125 

•369 


1 25 

.125 

•633 


1.75 

•25 

•738 



•25 

1.266 

.125 

•739 

•375 

1.108 

.125 

.528 

•375 

1.9 

•25 

1.479 

•5 

1.477 

•25 

1.056 

•5 

2-535 

•375 

2.218 

.625 

1.846 

•375 

1.584 

•625 

3 -i 68 

•5 

2-957 


* For weights of square bars see preceding page. 
























































WEIGHTS OF METALS, 


127 


Thickness. | 

Weight. 

Thickness. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 


1.75 


1.125 

.625 

3.696 

i -5 

10.772 

475 

4-435 

1.625 

11.67 

.875 

5.178 

i -75 

12.567 

1 

5 - 9 r 4 

i -875 

13-465 

1.125 

6.653 

2 

14.362 

1.25 

7-393 


2.25 

1-375 

8.132 


i -5 

8.871 

.125 

•95 

1.625 

9.61 

•25 

1.9 


1.875 

•375 

•5 

2.851 

3.802 

.125 

•792 

.625 

4-752 

.25 

1-584 

•75 

5-703 

•375 

2.376 

•875 

6-653 

•5 

3.168 

1 

7.604 

.625 

3-96 

1-125 

8-554 

•75 

4-752 

1.25 

9-505 

.875 

5-544 

1-375 

10.455 

1 

6.336 

i -5 

11.406 

1.125 

7.129 

1.625 

12.356 

1.25 

7.921 

i -75 

13-307 

1-375 

8.713 

i -875 

14-257 

i -5 

9-505 

2 

15.208 

1.625 

10.297 

2.125 

16.158 

i -75 

11.089 


2.375 


2 

.125 

1.003 

.125 

•845 

•25 

2.006 

.25 

1.689 

•375 

3.009 

•375 

2-534 

•5 

4.013 

•5 

3-379 

.625 

5.916 

.625 

4.224 

•75 

6.019 

•75 

5.069 

.875 

7.022 

•875 

5 - 9 I 4 

1 

8 025 

1 

6.758 

1-125 

9.028 

1.125 

7.604 

1.25 

10.032 

1.25 

8.448 

1-375 

n-035 

1-375 

9.294 

i -5 

12.038 

i -5 

10.138 

1.625 

13.042 

1.625 

10.983 

i -75 

14.045 

i -75 

11.828 

i -875 

15.048 

1-875 

12.673 

2 

16.051 


2.125 

2.125 

2.25 

17-054 

18.057 

.125 

.898 


2.5 

•25 

i -795 


•375 

2.693 

.125 

1.056 

•5 

3 - 59 i 

•25 

2.112 

.625 

4.488 

•375 

3- i6 8 

•75 

5-386 

•5 

4.224 

•875 

6.283 

.625 

5.28 

1 

7.181 

•75 

6.336 

1.125 

8.079 

.875 

7-392 

1.25 

8-977 

1 

8.448 

1-375 

9.874 

1.125 

9-504 


Thickness. 

Weight. 

Thickness. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 


2.5 


2.875 

I.25 

10.56 • 

.125 

1.215 

1-375 

11.616 

•25 

2.429 

i -5 

12.672 

•375 

3-644 

1.625 

13.728 

•5 

4.858 

i -75 

14.784 

•625 

6.072 

i-875 

15.84 

■75 

7.287 

2 

16.896 

•875 

8.502 

2.125 

I 7-952 

1 

9.716 

2.25 

19.008 

1-125 

10.931 

2-375 

20.064 

1-25 

12.145 


2.625 

i -375 

13-36 



i -5 

14-574 

.125 

1.109 

1.625 

15-789 

.25 

2.218 

i -75 

17.003 

•375 

3-327 

i-875 

18.218 

•5 

4-436 

2 

19.432 

.625 

5-545 

2.125 

20.647 

-75 

6.654 

2.25 

21.861 

■875 

7-763 

2-375 

23.076 

1 

8.872 

2-5 

24.29 

1.125 

9.981 

2.625 

25-505 

1.25 

11.09 

2-75 

26.719 

1-375 

12.199 


q 

i -5 

13-308 


O 

1.625 

14.417 

.125 

1.267 

i -75 

15-526 

•25 

2-535 

i-875 

16.635 

•375 

3.802 

2 

17-744 

•5 

5.069 

2.125 

18.853 

•625 

6.337 

2.25 

19.962 

•75 

7.604 

2-375 

21.071 

•875 

8.871 

2-5 

22.18 

1 

10.138 



1-125 

11.406 


<0. i 0 

1.25 

12.673 

.125 

1.162 

i -375 

13-94 

•25 

2.323 

i -5 

15.208 

•375 

3-485 

1.625 

16.475 

•5 

4.647 

i -75 

17.742 

•625 

5.808 

i-875 

19.01 

•75 

6.97 

2 

20.277 

•875 

8.132 

2.25 

22.811 

1 

9.294 

2-5 

25-346 

1-125 

10.455 

2-75 

27.881 

1.25 

11.617 


3 25 

1-375 

12.779 



i -5 

13-94 

.125 

1-373 

1.625 

15.102 

•25 

2.746 

i -75 

16.264 

•375 

4.H9 

i-875 

I 7-425 

•5 

5-492 

2 

18.587 

.625 

6.865 

2.125 

19.749 

•75 

8.237 

2.25 

20.91 

•875 

9.61 

2-375 

22.072 

1 

10.983 

2-5 

23-234 

1.125 

12.356 

2.625 

24-395 

1.25 

13-73 































128 


WEIGHTS OF METALS. 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 


3.25 


3.75 

4.5 


5 


1-375 

15.102 

.I.875 

23.762 

•75 

11.406 

3-25 

54 - 9 l 6 

i -5 

16.475 

2 

25-346 

1 

15.208 

3-5 

59- J 4 

1.625 

17.848 

2.25 

28.514 

1.25 

19.OI 

3-75 

63-365 

i -75 

19.221 

2-5 

31.682 

i -5 

22.812 

4 • 

67.589 

i-875 

20.594 

2-75 

34-851 

i -75 

26.614 

4-25 

71.813 

2 

21.967 

3 

38.019 

2 

30.415 

4-5 

76.038 

2.25 

24.712 

3-25 

41.187 

2.25 

34-217 

4-75 

80.262 

2-5 

27.458 

3-5 

44 355 

2-5 

38.019 

5 25 


2-75 

30.204 


A 

2-75 

41.82 



3 

32.95 



3 

45-623 

•25 

4436 


q k 

.125 

1.69 

3-25 

49.425 

•5 

8.871 



•25 

3-38 

3-5 

53.226 

•75 

13-307 

.125 

1.479 

•5 

6-759 

3-75 

57.028 

1 

17.742 

•25 

2.957 

•75 

10.138 

4 

60.83 

1-25 

22.178 

•375 

4-436 

1 

13-518 

4-25 

64.632 

I -5 

26.613 

•5 

5-914 

1-25 

16.897 

4 75 


i -75 

31.049 

.625 

7-393 

i -5 

20.277 

xlti 


2 

35-484 

•75 

8.871 

i -75 

23.656 

•25 

4.013 

2.25 

39-92 

.875 

10.35 

2 

27.036 

•5 

8.026 

2.5 

44-355 

1 

11.828 

2.25 

30-415 

•75 

12.036 

2-75 

48.791 

1.125 

13-307 

2-5 

33-795 

1 

16.052 

3 

53.226 

1.25 

14-785 

2-75 

37-174 

1.25 

20.006 

3-25 

57.662 

i -375 

16.264 

3 

40.554 

i -5 

24.079 

3-5 

62.097 

i -5 

17.742 

3-25 

43-933 

i -75 

28.092 

3-75 

66-533 

1.625 

19.221 

3-5 

47 - 3 I 3 

2 

32.105 

4 

70.968 

i -75 

20.699 

3-75 

50.692 

2.25 

36.118 

4-25 

75-404 

i-875 

22.178 


a ok 

2-5 

40.131 

4-5 

79-839 

2 

23.656 



2-75 

44.144 

4-75 

84.275 

2.25 

26.613 

.125 

i -795 

3 

48.157 

5 

88.71 

2-5 

29-57 

•25 

3 - 59 i 

3-25 

52.17 

e /r 


2-75 

32.527 

•5 

7.181 

3-5 

56.184 

5.5 


3 

35-485 

•75 

10.772 

3-75 

60.197 

•25 

4.647 

3-25 

38.441 

1 

14.364 

4 

64.21 

•5 

9.294 


3.75 

1.25 

17-953 

425 

68.223 

•75 

13-94 


i -5 

21-544 

4-5 

72.235 

1 

18.587 

.125 

1.584 

i -75 

25-135 

£ 


1.25 

23-234 

•25 

3.168 

2 

28.725 

O 


1.5 

27.881 

•375 

4-752 

2.25 

3 2 - 3 l6 

•25 

4.224 

i -75 

32.527 

•5 

6.336 

2-5 

35-907 

•5 

8.449 

2 

37-174 

.625 

7.921 

2-75 

39-497 

•75 

12.673 

2.25 

41.821 

•75 

9-505 

3 

43.088 

1 

16.897 

2-5 

46.468 

•875 

11.089 

3-25 

46.679 

1.25 

21.122 

2.75 

5 i-ii 4 

1 

12.673 

3-5 

50.269 

i -5 

25-346 

3 

55-76i 

1-125 

I 4-257 

3-75 

53-86 

i -75 

29-57 

3-25 

60.408 

1 25 

15.841 

4 

57-45 

2 

33-795 

3-5 

65-055 

1-375 

17-425 


4.5 

2.25 

38.019 

3-75 

69.701 

i -5 

19.009 


2-5 

42.243 

4 

74-348 

1.625 

20.594 

•25 

3.802 

2-75 

46.468 

4-25 

78.995 

i -75 

22.178 

•5 

7.604 

3 

50.692 

4-5 

83.642 


Illustration.— Wliat is weight of a bar of iron 5.25 ins. in breadth by .7*; inch 
in thickness? 

In column 7, as above, find 5.25; and below it, in column, .75; and opposite to 
that is 13.307, which is 13 lbs. and .307 of a pound. 

For parts of a pound and of a foot, operate according to rule laid down for table 
page 125. 



























WEIGHT OF SHEET AND HOOP IRON, 


129 


"Weight of Sheet Iron. (English. D. K. Clark.) 
Per Square Foot (at 480 lbs.per Cube Foot). 

As by Wire-gauge used in South Staffordshire, England. 


Thickness. 

Weight. 

Square 
Feet 
in 1 ton. 

Thickness. 

Weight. 

Square 
Feet 
in x ton. 

Thickness. 

Weight. 

Square 
Feet 
in 1 ton. 

No. 

Inch. 

Lbs. 

No. 

No. 

Inch. 

•Lbs. 

No. 

No. 

Inch. 

Lbs. 

No. 

32 

.OI25 

•5 

4480 

21 

•<=>344 

I.38 

1623 

IO 

.1406 

5-63 

398 

31 

.OI41 

.562 

3986 

20 

•0375 

i -5 

1493 

9 

•1563 

6.25 

358 

30 

.0156 

.625 

3584 

!9 

.0438 

i -75 

1280 

8 

.1719 

6.88 

326 

29 

.OI72 

.688 

3256 

18 

•05 

2 

1120 

7 

•1875 

7-5 

299 

28 

.0188 

•75 

2987 

J 7 

•0563 

2.25 

996 

6 

.2031 

8.13 

276 

27 

.0203 

.8x3 

2755 

l6 

.0625 

2-5 

896 

5 

.2188 

8-75 

256 

26 

.0219 

•875 

2560 

15 

•075 

3 

747 

4 

•2344 

9-38 

239 

25 

.0234 

•938 

2388 

14 

.0875 

3-5 

640 

3 

•25 

10 

224 

24 

.025 

1 

224O 

13 

.1 

4 

560 

2 

.2813 

11.25 

199 

23 

.0281 

I - I 3 

1982 

12 

.1125 

4-5 

4981 

1 

•3125 

12.5 

179 

22 

•0313 

1.25 

1792 

II 

.125 

5 

448 






Weight of* Hoop Iron. (English.) 
Per Lineal Foot. 


Width. 

W. G. 

Weight. 

Width. 

W. G. 

Weight. 

Width. 

W.G. 

Weight 

Ins. 

No. 

Lbs. 

Ins. 

No. 

Lbs. 

Ins. 

No. 

Lbs. 

.625 

21 

.067 

I-I 25 

17 

.21 

i -75 

14 

.484 

•75 

20 

.0875 

I.25 

l6 

.27 

2 

13 

•634 

•875 

19 

.I2l6 

1-375 

15 

•33 

2.25 

13 

.714 

1 

l8 

.1636 

i -5 

15 

•36 

2-5 

12 

.91 


"Weight of Black and. (Galvanized Sheet Iron. 

(Morton's Table, founded upon Sir Joseph Whitworth Sf Co.'s Standard Bir¬ 
mingham Wire-Gauged) (D. K. Clark.) 


Note.— Numbers on Holtzapffel’s wire-gauge are applied to thicknesses on Whit¬ 
worth gauge. 


Gauge and Weight of 
Black Sheets. 

Approxii 
of Sq. 1 ' 
Black. 

nate number 
t. in 1 ton. 
Galvanized. 

| Gauge and Weight of 
Black Sheets. 

Approximate number 
of Sq. Ft. in i ton. 
Black. 1 Galvanized. 

No. 

Inch. 

Lbs. 

Sq.'Ft. 

Sq. Ft. 

| No. 

Inch. 

Lbs. 

Sq. Ft. 

Sq. Ft. 

I 

•3 

12 

187 

185 

J 7 

.06 

2.4 

933 

876 

2 

.28 

II .2 

200 

197 

18 

•05 

2 

1120 

IO38 

3 

.26 

IO.4 

215 

212 

19 

.04 

1.6 

1400 

I274 

4 

.24 

9.6 

233 

229 

20 

.036 

1.4 

1556 

1403 

5 

.22 

8.8 

254 

250 

21 

.032 

1.28 

1750 . 

1558 

6 

.2 

8 

280 

275 

22 

.028 

1.12 

2000 

1753 

7 

.18 

7.2 

311 

304 

23 

.024 

.96 

2 333 

2004 

8 

.165 

6.6 

339 

331 

24 

.022 

.88 

2545 

2159 

9 

•15 

6 

373 

363 

25 

.02 

.8 

2800 

2339 

10 

•135 

5-4 

4 i 5 

403 

26 

•Ol8 

.72 

3 111 

2553 

11 

.12 

4.8 

467 

452 

27 

.Ol6 

.64 

35 oo 

2808 

12 

.11 

4.4 

509 

491 

28 

.OI4 

•56 

4000 

3122 

13 

•095 

3-8 

589 

566 

29 

.013 

•52 

4308 

3306 

14 

.085 

3-4 

659 

63O 

30 

.012 

.48 

4667 

3513 

15 

.07 

2.8 

800 

757 

31 

.OI 

•4 

5600 

4OI7 

16 

.065 

2.6 

862 

813 

32 

.009 

•36 

6222 

4327 

























































130 


WEIGHT OF ANGLE AND T IRON, 


^Weight of English ^Angle and. T Iron. (D. K. Clark.) 

ONE FOOT IN LENGTH. 


Note.—W hen base or web tapers in section, mean thickness is to be measured. 


Thick- 




Sum of 

Width 

and Depth in 

Inches. 




ness. 

'•5 

1.625 

'■75 

1.875 

2 

2.125 

2.25 

| 2.375 

2.5 

2.625 

2.75 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.125 

•57 

.62 

.68 

•73 

•78 

•83 

.88 

•94 

•99 

I.04 

I.09 

•1875 

.81 

.89 

•97 

1.05 

I - I 3 

1.21 

1.29 

i -37 

i -45 

I.52 

1.6 

•25 

I.04 

I * I 5 

1.25 

1.36 

1.46 

I.56 

1.67 

1.77 

1.88 

I.98 

2.08 

■3125 

I.24 

i -37 

i -5 

1.63 

1.76 

H89 

2.02 

2.15 

2.28 

2.41 

2-54 


2.875 

3 

3.125 

3.25 

3.375 

3.5 

3.625 

3-75 

3.875 

4 

4-25 

.125 

1.14 

1.2 

1.25 

i -3 

i -45 

I.4I 

1.46 

I - 5 I 

1.56 

1.62 

1.72 

•1875 

1.68 

1.76 

1.84 

1.91 

I -99 

2.O7 

2.15 

2.23 

2-3 

2.38 

2-54 

•25 

2.19 

2.29 

2.4 

2-5 

2.6 

2.71 

2.81 

2.92 

3.02 

3-13 

3-33 

•3125 

2.67 

2.8 

2-93 

3.06 

3 -i 9 

3-32 

3-45 

3-58 

3 - 7 i 

3-84 

4.1 

•375 

3 -i 3 

3.28 

3-44 

3-59 

3-75 

3 - 9 i 

4.06 

4.22 

4 - 3 8 

4-53 

4.84 

•4375 

3-57 

3-75 

3-93 

4.11 

4.29 

4.48 

4.66 

4.84 

5.02 

5-2 

5-56 


45 

4-75 

5 

525 

5.5 

575 

6 

6.25 

6.5 

6.75 

7 

•1875 

2.7 

2.85 

3.01 

3.16 

3 - 3 2 

3-48 

3-6 3 

3-79 

3-95 

4.1 

4.26 

•25 

3-54 

3-75 

3-96 

4.17 

4-38 

4-58 

4-79 

5 

5.21 

5 - 4 2 

5-63 

•3125 

4-36 

4.62 

4.88 

5 -i 4 

5-4 

5.66 

5 - 9 2 

6.18 

6-45 

6.71 

6.97 

•375 

5.16 

5-47 

5.78 

6.09 

6.41 

6.72 

7-03 

7-34 

7.66 

7-97 

8.28 

•4375 

5 - 9 2 

6.29 

6.65 

7.02 

7-38 

7-75 

8.11 

8.48 

8.84 

9.21 

9-57 

•5 

6.67 

7.08 

7-5 

7.92 

8-33 

8-75 

9.17 

9-58 

10 

10.42 

10.83 

•5625 

7-38 

7-85 

8.32 

8.79 

9.26 

9-73 

10.2 

10.66 

11.13 

12.6 

12.07 


7.25 

7-5 

7-75 

8 

8.25 

8-5 

8-75 

9 

9.25 

9.5 

9-75 

•25 

5-83 

6.04 

6.25 

6.46 

6.67 

6.88 

7.08 

7.29 

7-5 

7.71 

7.92 

•3125 

7-23 

7-49 

7-75 

8.01 

8.27 

8-53 

8-79 

9-05 

9 - 3 i 

9-57 

9-83 

•375 

8-59 

8.91 

9.22 

9-53 

9.84 

10.16 

IO -47 

10.78 

11.09 

11.41 

11.72 

•4375 

9-93 

10.3 

10.66 

11.03 

n -39 

11.76 

12.12 

12.49 

12.85 

13.22 

I 3-58 

•5 

11.25 

ix.67 

12.08 

12.5 

12.92 

13-33 

13-75 

14.17 

14.58 

i 5 

15.42 

•5625 

12.54 

13.01 

13.48 

13-94 

14.41 

14.88 

15-35 

15.82 

16.29 

16.76 

17.23 

.625 

13.8 

14.32 

14.84 

I 5-36 

15.89 

16.41 

16.93 

17-45 

17.97 

18.49 

19.01 


10 

10.5 

11 

"•5 

12 

12.5 

'3 

' 3-5 

'4 

' 4-5 

'5 

•375 

12.03 

12.66 

13.28 

i 39 i 

14-53 







•4375 

13-95 

14.67 

15-4 

16.13 16.86 

17-59 

l8 - 3 h 

19.04 

19.77 

20.5 

21.22 

•5 

15-83 

16.67 

17-5 

18.33 I9-I7 

20 

20.84 

21.67 

22.5 

23-34 

24.17 

•5625 

17.7 

18.63 

19-57 

20.51 

21.44 

22.38 

23-31 

24.25 

25.19 

26.12 

27.06 

.625 

19-53 

20.57 

21.61 

22.66 

23-7 

24.74 

25.78 

26.83 

27.87 

28.91 

29-95 

•75 

2 3- I 3 

24.38 

25-63 

26.88 

28.13 

29-37 

30.63 

31-88 

33-13 

34-38 

35-63 


12 

12.5 

'3 

' 3-5 

'4 

'5 

16 

'7 

18 

'9 

20 

.625 

23-7 

24.74 

25.78 

26.83 

27.87 129.95 

32.03 

34 - 12 * 

36.2 

38.28 

40.36 

•75 

28.13 

29-37 

30-63 1 

31.88 

33 -13 

35-63 

38.13 

40.63 41.13 

43-63 

46.13 

.875 

32.45 

33 - 9 i 

35-36 36.82 

38.28 

41.19 

44.12 

47.02 49.95 

52.87 

55-78 

1 

36.67 138.33! 4.0 1 

41.67 

43-33 

46.67 

50 

53 - 33 I 56.67 

60 

63-33 


Note.—A merican rolled is slightly heavier. 
























































































WEIGHT OF HOOP IRON.-CAST IRON.-METALS. I 3 I 


'W'eigh.t of Hoop Iron. ( D. K. Clark.) 

ONE FOOT IN LENGTH. 

As by Wire-gauge used in South Staffordshire (England). 


Thickness. 

.625 

•75 

•875 

1 

Width in Inches. 

• •» 25 | I.25 | I.375 

'•5 

1.625 

i -75 

2 

No. 

Inch. 

Lb. 

Lb. 

Lb. 

Lb. 

Lb 

Lb. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

21 

•0344 

.0716 

.0861 

.1 

•115 

.129 

.144 

.158 

.172 

• T 97 

.201 

.229 

20 

.0375 

.0781 

.0938 

.109 

.125 

.141 

.156 

.172 

.188 

.203 

.219 

•25 

19 

.0438 

.0911 

.109 

.128 

.146 

.164 

.182 

.2 

.219 

.238 

•257 

.292 

l8 

•05 

.IO4 

.125 

.146 

.167 

.188 

.208 

.229 

•25 

.271 

.292 

•333 

17 

.OS63 

.117 

.141 

.164 

.188 

.211 

•234 

.258 

.281 

•305 

.328 

•375 

l6 

.0625 

•13 

.156 

.182 

.208 

•234 

.26 

.286 

•313 

•339 

•365 

.417 

15 

•075 

.156 

.188 

.219 

•25 

.281 

•313 

•344 

•375 

•307 

•438 

•5 

14 

.0875 

.183 

.219 

.256 

•293 

•329 

.366 

.402 

•438 

•475 

.512 

.585 

13 

.1 

.208 

•25 

.292 

•333 

•375 

.416 

•458 

•5 

•543 

•584 

.667 

12 

.1125 

•234 

.281 

.328 

•375 

.422 

.469 

.516 

•563 

.609 

.656 

•75 

II 

.125 

.26 

•313 

•365 

.417 

.469 

.521 

•573 

.625 

.677 

.729 

•833 

IO 

.1406 

•293 

•352 

.41 

.469 

•527 

.586 

•645 

•703 

.762 

.82 

•938 

9 

•1563 

.326 

• 39 1 

•456 

.522 

•587 

.652 

.717 

•783 

.848 

•913 

1.04 

8 

.1719 

•358 

•43 

.501 

•573 

.644 

.716 

.788 

•859 

• 93 i 

I 

i-i 5 

7 

•1875 

•391 

.469 

•547 

•625 

•703 

.781 

•859 

•938 

1.02 

I.09 

1.25 

6 

.2031 

•423 

.508 

•593 

.677 

.762 

.836 

•931 

1.02 

1.1 

1.19 

i -35 

5 

.2188 

•456 

•547 

.638 

•729 

.82 

.912 

1 

1.09 

1.19 

1.28 

1.46 

4 

•2344 

.488 

.586 

.683 

.781 

•879 

•977 

1.07 

1.17 

1.27 

i -37 

1.56 


CAST IRON. 

To Compute "VHeight of a Cast Iron Bar or Bod.. 

Ascertain weight of a wrought iron bar or rod of same dimensions in 
preceding tables, or by computation, and from weight deduct th part. 

Or, As .1000 : .9257 ;; weight of a wrought bar or rod : to weight re¬ 
quired. Thus, what is weight of a piece of cast iron 4 x 3.7s X 12 inches? 

In table, page 128, weight of a piece of wrought iron of these dimensions 
is 50.692 lbs. Then, 1000 : .9257 :: 50.692 : 46.93 lbs. 

Braziers’ and Sheathing Copper. 

Braziers’ Sheets, 2X4 feet from 5 to 25 lbs., 2.5X5 feet from 9 to 150 lbs., and 
3X5 feet and 4X6 feet, from 16 to 300 lbs. per sheet. 

Sheathing Copper, 14 X 48 inches, and from 14 to 34 oz. per square foot. 

Yellow Metal, 14 X 48 inches, and from 16 to 34 oz. per square foot. 


'Weight of Corrngated Iron Roof Blates. 

per square foot. ( Birmingham Gauge .) 


No. 

Black. 

Galvanized. 

No. 

Black. 

Galvanized. 

No. 

Black. 

Galvanized. 


Oz. 

Oz. 


Oz. 

Oz. 


Oz. 

Oz. 

20 

26 

29 

23 

20 

22 

25 

16 

18 

22 

22 

24 

24 

18 

20 

26 

14 

16 


METALS. 

To Compute 'Weight of HMetals of any Dimen¬ 
sions or Form. 

By rules in Mensuration of Solids (page 360), ascertain volume of the 
piece, multiply it by weight of a cube inch, and product will give weight 
in pounds. 















































132 


WEIGHT OF CAST IRON PIPES. 


"Weight of Cast Iron IPipes or Cylinders. 

From i to 70 Inches in Internal Diameter. 


ONE FOOT IN EENGTH. 


Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

I 

•25 

3.06 

4-75 

•375 

18.84 

II 

•875 

101.85 


•375 

5-05 


•5 

25.72 

n -5 

•5 

58.81 

1-25 

•25 

3-68 


.625 

32-93 


•625 

74.28 


•3125 

4-79 


•75 

40-43 


•75 

90.06 


•375 

5-97 

5 

•375' 

19.76 


•875 

106.13 

i -5 

•375 

6.89 


•5 

26.95 

12 

•5 

61.26 


•4375 

8.31 


.625 

34-46 


.625 

77-34 


•5 

9.8 


•75 

42.27 


•75 

93-73 

i- 7 S 

.375 

7.81 

5-5 

•375 

21.59 


.875 

110.42 


•4375 

9-38 


•5 

29.4 

12.5 

•5 

63-71 


•5 

11.03 


.625 

37-52 


.625 

80.4 

2 

•375 

8-73 


•75 

45-95 


•75 

97-4 


•4375 

IO -45 

6 

•375 

23-43 


•875 

114.71 


•5 

12.25 


•5 

31.86 

13 

•5 

66.16 

2.25 

•375 

9-65 


.625 

40-59 


•625 

83-47 


•4375 

11.52 


•75 

49.62' 


•75 

101.08 


•5 

i 3-48 

6.5 

•375 

25.27 


•875 

119 

2.5 

•375 

10.57 


•5 

34-31 

I 3 < 5 

•5 

68.61 


•4375 

12.6 


.625 

43-65 


.625 

86.53 


•5 

14.7 


•75 

53-3 


•75 

104.76 

2-75 

•375 

n.49 

7 

•5 

36.76 


•875 

123.29 


•4375 

i 4-67 


•5625 

41.7 

14 

•5 

71.06 


•5 

1 5*93 


.625 

46.71 


.625 

89.6 

3 

•375 

12.4 


•75 

56-97 


•75 

108.43 


•5 

I 7 -I 5 

7-5 

•5 

39.21 


•875 

127.58 


.625 

22.2 


•5625 

44-45 

14-5 

•5 

73-51 


•75 

27-57 


.625 

49-77 


•625 

92.66 

3-25 

•375 

13-32 


•75 

60.65 


•75 

112.11 


•5 

18.38 

8 

•5 

41.66 


•875 

131.87 


.625 

23-74 


•5625 

47.21 

15 

•5 

75-96 


•75 

29.4 


.625 

52.84 


•625 

95-72 

3-5 

•375 

14.24 


•75 

64.32 


•75 

115.78 


•5 

19.6 

9 

•5 

46.56 


•875 

136.16 


.625 

25.27 


•5625 

52.72 

15-5 

•5 

78.47 


•75 

31.24 


.625 

58.96 


•625 

98.78 

3-75 

•375 

15.16 


•75 

71.67 


•75 

119.46 


•5 

20.83 

9-5 

•5 

49.01 


•875 

140.44 


.625 

26.8 


•5625 

55-48 

l6 

•625 

101.85 


•75 

33 -o 8 


.625 

62.06 


•75 

123.14 

4 

•375 

16.08 


•75 

75-35 


•875 

144-73 


•5 

22.05 

IO 

•5 

5 i -45 


I 

166.63 


.625 

28-33 


.625 

65.09 

16.5 

•625 

104.9 


•75 

34-92 


•75 

79-03 


•75 

126.75 

4-25 

•375 

17 


•875 

93-27 


•875 

149.02 


•5 

23.28 

10.5 

•5 

53 - 9 i 


I 

171-53 


.625 

29.86 


.625 

68.15 

17 

•625 

io 7-97 


•75 

36.76 


•75 

82.7 


•75 

130.48 

4-5 

•375 

17.92 


•875 

97-56 


•875 

153-3 


•5 

23.88 

II 

•5 

56.36 


I 

176-43 


.625 

3 I -4 


.625 

71.21 

17-5 

.625 

111.03 


•75 

38-59 


•75 

86.38 


•75 

134.16 









































WEIGHT OF CAST IRON PIPES. 1 33 


Diameter. 

Thick n. 

Weight. | 

Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Ins. 

Inch. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

17-5 

.875 

157-59 

29 

•75 

218.7 

40 

•875 

350.56 


I 

181.33 


•875 

256.23 


I 

4OI.86 

l8 

.625 

114.1 


1 

294.05 


1-125 

453-46 


•75 

I37-84 

30 

•75 

226.05 I 


I.25 

505-4I 


•875 

161.88 


.875 

264.8 

42 

•875 

367.69 


1 

186.23 


1 

303.86 


I 

42T-45 

19 

.625 

120.23 


1-125 

343-22 


1.125 

472.52 


•75 

I45-I9 

31 

•75 

233-4I 


I.25 

529.87 


.875 

170.46 


•875 

273-38 

44 

.875 

3S4.88 


1 

196.03 


1 

3 i 3-66 


I 

44 1 -! 

20 

.625 

126.35 


1.125 

354-24 


1-125 

497-58 


•75 

I52-54 

32 

•75 

240.75 


1.25 

554-42 


.875 

I79-03 


•875 

281.95 

46 

•875 

402.01 


1 

205.84 


1 

323-46 


I 

460.07 

21 

.625 

132.48 


1.125 

365-27 


1.125 

519-64 


•75 

159.89 

33 

•75 

248.11 


1-25 

578.88 


•875 

187.61 


•875 

290-53 

48 

.875 

419.17 


1 

215.64 


1 

333-26 


I 

480.29 

22 

.625 

138.61 


1.125 

376.29 


I-I25 

541.69 


•75 

167.24 

34 

•75 

255-46 


1.25 

603.44 


,875 

196.19 


•875 

299.11 

50 

•875 

436-43 


1 

225.44 


1 

343-o6 


I 

499.89 

23 

.625 

144-73 


1.125 

387-33 


1-125 

563-75 


•75 

174-59 

35 

•75 

262.81 


I.25 

627.93 


■875 

204.76 


.875 

307.68 

52 

•875 

453-49 


1 

235-24 


1 

352-87 


J 

5i9-5 

24 

.625 

150.86 


1.125 

398-35 


1.125 

585-81 


•75 

181.95 

36 

•75 

270.16 


I.25 

654.42 


•875 

213-34 


.875 

316.26 

55 

‘875 

479-23 


1 

245.04 


1 

362.67 


I 

548-9 

25 

.625 

156.98 


1-125 

409.28 


1.125 

618.91 


•75 

189.3 


1.25 

456-37 


I- 2 5 

689.21 


•875 

221.92 

37 

•75 

277-51 

58 

I 

578.29 


1 

254-85 


•875 

324.84 


1.125 

651.96 

26 

.625 

163.11 


1 

372.47 


I.25 

725-93 


•75 

196.65 


1-125 

420.4 


i-375 

800.22 


•875 

230.5 


1.25 

468.65 

60 

1 

597-92 


1 

264.65 

38 

•75 

284.86 


1-125 

674.01 

27 

.625 

169.23 


•875 

333-41 


1.25 

750-45 


•75 

204 


1 

382.27 


i-375 

827.17 


.875 

239.07 


1-125 

431.41 

65 

1 

646.93 


1 

274-45 


1.25 

480.89 


1-125 

729.18 

28 

.625 

!75-3 6 

39 

•75 

292.21 


1 25 

811.73 


•75 

211.35 


.875 

341-97 


i-375 

894.6 


•875 

247.65 


1 

392.08 

70 

1 

695.92 


1 

284.25 


1.125 

442.44 


i-45 

872.98 

29 

.625 

181.49 


1.25 

493-14 


i-5 

1051.25 


Equivalent Length of Pipe for a Socket. 

7 -j- — = l. d representing diameter of pipe and l length in inches. 


Additional weight of two flanges for any diameter is computed equal to a lineal 
foot of the pipe. 

Notk.— These weights do not include any allowance for spigot and socket ends. 
2 ._For rule to compute thicknesses of pipes, flanges, etc., see page 560. 

M 


































134 WEIGHT OF FLAT ROLLED BAR AND SQUARE STEEL 


AA^eiglit of UTlat Rolled. liar Steel. (D. K. Clark.) 
From .5 Inch to 8 Inches in Width, one foot in length. 
Width in Inches. 


Thick¬ 

ness. 

•5 

.625 

•75 

■875 

1 

'•25 

i -5 

-•75 

2 

2.25 

2-5 

2-75 

Inch. 

Lb. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbe. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

X 

•425 

•533 

.64 

•743 

•85 

1.06 

I.28 

I.49 

i -7 

I.91 

2.13 

2-34 


•S 31 

.665 

.8 

.929 

I.06 

i -33 

1-59 

1.86 

2.13 

2-39 

2.66 

2.92 

% 

.638 

.798 

.96 

1.11 

1.28 

i -59 

I.91 

2.23 

2-55 

2.87 

3 -i 9 

3 - 5 i 

Al6 

•744 

• 93 i 

1.12 

i -3 

I.49 

1.86 

2.23 

2.6 

2.98 

3-35 

3-72 

4.09 

X 

.85 

x.06 

1.28 

1.49 

i -7 

2.13 

2-55 

2.98 

3-4 

383 

4-25 

4.68 

%n 

— 

1.2 

1.44 

1.67 

1.91 

2-39 

2.87 

3-35 

3-83 

43 

4.78 

5.26 

% 

— 

i -33 

1.6 

1.86 

2.12 

2.66 

3-19 

3-72 

4-25 

4.78 

5 - 3 i 

5-84 

X 

— 

— 

1.76 

2.04 

2-34 

2.92 

3 - 5 i 

4.09 

4.68 

5.26 

5-84 

6-43 

% 

— 

— 

1.92 

2.23 

2-55 

3 -i 9 

3-8 3 

4.46 

5 -i 

5-74 

6.38 

7.01 

% 

— 

— 

— 

2.41 

2.76 

3-45 

4- J 4 

4-83 

5-53 

6.22 

6.91 

7.6 


— 

— 

— 

2.6 

2.98 

3 - 7 2 

4.46 

5-21 

5-95 

6.69 

7-44 

8.18 

% 

— 

— 

— 

— 

3 -i 9 

3-98 

4.78 

5-58 

6.38 

7.17 

7-97 

8.77 

I 

— 

— 

— 

— 

3-4 

4- 2 5 

5 -i 

5-95 

6.8 

7-65 

8-5 

9-35 


Width in Inches. 


Thick¬ 

ness. 

3 

3-25 

3-5 

4 

45 

5 

5-5 

6 

6.5 

7 

7-5 

8 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

X 

2-55 

2.76 

2.98 

3-4 

3 82 

4.26 

4.68 

5 -i 

5-52 

5 - 9 6 

6.38 

6.8 

X 

3-19 

3-45 

3-72 

4-25 

4.78 

5-32 

5-84 

6.38 

6.9 

7-44 

7-97 

8-5 

% 

3-83 

4 14 

446 

5 -i 

5-74 

6.38 

7.02 

7.66 

8.28 

8.92 

9-56 

10.2 

% 

4.46 

4-83 

5-21 

5-95 

6.7 

7-44 

8.l8 

8.92 

9.66 

10.4 

11.2 

11.9 

H 

5-1 

5'53 

5-95 

6.8 

7.66 

8-5 

9-36 

10.2 

11.1 

11.9 

12.8 

13.6 

SlQ 

5-74 

6 22 

6.69 

7-65 

8.6 

9-56 

IO.5 

n -5 

12.4 

13-4 

14-3 

I 5-3 

% 

6.38 

6.91 

7-44 

8-5 

9-56 

10.6 

n . 7 

12.8 

138 

14.9 

15-9 

17 

% 

7.01 

7.6 

8.18 

9-35 

10.5 

n. 7 

12.9 

14 

15.2 

16.4 

I 7-5 

18.7 

M 

7-65 

8.29 

8-93 

10.2 

II -5 

12.8 

!4 

15-3 

16.6 

17.9 

19.1 

20.4 

% 

8.29 

8.98 

9.67 

11.1 

12.4 

13.8 

15-2 

16.6 

18 

19-3 

20.7 

22.2 

y 8 

8-93 

9.67 

10.4 

11.9 

13-4 

14.9 

16.4 

17.9 

J 9 4 

20.8 

22.3 

23.8 

% 

9.56 10.4 

11.2 

12.8 

14-3 

15-9 

17-5 

19.1 

20.8 

22.4 

23-9 

25.6 

1 

10.2 

In.1 

11.9 

13.6 

15-3 

17 

18.7 

20.4 

22.1 

23.8 

25-5 

27.2 


“Weiglit of Rolled Sqiaare Steel. 

From .125 Inch to 6 Inches Square, one foot in length. 


Side. 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

125 

•053 

•75 

I.92 

1-375 

6-43 

2.125 

15-4 

3-75 

47.8 

1875 

.119 

.8125 

2.24 

1-4375 

7-03 

2.25 

17.2 

4 

54-4 

25 

.212 

.875 

2.6 

i -5 

7-65 

2-375 

19.2 

4-25 

61.4 

3 I2 5 

•333 

•9375 

3 06 

1-5625 

8-3 

2-5 

21.2 

4-5 

68.9 

375 

.478 

1 

3 4 

1.625 

8.98 

2.625 

23-5 

4-75 

76.7 

4375 

.651 

1.0625 

3-83 

1.6875 

9-79 

2-75 

25-7 

5 

85 

5 

•85 

1-125 

4-3 

i -75 

10.4 

2.875 

28.2 

5-25 

93-7 

5625 

1.08 

1.1875 

4-79 

1.8125 

11.2 

3 

30.6 

5-5 

102.8 

625 

i -33 

1.25 

5 - 3 i 

M 

bo 

•<1 

C/i 

11.9 

3-25 

35-9 

5-75 

112.4 

6875 

1.61 

I- 3!25 

5-86 

2 

13.6 

3-5 

41.6 

6 

122 4 








































































WEIGHT OF ROLLED STEEL, SHEET COPPER, ETC. 1 35 


^W'eigh.t of Round Rolled. Steel. 

From , .125 Inch to 12 Inches Diameter . one foot in length. 


Diam. 

Weight. | 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diam. 

Weight. 

Diam. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

125 

.0417 

•875 

2.04 

1.625 

7-05 

2.875 

22 

5-75 

88.3 

1875 

•0939 

•9375 

2.35 

1.6875 

7.61 

3 

24.I 

6 

96.1 

25 

.167 

I 

2.67 

i -75 

8.18 

3-25 

28.3 

6-5 

113-2 

3125 

.26 

1.0625 

3 

1.8125 

8.77 

3-5 

32 7 

7 

130.8 

375 

•375 

1.125 

3-38 

i -875 

9-38 

3-75 

34-2 

7-5 

136.8 

4375 

•511 

1.1875 

3 -76 

2 

IO.7 

4 

42.7 

8 

170.8 

5 

.667 

1.25 

4.17 

2.125 

12 

4-25 

48.3 

8-5 

193.2 

5625 

•845 

1-3125 

4.6 

2.25 

13.6 

4-5 

54-6 

9 

218.4 

625 

I.04 

i -375 

5-05 

2.375 

I 5 - 1 

4-75 

60.3 

9-5 

24I.2 

6875 

I.27 

1-4375 

5.18 

2.5 

16.7 

5 

66.8 

IO 

267.2 

75 

i -5 

i -5 

6.01 

2.625 

18.4 

5-25 

73-6 

II 

323 

8125 

1.76 

1-5625 

6.52 

2.75 

20.2 

5-5 

80.8 

12 

352.8 


"Weiglit of Hexagonal, Octagonal, and Oval Steel. 

ONE FOOT IN LENGTH. 



HEXAGONAL. 



OCTAGONAL. 


OVAL. 


Diam. 


Diam. 


Diam. 


Diam. 





over 

Sides. 

Weight. 

over 

Sides. 

Weight. 

over 

Sides. 

Weight. 

over 

Sides. 

Weight. 

Diam. 
over Sides. 

Area. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

So.In. 

Lbs. 

X 

.414 

I 

2.94 

% 

•396 

I 

2.82 

XxK 

.251 

•853 

X 

•736 

H 

\M 

oo\ 

3-73 

X 

.704 

iX 

3-56 

XxX 

•344 

1.17 

X 

1.15 


4.6 

X 

1.1 


4.4 

1 x X 

.446 

1.52 

% 

1.66 


5-57 

X 

1.58 

iX 

5-32 

iX x X 

.697 

2.37 

% 

2.25 


6.63 

X 

2.16 

*X 

6.34 

iX x % 

.884 

3 


"Weight of a Square Foot of Sheet Copper. 

Wire Gauge of Wm . Foster Sg Co . { England .) 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight 

W. G. 

Inch. 

Lbs. 

W. G. 

Inch. 

Lbs. 

W. G. 

Inch. 

Lbs. 

I 

.306 

14 

II 

.123 

5-65 

21 

•034 

1.55 

2 

.284 

13 

12 

.109 

5 

22 

.029 

i -35 

3 

.262 

12 

13 

.098 

4-5 

23 

.025 

I - I 5 

4 

.24 

II 

14 

.088 

4 

24 

.022 

I 

5 

.222 

10.15 

15 

.076 

3-5 

25 

•OI9 

.89 

6 

•203 

9-3 

16 

.065 

3 

26 

.017 

•79 

7 

.186 

8-5 

17 

•057 

2.6 

27 

•015 

•7 

8 

.168 

7-7 

18 

.049 

2.25 

28 

•013 

.62 

9 

•153 

7 

19 

.044 

2 

29 

.012 

•56 

IO 

.138 

6-3 

20 

.038 

i -75 

30 

.Oil 

•5 


'W'eigh.t of Composition Sh.eath.ing IKTails. 


No. 

Length. 

Number 
in a 
Pound. 

No. 

Length. 

Number 
in a 
Pound. 

No. 

Length. 

Number 
in a 
Pound. 

No. 

Length. 

Number 
in a 
Pound. 

I 

Inch. 

•75 

29O 

4 

Ins. 

1.125 

201 

7 

Ins. 

1-125 

184 

IO 

Ins. 

1.625 

IOl 

2 

•875 

260 

5 

1.25 

199 

8 

1.25 

168 

II 

i -75 

74 

3 

1 

212 

6 

I 

I90 

9 

i -5 

no 

12 

2 

64 



































































































WEIGHT OF IRON, STEEL, COPPER, ETC 


Weiglit of Cast and Wrought Iron. Steel, Copper, and 
Brass, of a given Sectional Area. 

Per Lineal Foot. 


Sectional ! 
Area. 

Wrought 

Iron. 

Cast Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun-metal. 

Sq. Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.1 

•336 

•313 

•339 

.385 

•492 

-357 

.38 

.2 

.671 

.626 

.677 

■ 77 1 

•984 

•713 

•759 

•3 

I.OO7 

•939 

1.016 

' 1-156 

I.476 

I.07 

1-139 

•4 

1-343 

1.251 

i -355 

1.542 

I.967 

I.427 

i- 5 i 9 

•5 

1.678 

1.564 

1.694 

1.927 

2.461 

I -783 

1.894 

.6 

2.014 

1.877 

2.032 

2.312 

2-953 

2.14 

2.279 

-7 

2-35 

2.19 

2.371 

2.698 

3 445 

2.497 

2.658 

.8 

2.685 

2.503 

2.71 

3083 

3 937 

2-853 

3-038 

•9 

3.021 

2.816 

3-049 

3 469 

4.429 

3-21 

3 4 i 8 

i 

3-357 

3.129 

3 387 

3 854 

4 922 

3 567 

3-798 

1.1 

3.692 

3 442 

3.726 

4 24 

5 - 4 I 4 

3-923 

4.177 

1.2 

4 028 

3 754 

4.065 

4.625 

5 906 

4.28 

4-557 

i -3 

4364 

4.067 

4 404 

5.01 

6398 

4.636 

4-937 

1.4 

4 699 

4 38 

4.742 

5-396 

6.89 

4-993 

5-317 

i -5 

5-035 

4693 

5.081 

5 - 78 i 

7 383 

5-35 

5.696 

i.6 

5 37 i 

5 006 

5 - 4 2 

6.167 

7-875 

5-707 

6.076 

i -7 

5.706 

5 - 3 i 9 

5-759 

6.552 

8367 

6 063 

6.456 

i.8 

6.042 

5 632 

6.097 

6-937 

8859 

6.42 

6.836 

1.9 

6.378 

5 945 

6.436 

7-323 

9 35 i 

6.777 

7-215 

2 

6 - 7 i 4 

6 258 

6-775 

7.708 

9843 

7-133 

7-595 

2.1 

7.049 

6-57 

7.H4 

8.094 

10.33 

7-49 

7-97 

2.2 

7-385 

6883 

7-452 

8.474 

1083 

7.847 

8-35 

2-3 

7.721 

7.196 

7.791 

8.864 

11.32 

8.203 

8-73 

2.4 

8.056 

7 509 

8.13 

9-25 

11.81 

8.56 

9.11 

2-5 

8.392 

7 822 

8 469 

9-635 

12.3 

8.917 

9.49 

2.6 

8.728 

8-135 

8.807 

10.02 

12.8 

9-273 

9.87 

2.7 

9063 

8.448 

9.146 

10.41 

13.29 

963 

10.25 

2.8 

9399 

8.76 

9 485 

10.79 

13-78 

9.98 

10.63 

2.9 

9-734 

9-073 

9.824 

11.18 

14.27 

10.34 

11.01 

3 

10.07 

9.386 

10.16 

11.56 

14.76 

10.7 

n -39 

3 -i 

10.41 

9.699 

10.5 

n -95 

15.26 

11.06 

11.77 

3-2 

10.74 

10.01 

10.84 

12.33 

15-75 

11.41 

12.15 

3-3 

11.08 

10.32 

11.18 

12.72 

16.24 

11 77 

12-53 

3-4 

11.41 

10.64 

11.52 

I 3 - 1 

16.73 

12.13 

12.91 

3-5 

n -75 

10.95 

11.86 

13-49 

17.22 

12.48 

13.29 

3-6 

12.08 

ii.26 

12.19 

13-87 

17.72 

12.84 

13.67 

3-7 

12.42 

11.58 

12.53 

14.26 

18.21 

13 2 

14.05 

3-8 

12.76 

11.89 

12.87 

14.64 

18.7 

13 55 

1443 

3-9 

13.09 

12.2 

13.21 

15-03 

19.19 

13 9 1 

14.81 

4 

13-43 

12.51 

13-55 

15.42 

19.69 

14.27 

15-19 

4.1 

I 3 - 7 6 

12.83 

13.89 

158 

20.18 

14.62 

15-57 

4.2 

14.1 

I 3 -I 4 

14 23 

16.19 

20.67 

14 98 

15-95 

4-3 

14 43 

13 45 

14-57 

1657 

21.16 

15-34 

16.33 

4.4 

14-77 

I 3-77 

I 4 - 9 I 

16.96 

21 65 

15.69 

16.71 

4-5 

15.11 

14 08 

15.24 

17-34 

22.15 

16 05 

17.09 

4.6 

15 44 

14-39 

I 5-58 

17.73 

22.64 

16.41 

17-47 

4-7 

I 5-78 

14.7 

15.92 

18.11 

23-13 

16.76 

17-85 

4.8 

16.11 

15.02 

16.26 

18.5 

23.62 

17.12 

18.23 

4.9 

1645 

I 5-33 

16.6 

18.88 

24.12 

17.48 

18.61 

S 

16.78 

15 64 

16.94 

19.27 

24.61 

17-83 

18.99 
























IRON BOILER TUBES, 


137 


Lap Welded. Charcoal Iron Boiler Tribes. 

Standard Dimensions. 


National Tube Works Company. 


Diameter. 

Ex- 1 In¬ 
ternal. J ternal. 1 

Thickness. 

Wire 

Gauge. 

1 

Circum: 

Ex¬ 

ternal. 

erence. 

In¬ 

ternal. 

Transverse Ar 

Ex- | In¬ 
ternal. j ternal. 

eas. 

Metal. 

Leng 
Sq. 
of Su 
Ex¬ 
ternal. 

h per 

Foot 

rface. 

In¬ 

ternal. 

Weight 

per 

Foot. 

Ins. 1 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Sq. Ins. 

Sq. Ins. 

Sq. Ins. 

Feet. 

Feet. 

Lbs. 

I 

.86 

.072 

15 

3-14 

2.69 

•78 

•57 

.21 

3.82 

4.46 

•71 

1.125 

.98 

.072 

15 

3-53 

3-08 

•99 

•76 

•24 

3-39 

3-89 

.8 

1-25 

1.11 

.072 

15 

3-93 

3-47 

1.23 

.96 

•27 

3.06 

3-45 

.89 

1 3 2 

I - I 5 

•083 

14 

4.12 

3-6 

i -35 

1.03 

•32 

2.91 

3-33 

I.08 

1-375 

1.21 

.083 

14 

4 - 3 2 

3-8 

1.48 

I - I 5 

•34 

2.78 

3.16 

I * I 3 

15 

i -33 

.083 

14 

4.71 

4.19 

1.77 

1.4 

•37 

2-55 

2.86 

1.24 

1.625 

i -43 

•095 

13 

5 -i 

4 5 i 

2.07 

1.62 

.46 

2-35 

2.66 

i -53 

1 75 

1.56 

•095 

13 

5-5 

4.9 

2 4 

1.91 

•49 

2.18 

2-45 

1.66 

1-875 

1.68 

•095 

13 

5-89 

5-29 

2.76 

2.23 

•53 

2.04 

2.27 

1.78 

2 

1.81 

•095 

13 

6.28 

5-69 

3 -i 4 

2-57 

•57 

1.91 

2.11 

1.91 

2.125 

i -93 

•095 

13 

6.68 

6.08 

3-55 

2.94 

.61 

1.8 

1.97 

2.04 

2.25 

2.06 

•095 

13 

7.07 

6 47 

398 

3-33 

.64 

i -7 

1.85 

2.16 

2-375 

2.16 

.109 

12 

7.46 

6.78 

4-43 

3-65 

•78 

1.61 

1.77 

2.61 

2-5 

2.28 

.109 

12 

7-85 

7.17 

4 9 1 

4.09 

.82 

i -53 

1.67 

2-75 

2-75 

2-53 

.109 

12 

8.64 

7-95 

5 94 

5-03 

•9 

I -39 

I * 5 I 

3-04 

2.875 

2.66 

.109 

12 

9-03 

8-35 

6.49 

5-54 

•95 

i -33 

1.44 

3-i8 

3 

2.78 

.109 

12 

9.42 

8.74 

7.07 

6.08 

•99 

1.27 

I -37 

3-33 

3-25 

3.0 1 

.12 

II 

10.21 

9.46 

8-3 

7.12 

1.18 

1.17 

1.26 

3 - 9 6 

3-5 

3.26 

.12 

II 

11 

10.24 

9.62 

8-35 

1.27 

1.09 

1.17 

4.28 

3-75 

n?- 5 i 

.12 

II 

SO 

H 

H 

11.03 

11.04 

9.68 

i -37 

1.02 

1.09 

4.6 

4 

3-73 

•134 

IO 

I2 -57 

11.72 

12 57 

10.94 

1.63 

•95 

1.02 

5-47 

4-25 

3 98 

! -134 

IO 

13-35 

12.51 

14.19 

12.45 

i -73 

•9 

.96 

5.82 

4-5 

4-23 

•134 

IO 

14.14 

13 29 

15-9 

14.07 

1.84 

•85 

•9 

6.17 

4-75 

448 

•134 

IO 

14.92 

14.08 

17.72 

I 5-78 

1.94 

.8 

•85 

6-53 

5 

4-7 

.148 

9 

i 5 - 7 i 

14.78 

19.63 

I 7-38 

2.26 

•76 

.81 

7-58 

5-25 

4-95 

.148 

9 

16.49 

I 5-56 

21.65 

I 9- 2 7 

2-37 

•73 

•77 

7-97 

5-5 

5-2 

.148 

9 

17.28 

16.35 

23.76 

21.27 

2.49 

•7 

•73 

8.36 

6 

5-67 

.165 

8 

18.85 

17.81 

28.27 

25-25 

3.02 

.64 

•67 

10.16 

7 

6.67 

.165 

8 

21.99 

20.95 

38.48 

34-94 

3-54 

•55 

•57 

11.9 

8 

7.67 

.165 

8 

25-13 

24.1 

50.27 

46.2 

4.06 

.48 

•50 

13-65 

9 

8.64 

.18 

7 

28.27 

27.14 

63.62 

58.63 

4-99 

.42 

•44 

16.76 

10 

9-59 

.203 

6 

31.42 

30.14 

78-54 

72.29 

6.25 

.38 

•4 

20.99 

11 

10.56 

.22 

5 

34-56 

33-17 

95-03 

87-58 7-45 

•35 

•36 

25-03 

12 

n -54 

.229 

4-5 

37-7 

36.26 

II 3 - 1 

104.63 

8 47 

•32 

•33 

28.46 

13 

12.52 

•238 

4 

40.84 

39-34 

132.73 123.19 

954 

.29 

•3 

32.06 

14 

13-5 

.248 

3-5 

43-98 

42.42 

I 53-94 I 43-22 

j 10.71 

•27 

.28 

36 

15 

14.48 

•259 

3 

47.12 

45 5 

176.71 

164.72 

11.99 

•25 

.26 

403 

16 

15-43 

.284 

2 

50.26 

48.48 

201.06 187.04 

14.02 

.24 

•25 

47.11 

17 

16.4 

•3 

1 

53-41 

5 I -52 

226.98 211.24 

15-74 

.22 

•23 

52.89 

18 

17.32 

•34 

0 

56-55 

54-41 

254 47 235-61 

18.86 

.21 

.22 

63-32 


Note. —In estimating effective heating or evaporating surface of Tubes, 
as heating liquids by steam, superheating steam, or transferring heat 
from one liquid or one gas to another, mean surface of Tubes is to be 
Computed. 

M* 





































138 


STEAM, GAS, AND WATER PIPE. 

Iron Welded. Steam, Gas, and Water Pipe. 

Standard Dimensions. 

National Tube Works Company. 


D 

In¬ 

ternal. 

iameter. 

Ex- | Actual 
ternal. Int’nal. 

Thickness. 

Circumference. 

Ex- In¬ 

ternal. ternal. 

^Tran 

Ex¬ 

ternal. 

sverse Ai 

In¬ 

ternal. 

eas. 

Metal 

Length 
per Sq 
Sur 

Ex¬ 

ternal. 

of Pipe 
Foot of 
face. 

In¬ 

ternal. 

I c ® 4S 

1 S 0 g 

G - -2 
0) — G 

50 

Weight 

per Foot. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

IllS. 

Sq. Ins. 

Sq. Ins. 

Sq.Ins 

Feet. 

Feet. 

Feet. 

Lbs. 

.125 

•4 

.27 

.07 

I.27 

.85 

• I 3 

.06 

•07 

9.44 

14-15 


.24 

•25 

•54 

•36 

.09 

i -7 

I.14 

•23 

.1 

.12 

7.07 

10.49 


.42 

•375 

.67 

.49 

.09 

2.12 

i -55 

•36 

.19 

.17 

5.66 

7-73 

75 i -2 

•56 

•5 

.84 

.62 

.11 

2.64 

1.96 

•55 

•3 

.25 

4-55 

6.13 

472.4 

.84 

•75 

1 05 

.82 

.11 

3-3 

2-59 

•87 

•53 

•33 

3-64 

4-63 

270 

I.II 

I 

I - 3 I 

1.05 

•13 

4- x 3 

3-29 

1.36 

.86 

•49 

2-9 

3-64 

166 9 

1.67 

1.25 

1.66 

1.38 

.14 

5.21 

4-33 

2.16 

i -5 

•67 

2-3 

2.77 

96.2 

2 24 

i -5 

1.9 

1.61 

.14 

5-97 

5.06 

2.83 

2.04 

.8 

2.01 

2-37 

70.7 

2.68 

2 

2.37 

2.O7 

•15 

7.46 

6.49 

4-43 

3-36 

1.07 

I.6l 

1.85 

42.9 

3 61 

2-5 

2.87 

2.47 

.20 

903 

7-75 

6-49 

4.78 

1.71 

i -33 

i -55 

30.1 

5-74 

3 

3 5 

3-°7 

.22 

II 

964 

9.62 

7-39 

2.24 

1.09 

1.24 

195 

7-54 

35 

4 

3-55 

•23,12.57 

11.15 

12.57 

9.89 

2.68 

•95 

1.08 

14.6 

9 

4 

45 

4-°3 

.24 

I 4- I 4 

12.65 

15 9 

12.73 

3 -i 7 

.85 

•95 

11.3 

10.66 

4 5 

5 

4 - 5 i 

•25 

I 5 - 7 I 

14.16 

19.63 

15 96 

3-67 

•76 

•85 

9 

1234 

5 

5-56 

5-04 

.26 

17.48 

15-85 

2431 

19.99 

4-32 

.69 

•76 

7.2 

14-5 

6 

6 62 

6.06 

.28 

20.81 

19.05 

3447 

28.89 

5 58 

•58 

•63 

5 

18.76 

7 

7.62 

7 02 

•30 

23-95 

22.06 

45.66 

38.74 

6-93 

•5 

•54 

3-7 

23 27 

8 

862 

798 

■32 

27.1 

25.08 

58.43 

50.04 

8 39 

•44 

.48 

2.9 

28.18 

9 

9 62 

8.94 

•34 

30.24 

28 08 

72.76 

62.73 

1003 

•4 

•43 

2-3 

33-7 

IO 

io -75 

10.02 

•37 

33-77 

3 i -48 

90.76 

78.84 

11.92 

•35 

•38 

1.8 

40.06 

II 

n -75 

II 

37 

36-91 

3456 

108.43 

95-03 

13-4 

•32 

•35 

i -5 

45.02 

12 

12.75 

12 

37 

40.05 

37-7 

127.68 

113.1 

14.58 

•3 

•32 

i -3 

48.98 

r 3 

14 

13 25 

•37 

43-98 

41.63 

153-94 

137-89 

16.05 

•27 

.29 

I 

53 92 

14 

15 

14-25 

•37 

47.12 

44-77 

176.71 

159.48 

17-23 

•25 

•27 

•9 

57-89 

15 

l6 

15 43 

.28 

50.26 

48.48 

201.06 

187.04 

14.02 

.24 

•25 

.8 

47.11 

l6 

17 

16.4 

•3 

53-4 

51-52 

226.98 

211.24 

15-74 

.22 

•23 

•7 

52.89 

17 

18 

17.32 

•34 

56-54 

5441 

254.46 

235.60 

18.86 

.21 

.22 

.6 

63-32 


STEEL LOCOMOTIVE TUBES. 

Ijap Welded Semi-Steel Locomotive Tubes. 

Standard Dimensions, 


National Tube Works Company. 




t/i 







Length per 


Diameter. 

o> 


Circumference. 

Transverse Areas. 

Sq. Foot 

-*» 

0 




<0 

® biD 






of Surface. 

.G O 

Ex- 

Ill- 

•J* 


Ex- 

In- 

Ex- 

In- 


Ex- 

In- 

® >- 

ternal. 

ternal. 

H 


ternal. 

ternal. 

ternal. 

ternal. 

Metal. 

ternal. 

ternal. 

a> 

> P-> 

Ins. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Sq. Ins. 

Sq. Ins. 

Sq. Ins 

Feet. 

Feet. 

Lbs. 

I 

•834 

•083 

14 

3.142 

2.62 

•785 

•546 

•239 

3.S2 

4-58 

.81 

1.25 

1.084 

.083 

14 

3-927 

3-405 

I.227 

•923 

•304 

3-056 

3-524 

1.03 

i -5 

I * 3 I 

•095 

13 

4.712 

4-115 

1.767 

1.348 

.419 

2.546 

2.916 

I.42 

i -75 

i- 53 2 

.IO9 

12 

5-498 

4.813 

2.405 

1.843 

.562 

2.183 

2-493 

I.9I 

2 

1.782 

.I09 

12 

6.283 

5-598 

3.142 

2.494 

.648 

1.91 

2.144 

2.2 

2.25 

2.032 

.IO9 

12 

7.069 

6.384 

3976 

3-243 

■733 

1.698 

1.88 

2.49 

2.5 

2.26 

.12 

II 

7-854 

7 -i 

4.909 

4.011 

.898 

1.528 

1.69 

3-05 

2.75 

2.51 

.12 

II 

8.639 

7.885 

5-94 

4.948 

•992 

1.389 

1.522 

3-37 

3 

2.76 

.12 

II 

9-425 

8.67 

7.069 

5-983 

1.086 

1.273 

1.384 

3-68 































































WEIGHT OF LEAD AND TIN PIPE AND TIN PLATES. I 39 


"Weiglit of Lead and. Tin Lined dPipe per Foot. 
From .375 Inch to 5 Inches in Diameter. (Tatham Bros.) 


WASTE-PIPE. \ BLOCK-TIN PIPE. 


Diam. 

Weight. 

Diam. 

Weight. 

Diam. 

Weight, i 

Diam. 

Weight. | 

Diam. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Inch. 

Lb. 

Inch. 

Lbs. 

Ins. 

Lbs. 

i -5 

2 

4 

8 

•375 

•3594 

.625 

•5 

1.25 

1.25 

2 

3 

4-5 

6 

•375 

•375 

.625 

.625 

1.25 

i -5 

3 

3-5 

4-5 

8 

-375 

•5 

•75 

.625 

i -5 

2 

3 

5 

5 

8 

•5 

•375 

•75 

•75 

i -5 

2.5 

4 

5 

5 

IO 

• 5 

•5 

X 

•9375 

2 

2-5 

4 

6 

5 

12 

I -5 

.625 

I 

1.125 

2 

3 


WATER-PIPE. 

From .375 Inch to 5 Inches in Diameter. 


Diam. 

Thick¬ 

ness. 

Weight. 

Diam. 

Thick¬ 
ness. 1 

Weight. 

uiam. 

Thick¬ 

ness. 

Weight. 

Diam. 

Thick¬ 

ness. 

Weight 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

•375 

.08 

.625 

.625 

•25 

3-5 

1.25 

.19 

4-75 

2.5 

•3125 

14 

•375 

.12 

I 

•75 

.1 

1.25 

1.25 

•25 

6 

2.5 

•375 

17 

.375 

.16 

1.25 

•75 

.12 

i -75 

i -5 

.12 

3 

3 

•1875 

9 

•375 

.19 

i -5 

•75 

.16 

2.25 

i -5 

.14 

3-5 

3 

•25 

12 

•375 

•34 

2-5 

■75 

.2 

3 

i -5 

•17 

4-25 

3 

•3125 

l6 

•5 

•°7 

•0545 

•75 

•23 

3-5 

i -5 

.19 

5 

3 

•375 

20 

•5 

.09 

•75 

•75 

•3 

4-75 

i -5 

•23 

6-5 

3-5 

•1875 

9-5 

•5 

.11 

I 

1 

.1 

i -5 

i -5 

•27 

8 

3-5 

•25 

15 

•5 

•13 

1.25 

I 

.11 

2 

I *75 

•13 

4 

3-5 

•3125 

18.5 

•5 

.l6 

i -75 

I I 

.14 

2-5 

i -75 

•17 

5 

3-5 

•375 

22 

•5 

.19 

2 

I 

•17 

3-25 

I, 75 

.21 

6-5 

4 

•1875 

12.5 

•5 

•25 

3 

I 

.21 

4 

I *75 

•27 

8-5 

4 

•25 

l6 

.625 

.08 

.O727 

I 

.24 

4-75 

2 

•15 

4-75 

4 

• 3 I2 5 

21 

.625 

.09 

I 

I 

•3 

6 

2 

.18 

6 

4 

•375 

25 

.625 

•13 

i -5 

1.25 

.1 

2 

2 

.22 

7 

4-5 

•1875 

14 

.625 

.l6 

2 

1-25 

.12 

2-5 

2 

•27 

9 

4-5 

•25 

18 

.625 

.2 

2.5 

1.25 

.14 

3 

2.5 

•1875 

8 

5 

•25 

20 

.625 

.22 

2.75 

1.25 

.16 

3-75 

2.5 

•25 

11 

5 

•375 

31 


nVLapLs and “W eiglit of Tin-plates. (English.) 


Mark 
or Brand. 

Plates 
per Box. 

Dimensions. 

Weight 1 
per Box. 

Mark 
or Brand. 

Plates 
per Box. 

Dimensions. 

Weight 
per Box. 


No. 

Ins. 

No. 


No. 

Ins. 

No. 

1 Cor x Com. 

225 

13.75x10 

112 

DXXXX. 

IOO 

16.75X12.5 

189 

2 C 

22l§ 

n.2^X Q.7S 

103 

SDC. 

200 

15 Xu 

168 

o c 

22 S 

12. 7 3 X Q- 3 

q8 

SDX. 

200 

15 X11 

188 

j . . 

H C 

225 

22 S 

IO. 7C x 10 

y 

I IQ 

SDXX. 

200 

15 X11 

20Q 

H X . 

I^7^XlO 

y 

I 37 

SDXXX. 

200 

15 X 11 

7 

230 

1 X . 

225 

13.75X10 

140 

SPXXXX. ... 

200 

15 XII 

251 

2 X .a...... 

225 

I 3 - 25 X 9-75 

133 

SDXXXXX.. 

200 

15 Xu 

272 

3 X . 

225 

12.75X 9 5 

126 

SDXXXXXX. 

200 

15 Xu 

293- 

1 XX. 

225 

13.75X10 

161 

Leaded IC. .. 

11 2 

20 X14 

112 

1 XXX. 

225 

13.75X10 

182 

“ IX... 

112 

20 X 14 

140 

x XXXX.... 

225 

13.75X10 

203 

ICW. 

225 

13.75X10 

112 

1 XXXXX.. 

225 

13.75X10 

224 

IX w. 

225 

13.75X10 

140 

I xxxxxx. 

225 

13.75X10 

245 

CSDW. 

200 

15 Xu 

168 

DC . 

TOO 

l6. 7^Xl2. ^ 

08 

CIIW. 

IOO 

;l6.75X 12.5 

105 

DX 

JOO 

16. 7 3 X 12. 3 

y 

126 

XIIW. 

IOO 

16.7SX 12.s 

126 

DXX 

TOO 

16. 7^Xl2. ^ 

147 

TT. 

4SO 

13.7S X10 

112 

DXXX. 

IOO 

16.75X12.5 

168 

XTT. 

450 

13.75X10 

126 


When the plates are 14 by 20 inches, there are 112 in a box. 







































































































140 


WEIGHT OF COPPER TUBES 


'W'eiglit of Seamless Drawn. Copper Tubes. 
American Tube Works. (Boston.) 

BY EXTERNAL DIAMETER. ONE FOOT IN LENGTH. 

Stubs' W, G. From, .25 Inch to 12 Ins*—/full , l light. 


No. 

| 20 

19 

18 

17 

16 

1 15 

14 

1 13 

1 12 

11 

Ins. 

j V32 / 

3/64 l 

3/64/ 

1/16 l 

1/16/ 

S/64 l 

5 / 64 / 

3/32 / 

7/64 

1/8 1 

Diamet’r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•25 

.09 

.1 

.12 

•13 

.14 

•15 

•17 

.18 

.19 

.19 

•375 

.14 

.16 

• I 9 

•23 

.24 

.26 

.29 

•32 

•35 

•37 

•5 

.2 

•23 

•27 

•31 

•34 

-37 

.42 

•47 

•52 

•56 

.625 

•25 

.29 

•34 

•4 

•44 

.48 

•55 

.61 

.69 

•74 

•75 

•3 

•36 

.42 

•49 

•54 

•59 

.67 

.76 

.85 

.92 

.875 

•36 

.42 

•49 

.58 

.64 

•7 

.8 

•9 

1.02 

1.11 

1 

.41 

.48 

•57 

.67 

•74 

.81 

•93 

1.05 

1.18 

1.29 

1-125 

.46 

•55 

.64 

.76 

.83 

.92 

1.05 

1.19 

i-35 

1.47 

1.25 

•52 

.61 

.71 

.84 

•93 

1.03 

1.18 

i-34 

1.52 

1.65 

1-375 

•57 

.68 

•79 

•93 

I,0 3 

1.14 

i-3 x 

1.48 

1.68 

1.84 

i-5 

.62 

•74 

.86 

1.02 

I * I 3 

1.25 

i-43 

1.63 

1.85 

2.02 

1.625 

.68 

.8 

•94 

1.11 

1.23 

1.36 

1.56 

1.77 

2.02 

2.2 

i-75 

•73 

•87 

1.01 

1.2 

I -33 

1.47 

1.69 

1.92 

2.18 

2.39 

1-875 

.78 

•93 

1.09 

1.29 

T -43 

1.58 

1.81 

2.06 

2-35 

2-57 

2 

.84 

1 

1.16 

i-37 

I *53 

1.69 

1.94 

2.21 

2.51 

2-75 

2.125 

.89 

1.06 

1.24 

1.46 

1.63 

1.8 

2.07 

2-35 

2.68 

2-93 

2.25 

•94 

X,I 3 

I -3 I 

i-55 

i-73 

1.91 

2.19 

2-5 

2.85 

3.12 

2-375 

1 

1.19 

i-39 

1.64 

1.82 

2.02 

2.32 

2.64 

3.01 

3-3 

2-5 

1.05 

1.25 

1.46 

i-73 

1.92 

2.13 

2-45 

2.79 

3.18 

3-48 

2.625 

1.1 

1.32 

i-54 

1.82 

2.02 

2.23 

2-57 

2-93 

3-35 

3-67 

2-75 

1.16 

1.38 

1.61 

1.9 

2.12 

2-34 

2.7 

3.08 

3-5i 

3-85 

2.875 

1.21 

i-45 

1.68 

I -99 

2.22 

2-45 

2.83 

3.22 

3.68 

4-03 

3 

1.26 

I -5 I 

1.76 

2.08 

2.32 

2.56 

2-95 

3-37 

3-84 

4.22 

3-25 

i-37 

1.64 

1.91 

2.26 

2.52 

2.78 

3.21 

3.66 

4.18 

4.58 

3-5 

1.48 

1.77 

2.06 

2-43 

2.72 

3 

3-46 

3-95 

4-5i 

4-95 

3-75 

1.58 

1.9 

2.21 

2.61 

2.92 

3.22 

3-7 1 

4.24 

4.84 

5-3i 

4 

1.69 

2.02 

2.36 

2.79 

3-n 

3-44 

3-97 

4-53 

5-i7 

5-68 

4-25 

1.8 

2.15 

2.51 

3-i4 

3 3i 

3.66 

4.22 

4.82 

5-5i 

6.05 

4-5 

1.9 

2.28 

2.65 

3-3 2 

3-5i 

3.88 

4-47 

5-n 

5-84 

6.41 

4-75 

2.01 

2.41 

28 

3-49 

3-7i 

4.1 

4-73 

5-4 

6.17 

6.78 

5 

2.12 

2-54 

2-95 

3-67 

3-9 1 

4-32 

4.98 

5-69 

6-5 

7- I 4 

5-25 

2.23. 

2.66 

3-i 

3.85 

4.11 

4-54 

5-23 

5-98 

6.84 

7-5i 

5-5 

2-34 

2.79 

325 

3.85 

4-3 

4.76 

5-49 

6.27 

7.17 

7.87 

5-75 

2.44 

2.92 

3-4 

4.02 

4-5 

4.98 

5-74 

6.56 

7-5 

8.24 

6 

2-55 

3-05 

3-55 

4.2 

4-7 

5-2 

5-99 

6.85 

7-83 

8.61 

6.25 

2.66 

3 -i 8 

3-7 

4.38 

4.9 

5-4i 

6.25 

7.14 

8.17 

8.97 

6-5 

2.76 

3-3i 

3-85 

4-55 

5-i 

5-63 

6-5 

7-43 

8-5 

9 34 

6-75 

2.87 

3-44 

4 

4-73 

5-3 

5-85 

6-75 

7.72 

8.83 

9-7 

7 

2.98 

3-56 

4-i5 

4.91 

5-49 

6.07 

7.01 

8.01 

9.16 

10.07 

7-25 

3-°9 

3-69 

4-3 

5.09 

5-69 

6.29 

7.26 

8.30 

9-5 

10.44 

7-5 

3-i9 

3.82 

4-45 

5.26 

5-89 

6.51 

7-5i 

8-59 

9-83 

10.8 

8 

3-4i 

4.08 

4-74 

5.62 

6.29 

6-95 

8.02 

9.17 

10.49 

u-53 

8-5 

3.62 

4-33 

5-04 

5-97 

6.68 

7-39 

8.52 

9-75 

11.16 

12.26 

9 

3-83 

4-59 

5-34 

6-33 

7.08 

7-83 

9-°3 

10.33 

11.82 

*3 

9-5 

4-05 

4-85 

5-64 

6.68 

7.48 

8.26 

9-54 

10.91 

12.49 

13-73 

10 

4.26 

5 -u 

5-94 

7-03 

7.87 

8.7 

10.05 

11.49 

I3-I5 

14.46 

10.5 

4-47 

5-37 

6.24 

7-39 

8.27 

9- x 4 

io-55 

12.07 

13.82 

I5-I9 

11 

4.69 

5.62 

6-54 

7-74 

8.67 

9-58 

11.06 

12.65 

14.48 

15.92 

n -5 

4.9 

5-88 

6.84 

8.1 

9.06 

10.02 

11.56 

13-23 

I5-I5 

16.66 

12 

5- 11 

6.13 

7 a 3 

8-45 

9.46 

10.45 

12.07 

13.81 

15.81 

17.29 













































WEIGHT OF COPPER TUBES 


141 


No. 

10 1 

9 

8 

7 

6 

5 

4 1 

3 1 

2 

1 

Ins. 

9/64 1 

9/64/ 

11 /64 l 

3/i6 l 

13/64 

V32 / 

*5/64 / 

V 4 / 

9/32/ 

*9/64 / 

Diamet’r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

•4 

.41 

.42 

•44 

— 

— 

— 

— 

— 

— 

•5 

.61 

. -6 4 

.67 

•7i 

•73 

•75 

.76 

— 

— 

— 

.625 

.81 

.86 

.92 

•99 

1.04 

1.09 

1.12 

1.13 

1.18 

— 

•75 

1.01 

1.09 

1.17 

1.26 

i-35 

1.42 

I.49 

i-53 

1.6l 

I.63 

•875 

1.22 

1-3i 

I.42 

I -53 

1.66 

1.76 

1.85 

1.92 

2.04 

2.09 

1 

1.42 

i-54 

I.67 

1.81 

1.97 

2.09 

2.21 

2.32 

2.48 

2-55 

1.125 

1.63 

1.78 

i-93 

2.08 

2.28 

2-43 

2.58 

2.71 

2.9I 

3 

1.25 

1.83 

2 

2.18 

2.36 

2-59 

2.76 

2.94 

3-11 

3-34 

3-46 

I, 375 

2.03 

2.22 

2-43 

2.63 

2.9 

3- 1 

3-3 

3-5 

3-77 

3-92 

i-5 

2.24 

2.44 

2.68 

2.91 

3- 21 

3-43 

3-67 

3-9 

4.21 

4.38 

1.625 

2.44 

2.67 

2.93 

3.18 

3-52 

3-77 

4-03 

4.29 

4.64 

4.83 

I *75 

2.65 

2.89 

3.18 

3-45 

3-83 

4.11 

4-39 

4.69 

5-07 

5-29 

1-875 

2.85 

3.12 

3-44 

3-7 3 

4- I 4 

4.44 

4.76 

5.08 

5-5i 

5-75 

2 

3.06 

3-34 

3-69 

4 

4 45 

4.78 

5-i2 

^5 48 

5-94 

6.21 

2.125 

3.26 

3-57 

3-94 

4 28 

4-75 

5-11 

5-48 

5-87 

6 37 

6.66 

2.25 

3-46 

3-8 

4.19 

4-55 

5.06 

5-45 

5-84 

6.27 

6.81 

7.12 

2 -375 

3-67 

4.02 

4.44 

4.82 

5-37 

5-78 

6.21 

6.66 

7.24 

7-57 

2-5 

3-87 

4-25 

4.69 

5-i 

5.68 

6.12 

6-57 

7.06 

7.67 

8.04 

2.625 

4.08 

4-47 

4-95 

5-37 

6 ‘ 

6-45 

6-93 

7-45 

8.1 

8 49 

2-75 

4.28 

4-7 

5-2 

5-65 

63 

6.79 

7.29 

7-85 

8-54 

8 95 

2.875 

4.48 

4.92 

5-45 

5-92 

6 61 

7.12 

7.66 

8.24 

8.97 

9 4i 

3 

4.69 

5-i5 

5-7 

6.2 

6 92 

7.46 

8 02 

8.64 

94 

9.87 

3-25 

5-i 

5-6 

6.2 

6.74 

7-54 

8.13 

8-75 

9-43 

10.27 

10.78 

3-5 

5-5i 

6.05 

6.71 

7.29 

8.16 

8.8 

9 47 

10.22 

11.14 

11.7 

3-75 

5-9 1 

6-5 

7.21 

7.84 

8.78 

9-47 

10.2 

11.01 

12 

12.61 

4 

6.32 

6-95 

7.71 

8-39 

9.4 

10.14 

10 92 

11 8 

12.87 

13.53 

4-25 

6 -73 

7-4 

8-22 

8.94 

10.02 

10 81 

11 65 

12.59 

13-73 

14.44 

4 5 

7.14 

7-85 

8.72 

9.49 

10.64 

11.48 

12-37 

13-38 

14.6 

15.36 

4-75 

7 55 

8-3 

9.22 

; 10.04 

11.26 

12.16 

13 1 

14.17 

15.46 

16.27 

5 

7.96 

8-75 

9-73 

j 10.58 

11.88 

12.83 

13-83 

14.96 

16.33 

17.19 

5-25 

8.36 

9.21 

10.23 

| n.13 

12-49 

13-5 

14-55 

15-75 

17.2 

18.1 

5-5 

8.77 

9.66 

10.73 

11.68 

I 3 11 

14.17 

15-28 

16.54 

18.06 

19 02 

5-75 

9.18 

10.11 

11.24 

12.23 

13-73 

14.84 

16 

17.33 

1893 

1993 

6 

9-59 

10.56 

11.74 

12.78 

1435 

15-51 

16-7 3 

18.12 

19.79 

20.85 

6.25 

10 

11.01 

12.24 

13-33 

14.97 

16.18 

17.46 

18.91 

20.66 

21.76 

6-5 

10.41 

11.46 

12.75 

,3.88 

15 59 

16.85 

18.18 

19.7 

21-53 

22.68 

6-75 

10.82 

11.91 

13-25 

14.42 

16.21 

I7-52 

18.91 

20.49 

22.39 

23.59 

7 

11.22 

12.36 

13-75 

14.97 

16.83 

18.19 

19.63 

21.28 

23.26 

24.51 

7-25 

11.63 

12.81 

14.26 

15-52 

17-45 

18.86 

20.36 

22.07 

24.13 

25.42 

7-5 

12.04 

13.26 

14.76 

16.07 

18.07 

19-54 

21.08 

22.86 

25 

26.34 

7-75 

12.45 

i3-7i 

15.26 

16.62 

18.68 

20.21 

21.81 

23.65 

25.86 

27.25 

8 

12.86 

14.17 

15-77 

17.17 

19-3 

20.88 

22.54 

2444 

26.72 

28.17 

8.25 

13-27 

14.62 

16-27 

17.71 

19 92 

2i-55 

23.26 

25-23 

27-59 

29.08 

8-5 

13-67 

15-07 

16.77 

18.26 

20.54 

22.22 

2399 

26.02 

28.45 

30 

8-75 

14.08 

I5-52 

17.28 

18.81 

21.16 

22.89 

24.71 

26.81 

29.32 

30.91 

9 

14.49 

15-97 

17.78 

19.36 

21.78 

23.56 

25-44 

27.6 

30.18 

31-83 

9-25 

14.9 

16.42 

18.28 

I 9-9 I 

22.4 

24.23 

26.17 

28.39 

31.05 

32.74 

9-5 

i5-3i 

16.87 

18.79 

20.46 

23.02 

24.9 

26.89 

29.18 

31.92 

33-66 

9-75 

15-72 

17.32 

19.29 

21.01 

23.64 

25.57 

27.62 

29.97 

32.78 

34-57 

10 

16.12 

17.77 

19.79 

21-55 

24.26 

26.24 

28.34 

30.76 

33.65 

35-49 

10.5 

16.94 

18.68 

20.8 

22.65 

25-5 

27-59 

29.79 

32-34 

35.38 

37-32 

11 

17.76 

I9-58 

21.81 

23-75 

26.73 

28.93 

3i-25 

33-92 

37-11 

39-15 

n-5 

18.57 

20.48 

22.81 

24.84 

27.97 

30.27 

32.7 

35-5 

38.84 

40.98 

12 

19-39 

21.38 

23.82 

25 94 

29.21 

31.61 

34-15 

37.08 

40.58 

42.81 




















































142 WEIGHT OF COPPER AND BRASS TUBES, ETC. 


By Internal Diameter. 

Add following Units to Weights for External Diameter in preceding tables. 


No. 

1 

2 

3 

4 

5 

6 

* 7 

8 

9 

10 


2.21 

1.97 

1.66 

1-38 

1.18 

1.01 

00 ! 

! 

1 

.67 

•53 

•43 

No. 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 


•35 

.29 

.22 

•17 

•13 

.11 

.08 

.06 

•05 

•03 


Illustration. —What is weight of a copper tube 6 ins. in internal diameter, 
No. 3 gauge, and one foot in length? 

By preceding table 6 ins. external, No. 3 gauge = 18.12, and 18.12 -f- 1.66 = 
19.78 lbs. 

WEIGHT OF BRASS TUBES. 

To Compute NVeiglit of Brass Tubes. 
American Tube Works. (Boston.) 

Rule. —Deduct 5 per cent, from weight of Copper tubes. 

Example. —What is weight of a brass tube 6 ins. in external diameter, No. 3 
gauge, and one foot in length? 

By preceding table 6 ins. = 18.12, from which deduct 5 per cent. = 17.21 lbs. 

By Internal Diameter. 

Rule. —Proceed as above for internal diameter of copper tube, and deduct 
5 per cent. 

Example. —Weight of a copper tube 6 ins. internal diameter, No. 3 gauge, and 
1 foot in length = 19.78 lbs. 

Hence, 19.78 — 5 per cent. = 18.79 ^s. 

Note.— Diameter of Tubes, as for Boilers, is given externally, and that for Pipes 
internally. 

Weights of English as given by D. K. Clark are essentially alike to the 
preceding. 

Brass Tubes Corresponding witli and Bitted for 
Iron Tubes or Pipes. 

American Tube Works. (Boston.) 

WEIGHT PER LINEAL FOOT. 


Diameter of Iron Pipe. 


Diameter of Iron Pipe. 


Diameter of Iron Pipe. 


Internal. 

External. 

Weight. 

Internal. 

External. 

W eight. 

Internal. 

External. 

Weight 

Inch. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

.125 

•375 

•25 

I 

I- 3 I 25 

i -7 

3 

3-5 

8-3 

•25 

•5625 

•43 

I.25 

I.625 

2-5 

3-5 

4 

IO.9 

•375 

.6875 

•63 

i -5 

1-875 

3 

4 

4-5 

12.7 

•5 

.8125 

•9 

2 

2-375 

4 

5 

5-5 

I 5-7 

•75 

1.0625 

1.25 

2-5 

2.875 

4.87 





“Weight of Street Brass. 
ONE SQUARE foot. ( HoltzgpffeVs Gauge.) 


Thickness. 

Weight. 

Thickness. 

tV eight. 

Thickness. 

Weight. 

Thickness. 

Weight 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

3 

•259 

IO.9 

9 

.148 

6.23 

15 

.072 

3-°3 

21 

.032 

i -35 

4 

.238 

IO 

10 

•134 

5-64 

16 

.065 

2.74 

22 

.028 

1.18 

5 

.22 

9.26 

11 

.12 

5-05 

17 

.058 

2.44 

23 

.025 

1.05 

6 

.203 

8-55 

12 

.IO9 

4-59 

18 

.049 

2.06 

24 

.022 

.926 

7 

.18 

7-58 

J 3 

•095 

4 

19 

.042 

1.77 

25 

.02 

.842 

8 

.165 

6-95 

14 

.083 

3-49 

20 

•035 

1.47 
























































































WEIGHT OF WROUGHT IRON TUBES. I43 

"Weiglit of ‘'W'l’OTTgb.t Iron Tubes. (English.) 

EXTERNAL DIAMETER. ONE FOOT IN LENGTH. 


HoltzapJJ'eVs Wire - Gauge , f full , l iglit . 


No. 

— 

— 

4 ! 

5 i 

6 

7 

8 

9 

Ins. 

•3125 

.281 

.238 

.22 | 

.203 

.18 

.165 

.I48 

5/16 

9/32 

. *5/64 / 

7/32 

I 3/64 

3/16 l 

n/64 l 

9/64 / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

7 

2I.9 

19.8 

16.9 

15.6 

14-5 

12.9 

11.8 

10.6 

7-5 

23-5 

21.3 

18.1 

16.8 

15-5 

13.8 

12.7 

II.4 

8 

25.2 

22.7 

193 

17.9 

16.6 

14.7 

I 3 > 5 

12.2 

8-5 

26.8 

24.2 

20.6 

19.I 

17.6 

i 5-7 

14.4 

I2.9 

9 

28.4 

25-7 

21.8 

20.2 

18.7 

l6.6 

I 5-3 

13-7 

9-5 

30.1 

27.I 

23.1 

2I.4 

19.8 

17.6 

l6.I 

14-5 

IO 

31-7 

28.6 

24-3 

22.5 

20.8 

18.5 

17 

15-3 


No. 

7 

8 

9 

10 

1 1 

1 2 

13 

'4 

15 


.18 

.165 

00 

M 

•*34 

. 12 

. 109 

•095 

.083 

.O72 

Ins. 

3/16 l 

n/64 l 

9/64/ 

9/64 1 

Vs 1 

7/64 

3/32/ 

5/64/ 

5/64 J 

Diarn. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

i -55 

1.44 

1.32 

1.22 

I.II 

1.02 

•9 

•797 

•7 

1.125 

1.78 

1.66 

1-51 

i -39 

1.26 

1.16 

i -3 

.906 

•794 

1.25 

2.02 

1.88 

1-71 

i -57 

I.42 

I *3 

I * I 5 

I.OI 

.888 

1.375 

2.25 

2.09 

1.9 

1.74 

1.58 

i -45 

1.27 

1.12 

•983 

i -5 

2.49 

2.31 

2.1 

1.92 

i -73 

i -59 

1.4 

1.23 

1.08 

1.625 

2.72 

2.52 

2.29 

2.09 

1.89 

i -73 

1.52 

i -34 

I * I 7 

i -75 

2.96 

2.74 

2.48 

2.27 

2.05 

1.87 

1.65 

i -45 

1.27 

1-875 

3-19 

2.96 

2.68 

2-45 

2.21 

2.02 

1.77 

1.56 

1.36 

2 

3-43 

3-17 

2.87 

2.62 

2.36 

2.16 

1.9 

1.67 

i -45 

2.125 

367 

3-39 

3.06 

2.8 

2.52 

2.3 

2.02 

1.78 

i -55 

2.25 

3-9 

3-6 

3.26 

2.97 

2.68 

2.44 

2.I4 

1.88 

1.64 

2.375 

4.14 

3.82 

3-45 

3-15 

2.83 

2.59 

2.27 

i -99 

1.74 

2-5 

4-37 

4'°4 

3-65 

3-32 

2.99 

2.73 

2-39 

2.1 

1.83 

2.625 

4.6l 

4-25 

3-84 

3-5 

3 -i 5 

2.87 

2.52 

2.21 

i -93 

2-75 

4.84 

4-47 

4-03 

3-67 

3 - 3 i 

3.02 

2.64 

2.32 

2.02 

2.875 

5.08 

4.68 

4-23 

3-85 

3-46 

3- i 6 

2.77 

2-43 

2 .II 

3 

5-32 

4.9 

4.42 

4.02 

3.62 

3-3 

2.89 

2-54 

2.21 

3-25 

5-79 

5-33 

4.81 

4-37 

3-94 

3-59 

3 -i 4 

2-75 

2.4 

3-5 

6.26 

5-76 

5-2 

4.72 

4-25 

3-87 

3-39 

2.97 

2.59 

3-75 

6-73 

6.I9 

5-58 

5-07 

4-57 

4.16 

3-64 

3 -i 9 

2.77 

4 

7.2 

6.63 

5-97 

5-43 

4.88 

4.44 

3-89 

3-4 

2.96 

4-25 

7.67 

7.06 

6.36 

5-78 

5-2 

4-73 

4 -i 3 

3.62 

3 -i 5 

4-5 

8.14 

7-49 

6-45 

6.13 

5 - 5 i 

5.01 

4-38 

3-84 

3-34 

4-75 

8.61 

7.91 

7-13 

6.48 

5.82 

5-3 

4-63 

4.06 

3-53 

5 

9.08 

8-35 

7-52 

6.83 

6.13 

5.58 

4.88 

4.27 

3-72 

5-25 

9-56 

8.79 

7 - 9 1 

7.18 

6.44 

5-87 

5 -i 3 

4.49 

3-9 

5-5 

IO 

9.22 

8-3 

7-53 

6.76 

6.15 

5-38 

4.71 

4.09 

5-75 

10.5 

965 

8.68 

7.88 

7.07 

6.44 

5-63 

4-93 

4.28 

6 

II 

IO.I 

9.07 

8.23 

7-39 

6-73 

5-87 

5 -i 4 

4-47 

6.25 

11.4 

10.5 

9.46 

8.58 

7-7 

7.01 

6.12 

5-36 

4.66 

6-5 

11.9 

10 9 

9-85 

8-93 

8.02 

7-3 

6-37 

5-58 

4-85 

6-75 

12.4 

11.4 

10.2 

9.28 

8-33 

7-58 

6.62 

5-79 

5-03 

7 

12.9 

H.8 

10.6 

9-63 

8.64 

7.87 

6.87 

6.01 

5.22 

7-2 5 

13-3 

12.2 

II 

999 

8.96 

8.15 

7.12 

6.23 

5 - 4 i 

7-5 

13.8 

12.7 

11.4 

10.3 

9.27 

8.44 

7-37 

6.45 

5-6 

7-75 

14-3 

13-1 

11.8 

10.7 

9-59 

8.72 

7.62 

6.66 

5-79 

8 

14.7 

13-5 

12.2 

II 

99 

9.01 

7.86 

6.88 I 

5 98 
































































144 


WEIGHT OF COPPER TUBES, 


'Weight of Seamless Drawn Copper Tubes. (English.) 
For Diameters and Thicknesses not given in preceding Tables. (D. K. Clark.) 

INTERNAL DIAMETER. ONE FOQT IN LENGTH. 


Holtzapjfel's Wire-Gauge, f full, l light. 
Specific Weight = 1.16. Wrought Iron = i. 


No. 

0000 

000 

00 

0 

No. 

0000 

000 

00 

0 

Ins. 

•454 

•425 

•38 

•34 

Ins. 

•454 

•425 

•38 

•34 

29/64 

27/64 f 

3/8 / 

11 / 32 

29/64 

2 7 / 6 4 / 

3 / 8 / 

°/32 

Diain. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•75 

— 

— 

— 

4-5 

5-75 

34-2 

31-9 

28.3 

25.2 

•875 

— 

— 

5-79 

5-02 

6 

35-6 

33-2 

29-5 

26.2 

1 

8.02 

7-36 

6-37 

5-53 

6-5 

38-4 

35-8 

31.8 

28.3 

1-125 

8.71 

8 

6-95 

6.05 

7 

41.1 

38-3 

34-i 

30.3 

1.25 

9.4 

8.65 

7-52 

6-57 

7-5 

43-9 

40.9 

36.4 

32.4 

1-375 

IO.I 

9-3 

8.1 

7.08 

8 

46.6 

43-5 

38-7 

34-5 

i-5 

10.8 

994 

8.68 

7.6 

9 

52.1 

48.7 

43-3 

38.6 

1.625 

i*-5 

10.6 

9.26 

8.12 

10 

57-7 

53-8 

47-9 

42.7 

i-75 

12.1 

11.2 

983 

8.63 

11 

63.2 

59 

52-5 

46.8 

i -875 

12.8 

11.9 

10.4 

9-i5 

12 

68.7 

64.2 

57-2 

5i 

2 

13-5 

12.5 

11 

9.66 

13 

74.2 

693 

61.8 

55-i 

2.125 

14.2 

I 3-3 

11.6 

10.2 

14 

79-7 

74-5 

66 4 

59-2 

2.25 

14.9 

13.8 

12.1 

10.7 

i5 

85.2 

79.6 

7i 

63-4 

2-375 

15.6 

14-5 

12.7 

11.2 

16 

90.7 

84.8 

75 6 

67.7 

2.5 

16.3 

I 5- 1 

13-3 

n . 7 

17 

96-3 

90 

80.2 

71.8 

2.625 

17 

15.8 

13 9 

12.2 

18 

101.8 

95-i 

84 9 

76 

2-75 

17.7 

16.4 

14-5 

12-8 

19 

107.3 

100.3 

89-5 

80.1 

3 

19.1 

17.7 

15.6 

13.8 

20 

112.8 

105-5 

94.1 

84.2 

3-25 

20.4 

x 9 

16.8 

14.8 

21 

118.3 

110.7 

98.7 

88.3 

3-5 

21.8 

20.3 

17.9 

15-9 

22 

123.8 

115.8 

103-3 

9 2 -5 

3-75 

23.2 

21.6 

19.1 

16.9 

23 

129.3 

120.9 

107.9 

96.6 

4 

24.6 

22.9 

20.2 

17.9 

24 

134.8 

126.1 

112.6 

100.6 

4-25 

25-9 

24 2 

21.4 

19 

26 

146 

136.4 

121.8 

108.8 

4-5 

2 7-3 

25-4 

22.5 

20 

28 

157-2 

146.7 

131 

117.1 

4-75 

28.7 

26.7 

23-7 

21 

30 

168.4 

I 57- 1 

140.2 

125.4 

5 

30.1 

28 

24.8 

22.1 

32 

179.6 

167.4 

149-5 

133-6 

5-25 

31-5 

29-3 

26 

23.I 

34 

190.7 

177.7 

158-7 

i4i-9 

5-5 

32.8 

30.6 

27.1 

24.I 

3 6 

201.9 

188 

167.9 

150.1 


For Diameters from 13 to 24 Inches. 


No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Ins. 

•3 

j 9/64 / 

. 284 
9/32 / 

•259 

Gif 

.238 
I 5 /64 / 

.22 

7/32/ 

.203 

13/64 

.18 

3/16 1 

.165 
Ir / 64 1 

. 148 
9 / 64 / 

•134 

9/64 1 

Diain. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

13 

48.5 

45-8 

4I.7 

38.3 

35-3 

32.6 

28.8 

26.4 

23.6 

2I.4 

14 

52.1 

49-3 

44.9 

41.2 

38 

35 -i 

31 

28.4 

25-4 

23 

15 

55-8 

52.7 

48 

44.I 

40.7 

37-6 

33-2 

30-4 

27.2 

24.6 

16 

59-4 

56.2 

51.2 

46.9 

43-4 

40 

35-4 

32.4 

29 

26.3 

17 

63 

59-6 

54-3 

49.8 

46 

42.5 

37-5 

34-4 

30.8 

27.9 

l 8 

66.7 

63.1 

57-4 

52.7 

48.7 

45 

39-7 

36.4 

32.6 

29-5 

*9 

70-3 

66.5 

60.6 

55-6 

5 i -4 

47-4 

41.9 

38-4 

34-4 

3 1 - 2 

20 

74 

70 

63-7 

58-5 

54 

49.9 

44.1 

40.4 

36.2 

32.8 

21 

77.6 

73-4 

66.9 

61.4 

56.7 

52.4 

46.3 

42.4 

38 

34-4 

22 

81.3 

76.9 

70 

643 

59-4 

54-9 

48.5 

44.4 

39-8 

36 

23 

84.9 

80.3 

73 2 

67.2 

62.1 

57-3 

50.7 

46.4 

41.6 

37-7 

24 

88.6 

838 

763 

70.1 

64.7 

59-8 

52 9 

48-5 

43-4 

39-3 


















































WEIGHT OF COPPER AND WROUGHT IRON TUBES. I45 
For Diameters from, 13 to 24 Inches. 


No. 

II | 12 

13 

'4 

• 5 

16 

'7 

18 

19 

20 


. 12 

. IO9 

•095 

.083 

.O72 

.065 

.058 

.049 

.O42 

•°35 


Vs 1 

7/64 

3/32/ 

5 / 64 / 

s/64 1 

r /i6 / 

V16 l 

3/64/ 

3/64 l 

V32 / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

13 

IQ.I 

17.4 

i5-i 

13.2 

II.4 

IO.3 

9.2 

7-77 

6.65 

5-55 

14 

20.6 

18.7 

16.3 

I4.2 

12.3 

II.I 

9.9 

8-37 

7.16 

5-98 

15 

22.1 

20 

17.4 

15.2 

13.2 

II.9 

10.6 

8.96 

7.67 

6.4 

16 

23-5 

2I.3 

1S.6 

l6.2 

I4.I 

I2.7 

n-3 

956 

8.18 

6.82 

17 

25 

22.7 

*9-7 

I7.2 

I4.9 

13-5 

12.1 

10.2 

8.69 

7.27 

l8 

26.4 

24 

20.9 

l8.2 

15-8 

14-3 

12.7 

10.7 

9.2 

7.69 

19 

27.9 

25-3 

22 

I9.2 

16.7 

I 5 - 1 

13 4 

11 "3 

9.71 

8.12 

20 

29-3 

26.6 

23.2 

20.2 

17.6 

i5-9 

14.1 

11.9 

10.2 

8-54 

21 

3 °.S 

27.9 

24-3 

2I.3 

18.4 

16.6 

14.8 

12.5 

IO.7 

8.96 

22 

32-3 

29-3 

25-5 

22.3 

19-3 

17.4 

15 5 

131 

II.2 

9-39 

23 

33-7 

30.6 

26.7 

23 3 

20.2 

18.2 

16.2 

13-7 

11.8 

9.81 

24 

35-2 

3i-9 

27.8 

24 3 

21.1 

T 9 

16.9 

14-3 

12.3 

10.2 


NVeiglit of NVrought Iron Tubes. (English.) 

For Diameters and Thicknesses not given in preceding Tables. (D. K. Clai'k.) 

INTERNAL DIAMETER. ONE FOOT IN LENGTH. 


Holtzapjfel's Wire-Gauge, f full , l light. 


No. 








j V4 

4 

5 

1 6 

7 

Ins. 

5/8 

9/16 

Thickn 

X /2 

ESS IN II 

7/l6 

s’CHES, 

3/8 

5/16 

.238 
I 5/64 / 

.22 

7/3 2/ 

.203 

13/64 

.18 
3/16 l 

Diam. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

19 

128.5 

II5.2 

102.1 

89.I 

76.1 

63 7 

50-4 

48 

44.2 

40.8 

36.2 

20 

135 

I2I.I 

IO7.3 

93-6 

80 

665 53 

50-4 

46-5 

42.9 

38 

21 

i4i-5 

I27 

112.6 

98.2 

83-9 

69.7 55.6 

52.9 

48.8 

45-i 

39-9 

22 

148.1 

I32.9 

117.8 

102.8 

87.9 

73 

! 58.3 

55-4 

5 1 - 1 

47.2 

41.8 

23 

154.6 

I38.8 

123.1 

107.4 

91.8 

76.3 60.9 

57-9 

53-4 

49 3 

43-7 

24 

161.2 

I44.7 

128.3 

112 

95-7 

79-6 63.5 

60 4 

55-7 

5i-5 

45 6 

26 

174-3 

I56-5 

138.8 

121.1 

103.6 

86.1 68.7 

65-4 

60.3 

55-7 

49-3 

28 

187.4 

168.3 

149.2 

i3°-3 

111.4 

92.7 74 

70.4 

64.9 

6° 

53-i 

30 

200.4 

l8o 

159-7 

1395 

ii9-3 

99.2 j 79.2 

75-4 

69-5 

64.2 

56.8 

32 

213-5 

191.8 

170.2 

148.6 

127.1 

105.7.84-4 

80 4 

74.1 

68.5 

60.6 

34 

226.6 

203.6 

180 6 

157-8 

135 

112.3-89.7 

85-4 

78.7 

72.8 

64.4 

36 

239-7 

215-4 

191.1 

167 

I 142-9 

118.8 949 

90.4 

83-4 

77 

68.1 

No. 


8 

9 

10 

11 

12 

13 

14 

'5 

16 

17 

18 



.165 

.148 

•134 

.12 

. IO9 

.095. 

.083 

.O72 

.065 

.058 

.049 

Ins. 


11 / 64 l 

9/64/ 

9/64 l 

!/8 l 

7/64 

3/32/ 

s/64/ 

5/64 1 

V 16 / 

V16 J 

3/64/ 

Diam 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

19 


33-i 

29.7 

26.9 

24 

21.8 

19 

l6.6 

I4.4 

13 

11.6 

9.78 

20 


34-8 

3 1 -2 

28.3 

25-3 

22.9 

20 

17-5 

I 5- 1 

13-7 

12.2 

IO.3 

21 


36.6 

^2.8 

29.7 

26.5 

24.I 

21 

18.3 

15-9 

14-3 

12.8 

10.8 

22 


38-3 

34 3 

3 1 - 1 

27.8 

25.2 

22 

19.2 

16.6 

i5 

I 3 4 

n -3 

23 


40 

35-9 

32.5 

29.1 

26.4 

23 

20.1 

17.4 

15-7 

14 

11.8 

24 


41.8 

37-4 

33-9 

30-3 

27-5 

24 

20.9 

18.1 

16.4 

14 6 

12.6 

26 


45-2 

40-5 

36.7 

32.8 

29.8 

26 

22.6 

19.7 

17.7 

15-8 

i3 4 

28 


48.7 

43 6 

39 5 

35-3 

32. 1 

28 

24.4 

21.2 

19.1 

17 

14.4 

30 


52.1 

46.7 

42-3 

37-8 

34-4 

3° 

26.1 

22.7 

20.5^ 

18.3 

15-4 

32 


55-5 

49.8 

45-i 

40.4 

36-7 

32 

27.9 

24.2 

21.8 

19-5 

16.5 

34 


59 

52-9 

48 

42.9 

39 

34 

29.7 

25.8 

23 2 

20.7 

17-5 

36 


62.4 

56 

50.8 

45-4 

4 i -3 

36 

3 M 

27-3 

24.6 

21.9 

18.6 


N 








































































I46 WEIGHT OF IKON, STEEL, COPPER, ETC. 


'W'eigh.t of a Square Foot of 'W'ronglit and. Cast 
Iron, Steel, Copper, Lead, Brass, and Zinc JPlates. 
From .0625 to 1 Inch in Thickness . 


Thickness. 

Wrought 

Iron. 

j Cast Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun- 

metal. 

Zinc. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.0625 

2.517 

2.346 

2.541 

2.89 

3.691 

2.675 

2.848 

2-34 

.125 

5-035 

4-693 

5.081 

5 - 78 l 

7.382 

5-35 

5.696 

4.68 

•1875 

7-552 

7-039 

7.622 

8.672 

II.074 

8.025 

8-545 

7.02 

•25 

IO.O7 

9.386 

IO.163 

II.562 

I 4-765 

10.7 

n -393 

9-36 

•3125 

12.588 

n -733 

I2.703 

14-453 

18.456 

13-375 

14.241 

n -7 

•375 

15.106 

14.079 

15.244 

17-344 

22.148 

16.05 

17.089 

14.04 

•4375 

17.623 

16.426 


20.234 

25-839 

18.725 

19.938 

16.34 

•5 

2O.I4I 

18.773 

20.326 

23-125 

29-53 

21.4 

22.786 

18.72 

•5625 

22.659 

21.119 

22.866 

26.016 

33.222 

24.075 

25-634 

21.06 

.625 

25.176 

23.466 

25.407 

28.906 

36 - 9 I 3 

26.75 

28.483 

23-4 

.6875 

27.694 

25.812 

27.948 

31-797 

40.604 

29.425 

3 i- 33 i 

25-74 

•75 

30.211 

28.159 

30.488 

34.688 

44.296 

32.1 

34-179 

28.68 

.8125 

32.729 

30-505 

33.029 

37-578 

47.987 

34-775 

37.027 

30.42 

.875 

35-247 

32-852 

35-57 

40.469 

51.678 

36.656 

39-875 

32.76 

•9375 

37-764 

35-199 

38.11 

43-359 

55-37 

39-331 

42.723 

35 -i 

1 

40.282 

37-545 

40.651 

46.25 

59.061 

42.8 

45-572 

37-44 


From One Twentieth Inch to Two Inches in Thickness. 


Thickness. 

W rouglit 
Iron. 

j Cast Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun- 

metal. 

Zinc. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•05 

2.014 

I.877 

2.033 

2.312 

2-593 

2.14 

2.279 

I.872 

.1 

4.028 

3-754 

4.065 

4625 

5.906 

1 f 28 

4-557 

3-744 

•15 

6.042 

5-632 

6.098 

6.938 

8.859 

6.42 

6.836 

5.616 

.2 

8.056 

7-509 

8.13 

9- 2 5 

II. 8 l 2 

8.56 

9.H4 

7.488 

.25 

IO.071 

9.386 

IO.163 

11.562 

I 4-765 

IO.7 

n -393 

9-36 

•3 

12.085 

11.264 

I2.195 

13 875 

17.718 

12.84 

13.672 

11.232 

•35 

14.099 

13.141 

14.228 

16.187 

20.671 

14.98 

15-95 

I 3- I °4 

•4 

16.113 

15.018 

16.26 

18.5 

23.624 

17.12 

18.229 

14.976 

•45 

18.127 

16.895 

18.293 

20.812 

26.577 

19.26 

20.507 

16.848 

•5 

20.I4I 

18.773 

20.325 

23-125 

29-53 

21.4 

22.786 

18.72 

•55 

22.155 

20.65 

22.358 

25-437 

32.484 

23-54 

25.065 

20.592 

.6 

24.169 

22.527 

24.39 1 

27-75 

35-437 

25.68 

27-343 

22.464 

•65 

26.183 

24.409 

26.423 

30 063 

38-39 

27.82 

29.622 

24-336 

•7 

28.X97 

26.281 

28.456 

32-375 

41-343 

29.96 

31-9 

26.208 

•75 

30.211 

28.154 

30.488 

34.687 

44.296 

32.I 

34-179 

28.08 

.8 

32.226 

30.035 

32.521 

37 

47.249 

34-24 

36.458 

29-95 

•85 

34-24 

31.912 

34-553 

39-312 

50.202 

36.38 

38.736 

31.824 

•9 

36-254 

33-79 

36.586 

41.625 

53 -I 54 J 

38.52 

41.015 

33.696 

•95 

38.268 

35.668 

38.628 

43-937 

56.108 

40.66 

43-293 

35-568 

X 

40.282 

37-545 

40.651 

46.25 

59.061 

42.8 

45-572 

37-44 

1.125 

45 - 3 I 7 

42.238 

45-732 

52-031 

66.443 

48.15 

51.268 

42.12 

1.25 

50.352 

46.931 

50.814 

57 - 8 i 3 

73.826 

53-5 

56.965 

46.8 

1-3125 

52.87 

49.278 

53-354 

60.703 

77 - 5 I 7 

56.17 

59 - 8 i 3 

49.14 

1-375 

55-387 

51.624 

55-895 

63-594 

81.209 

58.85 

62.661 

51.48 

1-4375 

57-905 

53-971 

58.436 

66.484 

84.9 

61 -53 

65 - 5 i 

53-82 

i -5 

60.422 

56.317 

60.976 

69-375 

88.591 

64.2 

68.358 

56.16 

1-5625 

62.94 

58.663 

63-517 

72.266 

92.283 

66.88 

71.206 

58.5 

1.625 

65-458 

61.011 

66.058 

75 -I 56 

95-974 

69-55 

74-054 

60.84 

i -75 

70-493 

65.704 

7 I-L 39 

80.938 

103.356 

74-9 

79-751 

65-52 

1-875 

75-528 

70.397 

76.22 

86.719 

no. 739 

80.25 

85-447 

70.2 

2 

80.564 1 

75-09 

81.3 

9 2 -5 

118.122 

85.6 

91.144 

74.88 















































WEIGHT OF CAST IROUST WATER PIPES AND TUBES. I47 


Standard. Cast Iron Water 3 ?ipes. ( English .) 
For a Head of 200 Feet . 


1 

5 

Thickness. 

Depth 

of 

Socket. 

Thickness 

of 

Socket. 

Packing. 

Weight 

per Yard* 

Lead 

Joint. 

Diameter. 

Thickness. 

Depth 

of 

Socket. 

Thickness 

of 

Socket. 

Packing. 

W eight 

per Yard.* 

Lead 

Joint. 

Ins. 

Inch. 

Ins. 

Inch. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Ins. 

Inch. 

Inch. 

Lbs. 

Lbs. 

3 

•3125 

3-5 

.625 

•25 

36 

.8 

8 

•4375 

3-75 

.625 

•375 

113 

3-3 

4 

•3125 

3 

.625 

•25 

51 

1.2 

9 

•4375 

3-75 

•75 

•375 

128 

4.6 

5 

•375 

3 

.625 

•375 

6l 

2 

10 

•5 

4 

•75 

•375 

168 

4.9 

6 

•375 

3-75 

•625 

•375 

75 

2.7 

11 

•5 

4 

•75 

•375 

175 

5-3 

7 

•375 

3-75 

.625 

•375 

85 

2.9 

12 

•5625 

4 

•875 

•375 

213 

5-7 


* Measured as laid. 


To Compute AVeight of Metal IPipes. 

I) 2 — (PC. D and d representing external and internal diameters in inches , 
and C coefficient . 

Cast Iron 2.45. Wrought Iron 2.64. Brass 2.82. Copper 3.03. Lead 3.86. 


To Compute 'Weiglit of Metal Tubes and IPipes 
per Lineal ZEPoot. 

From .5 Inch to 6 Inches Internal Diameter . 


Diam. 

Area of Plate. 

Diam. 

Area of Plate. 

Diam. 

Area of Plate. 

Diam. 

Area of Plate. 

Ins. 

Sq. Foot. 

Ins. 

Sq. Foot. 

Ins. 

Sq. Feet. 

Ins. 

Sq. Feet. 

•5 

.1309 

I- 3 I 25 

•3436 

2-75 

.7199 

4-5 

I.1781 

•5625 

•1473 

i -375 

.36 

2.875 

•7526 

4.625 

1.2108 

.625 

.1636 

1-4375 

•3764 

3 

•7854 

4-75 

1-2435 

.6875 

.18 

i -5 

•3927 

3-125 

.8l8l 

4875 

1.2763 

•75 

.1964 

1.625 

•4254 

3-25 

.8508 

5 

1.309 

.8125 

.2127 

i -75 

.4581 

3-375 

.8836 

5-125 

I- 34 I 7 

•875 

.2291 

i -875 

.4909 

3-5 

• 9 i6 3 

5-25 

I -3744 

•9375 

•2454 

2 

•5236 

3-625 

•949 

5-375 

1.4072 

1 

.2618 

2.125 

•5543 

3-75 

.9818 

5-5 

1-4399 

1.0625 

.2782 

2.25 

.587 

4 

1.0472 

5-625 

1.4726 

1-125 

•2945 

2-375 

.6198 

4.125 

1.0799 

5-75 

1-5053 

1.1875 

• 3 io 5 

2-5 

•6545 

4-25 

1.1126 

5-875 

i- 538 i 

1-25 

.3272 

2.625 

.6872 

4-375 

1 -1454 

1 6 

1.5708 


Application of Table. 

When Thickness of Metal is given in Divisions of an Inch. 

To internal diameter of tube or pipe add thickness of metal; take 
area of the plate in square feet, from table for a diameter equal to 
sum of diameter and thickness of tube or pipe, and multiply it by 
weight of a square foot of metal for given thickness (see table, page 
146), and again by its length in feet. 

Illustration.—R equired weight of 10 feet of copper tube 1 inch in diameter and 
125 of an inch in thickness. 

1 -f. 125 = 1.125 X 3-1416 -r- 12 = ,2945 square feet for 1 foot of length. 

Weight of 1 square foot of copper . 125th of an inch in thickness, per table, page 
135, =: 5.781 lbs .; then, .2945 (from table above) X 5 - 7 81 X 10 = 17.025 lbs. 

When Thickness of Metal is given in Numbers of a Wire-Gauge. 

To internal diameter of tube or pipe add thickness of number from 
table, pp. 120 or 121; multiply sum bv 3.1416, divide product by 12, and 
quotient will give area of plate in square feet. Then proceed as before. 













































I48 WEIGHT OF IRON AND COPPER flPES, BOLTS, ETC. 


Illustration. —Required weight of 10 feet of copper pipe 2 inches in diameter 
and No. 2 American wire-gauge in thickness. 

2 -f- .257 63 X 3-1416-1-12== 2.257 63 X 3.1416 4 -12 =. 591 square feet ; then, . 591 
X 11.6706 (weight from table, page 118) = 6.897 lbs. * 

"Weiglit of Riveted. Iron and Copper Ripes, 
From 5 to 30 Inches in Diameter. 


ONK FOOT IN LENGTH. 


Diameter. 

Thickness. 

Iron. 

Copper. 

Diameter. 

Thickness. 

Iron. 

Copper. 

Ins. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs 

Lbs. 

5 

.125 

7.12 

8.14 

9 

•25 

25.OI 

28.58 


•1875 

10.68 

12 21 


•25 

26.33 

3°.°9 


•25 

14.25 

16.28 

10 

•25 

27-75 

3 I - 7 I 

5*5 

.125 

7.78 

8 89 

10.5 

•25 

29.19 

33-22 


•1875 

11.66 

13 33 

11 

•25 

30-49 

34-85 


•25 

I 5-56 

17.78 

12 

•25 

33 -L 3 

37.86 

6 

.125 

8.44 

9.64 

13 

•25 

35-88 

41 


■1875 

12.65 

14.46 

14 

•25 

38-52 

44 02 


•25 

16.88 

19.29 

15 

■25 

41.26 

47-15 

6-5 

.125 

9.1 

IO.4 


•3125 

5 i -57 

58.94 


■1875 

i 3 6 5 

15.6 

16 

•25 

43-9 

So 17 


•25 

18.2 

20.8 


•3125 

54-87 

62.71 

7 

.125 

9.78 

II.18 

17 

•25 

46.53 

53 -18 


•1875 

14 68 

16.78 


•3125 

58.17 

66.48 


•25 

19-57 

22.37 

18 

■25 

49.17 

56.2 

7-5 

.125 

10.49 

11.99 


•3125 

61.47 

70.25 


•1875 

15-73 

17.98 

20 

• 3 I2 5 

68.07 

77-79 


•25 

20.89 

23.87 

24 

•3125 

81 -33 

92-95 

8 

•I875 

16.7 

19 08 

25 

•3125 

84-57 

96.65 


•25 

22.26 

25-44 

28 

•3125 

94-56 

107.95 

8-5 

•25 

23-59 

26.96 

30 

•3125 

101.14 

H 5-59 


Above weights include laps of sheets for riveting and calking. 

Weights of the rivets are not added, as number per lineal foot of pipe depends 
upon the distance they are placed apart, and their diameter and length depend 
upon thickness of metal of the pipe. 


"Weight of Copper Rods or Holts, 
From .125 Inch to 4 Inches in Diameter. 


ONE FOOT IN LENGTH. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

•047 

.8125 

I.998 

i -5 

6.8ll 

2-75 

22.891 

•1875 

.106 

.875 

2.318 

•5625 

7-39 

•875 

25.019 

•25 

.189 

•9375 

2.66 

.625 

7-993 

3 

27.243 

.3125 

.296 

1 

3-03 

•75 

9.27 

.125 

29-559 

•375 

.426 

1.0625 

3-42 

•875 

10.642 

•25 

3 I - 97 2 

•4375 

•579 

.125 

3-831 

2 

12.108 

•375 

34.481 

•5 

•757 

-1875 

4.269 

.125 

13.668 

•5 

37.081 

•5625 

•958 

•25 

4-723 

•25 

I 5-325 

.625 

39-777 

‘625 

1.182 

• 3 I2 5 

5.21 

•375 

1 7-075 

•75 

42.568 

•6875 

I - 43 I 

•375 

5-723 

•5 

18.916 

.875 

45-455 

•75 

i- 7°3 1 

•4375 

6.255 

•625 

20.856 

4 

48.433 








































WEIGHT OF METALS 


149 


Weiglit of Metals of a Griven. Sectional Area. 

From .1 Square Inch to 10 Square Inches. 

PER LINEAL FOOT. ( D . K. Clark.) 


Sect. 

Wrought 

Iron. 

Cast 

Iron. 

Steel. 

Brass. 

Gun- 

metal. 

Sect, j 

Wrought 

Iron. 

Cast 

Iron. 

Steel. 

Brass. 

Gun- 

metal. 

Area. 

1. 

• 9375 - 

1.02. 

1.052. 

1.092. 

Area. 

1 

1. 

• 9375 - 

1.02. 

4 

1.052. 

1.092. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

• I 

•33 

•31 

•34 

•35 

•36 

5.1 1 

17 

15-9 

17-3 

17.9 

186 

• 2 

.67 

.62 

,.68 

•7 

•73 

5-2 

I 7-3 

16.3 

17.7 

18.2 

18.9 

•3 

I 

.94 

1.02 

1.05 

I.09 

5-3 1 

17.7 

16.6 

18 

18.6 

19-3 

•4 

i -33 

1.25 

i- 3 6 

M 3 

1.46 

54 

18 

16.9 

18.4 

18.9 

19.7 

.5 

1.67 

1.56 

i -7 

I *75 

1.82: 

5-5 

18.3 

17.2 

18.7 

19-3 

20 

.6 

2 

1.88 

2.04 

2 .II 

2.18 

5-6 

18.7 

i 7-5 

19 

19.6 

2O.4 

•7 

2-33 

2.19 

2.38 

2.46 

2.55 

5-7 

19 

17.8 

19.4 

20 

20.8 

.8 

2.67 

2.5 

2.72 

2.81 

2 . 9 1 

5-8 

19 -3 

18.1 

19.7 

20.3 

21.1 

•9 

3 

2.81 

3.06 

3-i6 

3-281 

5-9 

19.7 

18.4 

20.1 

2O.7 

21.5 

I 

3 33 

3 -i 5 

3-4 

3 - 5 i 

3 .6 4 ! 

6 

20 

18.8 

2O.4 

21 

21.8 

I.I 

3 67 1 

3 44 

3-74 

3-86 

4 

6.1 

20.3 

19.1 

20.7 

2I.4 

22.2 

1.2 

4 

3-75 

4.08 

4.21 

4 - 37-1 

6.2 

2O.7 

19.4 

21.1 

21.7 

22.6 

i -3 

4-33 ! 

4.06 

4.42 

4-56 

4 - 73 ’ 

6-3 

21 

19.7 

2I.4 

22.1 

22.9 

1.4 

4.67 

4-38 

4.76 

4 - 9 1 

5 -i 

6.4 

21.3 

20 

21.8 

22.4 

23-3 

i -5 

5 

4.69 

5 -i 

5.26 

54 6 

6.5 

2I.7 

20.3 

22.1 

22.8 

2 . 3-7 

1.6 

5-33 

5 

544 

5 - 6 i 

5.82 

6.6 

22 

20.6 

22.4 

23.1 

24 

i -7 

567 

S- 3 i 

5-78 

5 - 9 6 

6.19 

6.7 

22.3 

2O.9 

22.8 

23-5 

24.4 

1.8 

6 

5-63 

6.12 

6.31 

6-55 

6.8 

22.7 

21.3 

23.1 

23-9 

24.8 

1.9 

633 

5-94 

6,46 

6.66 

6.92 

6.9 

23 

21.6 

23-5 

24.2 

25.1 

2 

6.67 

6.25 

6.8 

7.01 

7.28 

7 

23-3 

21.9 

23.8 

24.6 

25-5 

2.1 

7 

6.56 

7.14 

7-36 

7-64 

7 -i 

23-7 

22.2 

24.I 

24.9 

25.8 

2.2 

7-33 

6.88 

7 48 

7.72 

8.01 

7.2 

24 

22.5 

24-5 

25-3 

26.2 

2.3 

7.67 

7.19 

7.82 

8.07 

8-37 

7-3 

24-3 

22.8 

24.8 

25.6 

26.6 

2.4 

8 

7-5 

8.l6 

8.42 

8.74 

74 

24.7 

23.1 

25.2 

26 

26.9 

2.5 

8-33 

7.81 

8-5 

8.77 

9.1 

7-5 

25 

! 23-4 

25-5 

26.3 

27 -3 

2.6 

8.67 

8.13 

8.84 

9.12 

9.46 

7.6 

25-3 

23.8 

25-9 

26.7 

27.7 

2.7 

9 

8.44 

9.18 

947 

9-83 

7-7 

25-7 

1 24. 1 

26.2 

27 

28 

2.8 

9 33 

8-75 

9-52 

9.82 

10.2 I 

7.8 

26 

24.4 

26.5 

274 

28.4 

2.9 

9.67 

9.06 

9.86 

10.2 

10.6 

7-9 

26.3 

24.7 

26.9 

27.7 

28.8 

3 

1 10 

9 38 

10.2 

10.5 

IO.9 

8 

26.7 

! 25 

27.2 

28.1 

29.I 

3 -i 

103 

969 

10.5 

IO.9 

n -3 

8.1 

27 

25-3 

27-5 

28.4 

29-5 

3-2 

! 10.7 

IO 

IO.9 

II .2 

11.7 

8.2 

27-3 

25.6 

27.9 

28.8 

29.9 

3-3 

11 

10.3 

11.2 

11.6 

12 

8.3 

27.7 

25.9 

28.2 

29.1 

30.2 

3-4 

11 *3 

10.6 

11.6 

11.9 

I2.4 

8.4 

28 

26.3 

28.6 

29-5 

30.6 

3-5 

n -7 

10.9 

11 .9 

12.3 

I2.7 

8.5 

28.3 

26.6 

28,9 

29.8 

30-9 

3-6 

12 

i n -3 

12.2 

12.6 

I 3 - 1 

8.6 

28.7 

26.9 

29.2 

30.2 

3 i -3 

3-7 

12.3 

11.6 

12.6 

13 

13-5 

8.7 

29 

27.2 

29.6 

3°-5 

3 i -7 

3-8 

I2.7 

11.9 

12.9 

13-3 

13.8 

8.8 

29-3 

27-5 

29.9 

30-9 

32 

3-9 

13 

12.2 

13-3 

I 3*7 

14.2 

8.9 

29.7 

27.8 

30-3 

31.2 

324 

4 

13-3 

12.5 

13.6 

14 

14.6 

9 

30 

28.1 

30.6 

31.6 

32.8 

4-i 

13-7 

12.8 

13-9 

144 

14.9 

9.1 

30-3 

28.4 

30-9 

3 J -9 

33 -i 

4.2 

14 

13-1 

14-3 

14.7 

15-3 

9.2 

30-7 

28.8 

31-3 

32 -3 

33-5 

4-3 

| J 4-3 

13-4 

14.6 

i 5 -i 

15-7 

9-3 

3 i 

29.I 

3 1 - 6 

32.6 

33-9 

4.4 

i i 4-7 

13.8 

i 5 

154 

l6 

94 

3 I, 3 

294 

32 

33 

34-2 

4-5 

15 

I4.I 

15-3 

15.8 

16.4 

9-5 

3 i -7 

29.7 

3 2 -3 

33-3 

34-6 

4.6 

15-3 

14.4 

15-6 

16.1 

16.7 

9.6 

32 

30 

32.6 

33-7 

34-9 

4-7 

15-7 

14.7 

l6 

16.5 

1/.1 

9-7 

3 2 -3 

30-3 

33 

34 

35-3 

4.8 

l6 

i5 

16.3 

16.8 

17-5 

9.8 

32-7 

30.6 

33-3 

34-4 

35-7 

4.9 

16.3 

I 5-3 

16.7 

17.2 

17.8 

9.9 

33 

30-9 

33-7 

34-7 

36 

5 

16.7 

15-6 

17 

" 

17-5 

18.2 

IO 

33-3 

3 I -3 

34 

35- 1 

36-4 


N* 




























































150 LEAD PIPES.-COPPER PIPES AND COCKS. 

■Weigh, t of Lead. IPipe. (English.) 


ONE FOOT IN LENGTH. 


Diam. 

Thick¬ 

ness. 

Weight. 

Diam. 

Thick¬ 

ness. 

Weight. 

Diam. 

Thick¬ 

ness 

Weight. 

Diam. 

Thick¬ 

ness. 

Weight. 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

•5 

.097 

•93 

I 

.136 

2.4 

i -75 

.166 

5 

3 

•275 

14 


.112 

1 .07 


.156 

2.8 


.199 

6 

3-5 

.225 

13 


.124 

1.2 


.2 

3-73 


.228 

7 


•273 

l6 


.146 

1.47 


.225 

4.27 


.256 

8 

4 

•257 

17 

.625 

.089 

I 

1.25 

•139 

3 

2 

.178 

6 

b 


• 3 I2 5 

20.5 


.IOI 

1-13 


.16 

3-5 


.204 

7 


•327 

22 


.121 

1.4 


.18 

4 


.231 

8 

4-25 

•3125 

22.04 


.14 

2 


•193 

4-33 


.266 

9-33 

4-5 

.232 

17 

•75 

.112 

1.6 

i -5 

.156 

4 

2-5 

.2 

8.4 


•295 

22 


.147 

1.87 


.179 

4.67 


.227 

9.6 


•3125 

23-25 


.181 

2.13 


.224 

6 


.261 

II .2 

4-75 

• 3 I2 5 

24-45 


.215 

2.4 


•257 

7 

3 

.218 

II .2 

5 

• 3!25 

25.66 


Dimensions of Copper IPipes and Composition 

Cocks. 

From i Inch to 23 Inches in Diameter . 


0 'o 
• © 0 

B 

Flange Diameter. 

Thick- 

Bolts. 

0 0 

.(DO 

Flange 

Diam. 

Thick- 

Bolts. 

.SS^3 



ness. 



.2 Ch T3 


ness. 



O § 

Pipe. 

Cock. 


No. 

Diam 

0 s 

Pipe. 


No. 

Diam. 

Ins. 

Ins. 

Ins. 

Inch. 


Inch. 

Ins. 

Ins. 

Inch. 


Inch. 

I 

3-375 

3-5 

•375 

3 

•5 

9 

12.75 

•625 

9 

.625 

1-25 

3-625 

3-75 

•375 

3 

•5 

9-25 

I 3- I2 5 

•625 

IO 

.625 

i -5 

3-875 

4-25 

•375 

3 

•5 

9-5 

13-375 

.6875 

IO 

.625 

i -75 

4.125 

4-375 

•4375 

4 

•5 

9-75 

13-625 

.6875 

IO 

.625 

2 

4-375 

4-75 

•4375 

4 

•5 

IO 

i 3-875 

.6875 

IO 

.625 

2.25 

4625 

5-25 

•4375 

5 

•5 

10.5 

14-5 

.6875 

IO 

.625 

2-5 

4-875 

5-5 

•4375 

5 

•5 

II 

i 5 

.6875 

IO 

.625 

2.75 

5-25 

5-75 

•4375 

5 

•5 

n -5 

15-625 

•75 

IO 

•75 

3 

6 

6.25 

•5 

5 

•625 

12 

16.125 

•75 

IO 

•75 

3-25 

6.125 

6.625 

•5 

6 

•625 

12.5 

16.625 

•75 

IO 

•75 

3-5 

6-375 

6.875 

•5 

6 

•625 

13 

17-25 

•75 

IO 

.75 

3-75 

6.625 

7-25 

•5 

6 

•625 

13-5 

17-875 

•75 

IO 

•75 

4 

6.875 

7-375 

•5 

6 

•625 

14 

i8.375 

•75 

IO 

•75 

4-25 

7-125 

7.625 

•5 

6 

•625 

14-5 

18.875 

•75 

IO 

•75 

4-5 

7-375 

8.25 

•5 

6 

•625 

15 

19 5 

•75 

IO 

•75 

4-75 

7.625 

8-5 

•5 

6 

•625 

15-5 

20 

•75 

IO 

•75 

5 

8 

9 

•5 

6 

•625 

l6 

20.5 

•75 

IO 

•75 

5-25 

8.25 

9-25 

•5 

6 

•625 

16.5 

21.125 

•75 

IO 

•75 

5-5 

8-5 

9-5 

•5 

6 

•625 

17 

21.625 

•75 

II 

•75 

5-75 

9 

9875 

•5 

6 

•625 

i 7-5 

22.125 

•75 

II 

•75 

6 

9-25 


•625 

8 

•625 

18 

22.75 

•75 

II 

•75 

6 25 

9-75 


•625 

8 

•625 

18.5 

23-25 

•75 

II 

•75 

6-5 

IO 


.625 

8 

.625 

19 

23-75 

•75 

12 

•75 

6 -75 

IO 


.625 

8 

•625 

19-5 

24-375 

•75 

12 

•75 

7 

10.5 


•625 

8 

.625 

20 

24.875 

•75 

12 

•75 

7-25 

10.75 


•625 

8 

•625 

20.5 

25-375 

•75 

!3 

•75 

7-5 

11.125 


•625 

8 

•625 

21 

26 

•75 

13 

•75 

7-75 

n -375 


•625 

8 

•625 

21.5 

26.5 

•75 

13 

•75 

8 

11.625 


•625 

9 

•625 

22 

27 

•75 

13 

•75 

8.25 

12 


•625 

9 

•625 

22.5 

27.625 

•75 

14 

•75 

8-5 

12.25 


•625 

9 

•625 

23 

28.125 

•75 

14 

•75 

8-75 

12.5 


.625 

9 

•625 







































































WEIGHT OF SHEET LEAD, LEAD AND TIN PIPES, ETC. I 5 I 


'W'eiglrt of SHeet Lead., 

PEK SQUARE FOOT. 


Thickness. 

Weight. 

Thickness. 

Weight. 1 

Thickness. 

Weight. 

Thickness. 

Weight 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

.OI7 

I 

.068 

4 

• 

H 

W 

CO 

7 

.169 

IO 

•034 

2 

.085 

5 

•135 

8 

.186 

II 

.051 

3 

.IOI 

6 

.152 

9 

.203 

12 


W'eiglit of Tin. 3?ipe. 





ONE 

FOOT IN LENGTH. 




Diam. 

THICKNESS. 

Diam. 

THICKNESS. 

Diam. 

THICKN. 

Diam. 

THICKN. 

External. 

% inch. 

% inch. 

External. 

?S 2 inch - 

% inch. 

External. 

% inch. 

External. 

% inch. 

Inch. 

Lb. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

•25 

.148 

— 

1.25 

1.095 

1-417 

2.25 

5-04 

3-25 

7-56 

•5 

•384 

.472 

i -5 

1.328 

1-732 

2.5 

5-67 

3-5 

8.19 

•75 

.62 

.787 

i -75 

1.564 

2.047 

2.75 

6-3 

3-75 

8 82 

I 

.856 

1-103 

2 

1.802 

2.362 

3 

6-93 

4 

9-45 


"Weiglit of Lead Encased Tin Pipes. 


Diameter. I Light Weights. 


Ins. 

Lbs. 

Lbs. 

Lbs. 

•375 

I 

i-5 

2 

•5 

2 

2-5 

3 

.625 

3 

3-5 

4 

•75 

3-5 

4 

4-5 

I 

4-5 

5 

5-5 

1.25 

6-5 

7 

8 

i-5 

8 

9 

IO 

2 

II 

13 

— 




For Supply of Water Head 



feet. 

50 feet and 

under. 

51 to 

250 

feet. 

251 to 500 

L 

bs. 


Lbs. 


Lbs. 


2-5 

to 

4 

3 

to 

4-5 

3-5 

to 

s 

3-5 

u 

5 

4 

u 

6 

4-5 

u 

7 

4-5 

a 

7 

5-25 

a 

8 

6 

a 

9 

5-5 

u 

8 

6 

u 

9 

7 

u 

IO 

7- 2 5 

a 

IO 

8 

u 

II 

9 

u 

12 

9 

u 

12.5 

IO 

u 

14 

12 

a 

l6 

II 

a 

16 

12.5 

u 

18 

14 

c. 

21 

l6 

u 

23 

18.5 

u 

26 

21 

u 

30 


* The extreme weights are for extra heavy pipe with less proportion of tin. 


Diinensions and "Woiglit of Slieot Zinc* (Vielle-Montague.) 

PER SQUARE FOOT. 


No. 

Thickness. 

2X.5 metres; 
area, i square metre. 

6.56X1.64 feet; area, 
10.76 square feet. 


Millim. 

Inch. 

Kilom. 

Lbs. 

9 

.41 

.0161 

2.9 

6.39 

IO 

•51 

.0201 

3-45 

7.61 

II 

.6 

.0236 

4 -oS 

8.93 

12 

.69 

.0272 

4-65 

10.25 

13 

.78 

.0307 

5-3 

11.68 

14 

.87 

.0343 

5-95 

13.12 

15 

.96 

.0378 

6-55 

14.44 

l6 

I.I 

.0433 

7-5 

16.53 

J 7 

1.23 

.0485 

8-45 

18.63 

18 

1.36 

■0536 

9-35 

20.61 

19 

I.48 

•0583 

10.3 

22.71 

20 

1.66 

•0654 

11.25 

24.8 

21 

1.85 

.0729 

12.5 

27.56 

22 

2.02 

.0795 

13-75 

30-31 

23 

2.19 

.0862 

15 

3307 

24 

2-37 

•0933 

16.25 

35-82 

25 

2.52 

.0992 

17-5 

38-58 

26 

2.66 

.!047 

18.8 

41.44 


2X.65 metres; 
area, 1.3 sq. metres. 

6.56X2.13 feet; area, 
13 99 square feet. 

2X.8 metres; 
area, 1.6 sq. metres. 

6 56X2.62 ft.; area, 
17.22 square feet. 

Weight. 

Kilom. 

Lb 9 . 

Kilom. 

Lbs. 

Lbs. 

3-7 

8.16 

4.6 

IO.I4 

•589 

4-45 

9.81 

5-5 

12.12 

•704 

5-3 

11.68 

6-5 

14-33 

.832 

6.1 

13-45 

7-5 

16.53 

.96 

6.9 

15.21 

8-5 

18.74 

1.088 

7-7 

16.94 

9-5 

20-94 

1.216 

8-55 

18.85 

10.5 

23-15 

1-344 

9-75 

21.5 

12 

26.46 

i -536 

10.95 

24.14 

13-5 

29.97 

1.74 

12.2 

26.9 

15 

33-07 

1.92 

13-4 

29-54 

16.5 

36-38 

2.112 

14.6 

32.19 

18 

39.68 

2.304 

16.25 

35-82 

20 

44.09 

2.56 

17.9 

39-46 

22 

48.5 

2.816 

19-5 

42.99 

24 

52.91 

3-073 

21.1 

46.52 

26 

57-32 

3-329 

22.75 

50.15 

28 

61 -73 

3-585 

24.4 

53-79 

3 1 

68.34 

3-969 









































































152 WEIGHT OF SHEET ZINC.-SPIKES, HORSESHOES. 


f , Table — (Continued). 

Special Sizes for Sheathing Ships. 






Dimensions of Sheets. 





1.15 X .35 metres ; 

1.3 X .4 metres ; 

Weight 

per 

No. 

Thickness. 

area, .402 sq. metre. 

area, .52 sq. metre. 




3.77 X 1.IS feet; area. 

4.26 X 1.31 feet; area, 

Sq. Foot. 




4.33 sq. feet. 

5.6 sq. feet. 



Millim. 

Inch. 

Kilom. 

Lbs. 

Kilom. 

Lbs 

Lbs. 

15 

.96 

.0378 

2.65 

5-84 

-3-4 

7*5 

1-344 

l6 

1.1 

•0433 

3 

6.61 

3-9 

8.6 

1-536 

17 

I.23 

.0485 

3-4 

7-5 

4.4 

9-7 

1.74 

18 

1.36 

•0536 

3-75 

8.27 

4.9 

10.8 

X.92 

19 

1.48 

•0583 

4-i5 

9- J 5 

5-35 

11.79 

2.112 

20 

1.66 

.0654 

4-55 

10.03 

5-85 

12.9 

2.304 


Note.—A deviation of 25 dekagrammes, or about half a pound, more or less, from 
the proper weight of each number of sheet, is allowed. 

Nos. 1 to 9 are employed for perforated articles, as sieves, and for articles de 
Paris. Nos. 10 to 12 are used in manufacture of lamps, lanterns, and tin-ware gen¬ 
erally, and for stamped ornaments. The last numbers are used for lining reservoirs, 
and for baths and pumps. 

Skip and. Railroad Spikes. 

DIMENSIONS AND NUMBER PER POUND. (P. C. Page , Mass.) 

Siiip Spikes. 


y In. Sq. | 

X In. Sq. 

Xe k. Sq. 

y In. Sq. 

Xe In. Sq. 

% In. Sq. 

% In.Sq. 

rd 



B'O 

rB 

B £ 

■5 

B £ 

rB 

£ £ 

■a 

E £ 

£ 

B 

c 

<£> 

O 0 

a 

d 

bJ 

og 

£0h 

B 

d 

hJ 

4 ° 

B 

D 

l-H 

P 1 


o' 0 
ZPh 

B 

a> 

b-3 

°1 

ZP* 

tx 

B 

a» 

b-3 

0 * 0 
iZPu 

Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


3 

19 

3 

10 

4 

5-4 

5 

3-4 

6 

2.2 

8 

1.4 

IO 

.8 

3-5 

15.8 

3-5 

9.6 

4-5 

5 

5-5 

3 -i 

6-5 

2 

9 

1.2 

15 

.6 

4 

13.2 

4 

8 

5 

4.6 

6 

3 

7 

1.9 

10 

1.1 


— 

4-5 

12.2 

4-5 

6 

5-5 

4-2 

6-5 

2.8 

7-5 

1.8 

11 

I 

— 

— 

5 

10.2 

5 

5-8 

6 

4 

7 

2.6 

8 

i-7 

— 

— 

— 

— 

— 

— 

6 

5-2 

6-5 

3-2 

7-5 

2.4 

8.5 

1.6 

— 

— 

— 

— 

“ 

— 

—“ 

— 

— 

— 

8 

2.2 

9 

i-5 

— 

— 

— 

— 


— 

— 

— 

— 

— 

— 

— 

10 

1.4 

— 

— 

— 

— 


Railroad Spikes.5 inch square x 5.5 ins. 2 per lb. 

“ “ .5625 “ “ X 5-5 “ 1.6 “ 


Spikes and Horseshoes. 


LENGTH AND number per pound. {II. Burden, Troy , N. Y.) 


* b £ 

B 

D 

Boat £ 

O —3 

Z 

Spikes. 

fcXi 

B 

D 

4= 

C b-4 
£ 

rB 

bfi 

B 

<D 

Ship 

a . 

O 

£ 

Spikes. 

^3 

be 

B 

CD 

B . 

c *-3 
£ 

Hook Hea 

Length. 

d. 

d . 

— ^0 

© 

5s 

Horse 

rB 

b£ 

B 

iD 

bj 

shoes. 

— A 

cJ 

Z 

Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


3 

17-5 

6-5 

4.78 

4 

8 

7-5 

2.5 

4 X.375 

5-55 

I 

.84 

3-5 

14.68 

7 

3.62 

4-5 

6-5 

8 

1.74 

4-5 X.4375 

4.14 

2 

•75 

4 

12.57 

7-5 

3-37 

5 

4-37 

8-5 

1.63 

5 X .5 

2.52 

3 

•65 

4-5 

9.2 

8 

2-95 

5-5 

4-3 

9 

I -55 

5-5 x .5 

2.41 

4 

•56 

5 

7.2 

8-5 

2.9 

6 

4.2 

10 

I - I 5 

5.5 X.5625 

1.87 

5 

•39 

5-5 

6-3 

9 

2.1 

6-5 

3-77 

— 

— 

6 X .5625 

1.72 



6 

4-97 

10 

1.98 

7 

2-75 

— 

— 

| 6 x .625 

1.38 

— 

— 
























































































CAST IRON AND LEAD BALLS.-NAILS. I 53 


Weigh, t and Volmrie of Cast Iron and. Lead Balls. 

From 1 Inch to 20 Inches in Diameter. 


Diameter. 

Volume. | 

Cast Iron. 

Lead. 

Diameter. 

Volume. 

Cast Iron. 

Lead. 

Ins. 

Cube Ins. 

Lbs. 

Lbs. 

Ins. 

Cube Ins. 

Lbs. 

Lbs. 

I 

•523 

.136 

.215 

9 

381.703 

99-51 

* 56-553 

i -5 

I.767 

.461 

•725 

9-5 

448.92 

117.034 

X84.121 

2 

4.189 

I.O92 

I.718 

10 

523-599 

136.502 

2x4.749 

2-5 

8.l8l 

2.133 

3-355 

10.5 

606.132 

158.043 

248.587 

3 

14-137 

3- 6 85 

5-798 

if 

696.91 

181.765 

285.832 

3-5 

22.449 

5-852 

9.207 

n -5 

796-33 

207.635 

326 . 59 X 

4 

33-51 

8.736 

13-744 

12 

904.778 

235.876 

371.096 

4-5 

47 - 7*3 

12.439 

19.569 

12.5 

IO22.656 

266.647 

4 I 9 - 5 I 2 

5 

65 45 

17.063 

26.843 

*3 

II50.346 

299.623 

471.806 

5-5 

87.114 

22.721 

35-729 

14 

1436.754 

374-563 

589273 

6 

113.097 

29.484 

46-385 

15 

1767.145 

460.696 

724.781 

6-5 

143-793 

37-453 

58.976 

16 

2144.66 

559 - 1 *4 

879.616 

7 

179-594 

46.82 

73-659 

17 

2572.44 

670.717 

1055.066 

7-5 

220.893 

57-587 

90.598 

18 

3053.627 

796.082 

1252.422 

8 

268.082 

69.889 

109.952 

19 

359 1 -363 

936.271 

1472.97 

8-5 

321-555 

83.84 

131.883 

| 20 

4188.79 

1 1092.02 

I 7 I 7-995 


Note.— To compute weight of balls of other metals, multiply weight given in 
table by following multipliers: 

For Wrought Iron.1.067. Brass.1.12. 

Steel.x.088. Gun-metal.1.165. 


"Weiglit and Diameter of Cast Iron Balls. 


Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

I 

I.94 

12 

4-45 

50 

7.16 

224 

11.8 

1344 

21.44 

2 

2-45 

14 

4.68 

56 

7-43 

336 

13-51 

1568 

22.57 

3 

2.8 

l6 

4.89 

60 

7.6 

448 

14.87 

1792 

23.6 

4 

3.08 

l8 

5-09 

70 

8.01 

560 

16.02 

2016 

24-54 

5 

3-32 

20 

5-27 

80 

8-37 

672 

17.02 

2240 

25.42 

6 

3-53 

25 

5.68 

90 

8.71 

784 

17.91 

2800 

27.38 

7 

3-72 

28 

5-9 

IOO 

9.02 

896 

18.73 

3360 

29.1 

8 

3-89 

30 

6.04 

112 

9-37 

IO08 

19.48 

392° 

30.64 

9 

4.04 

40 

6.64 

l68 

10.72 

1120 

20.17 

4480 

32-03 


No. 5 
“ 6 


Length, of Horseshoe 
By Numbers. 


1.5 Ins. No. 7 
1.75 “ “ 8 


1.875 I ns * 

o 


USTails. 

No. 9. . 

^ TA 


2.25 Ins, 
2.5 “ 


Lengths of Iron INTails, and Number in a L“b. 


Size. 

L’gth. 

No. 

Size 

L’gth. 

No. 

Size. 

L’gth. 

No. 

Size. 

L’gth. 

No. 

Size. 

L’gth. 

3d. 

Ins. 

I.25 

42O 

Sd. 

Ins. 

i -75 

220 

8 d. 

Ins. 

2-5 

IOO 

12 d. 

Ins. 

3-25 

52 

3 ° d. 

Ins. 

4 

4 

^•5 

270 

6 

2 

175 

10 

3 

65 

20 

3-5 

28 

40 

4-25 























































































154 NAILS, SPIKES, TACKS, ETC. 


'Wrought Iron. Cut IN'ails. Tacks, Spikes, etc. 


('Cumberland Nail and Iron Co .) 
Lengths and Number per Lb . 
Finishing. 


Ordinary 


Size. 

Length. 

No. per Lb. 

2 d 

Ins. 

•875 

716 

3 fine 

1.0625 

588 

3 

1.0625 

448 

4 

i -375 

336 

5 

i -75 

2l6 

6 

0 

166 

7 

2.25 

Il8 

8 

2-5 

94 

10 

2-75 

72 

12 

3-5 

50 

20 

3-75 

32 

30 

4-25 

20 

40 

4-75 

17 

So 

5 

14 

60 

5-5 

10 


Light. 

4 d 

1-375 

373 

5 

i -75 

272 

6 

2 

196 


Brads. 

6 d 

2 

163 

8 

2-5 

96 

10 

2-75 

74 

12 

3-125 

50 


3 Ten.ce. 

6 d 

2 

96 

7 

2.25 

66 

8 

2-5 

56 

10 

2-75 

50 

— 

3 

40 


Size. 

Length. 

No. per Lb. 


Ins. 


4 d 

i -375 

384 

5 

i -75 

256 

6 

2 

204 

8 

2-5 

102 

10 

3 

80 

12 

3-625 

65 

20 

3-875 

46 


Core. 

6 d 

2 

143 

8 

2-5 

68 

10 

2-333 

60 

12 

3-125 

42 

20 

3-75 

25 

30 

4-25 

18 

40 

4-75 

14 

WH 

2-5 

69 

WHL 

22J 

72 


Clinch. 

6 d 

2 

152 

7 

2.25 

i 33 

8 

2-5 

92 

10 

2-75 

72 

— 

3 

60 

— 

3-25 

43 


Slate. 

3 d 

1.625 

288 

4 

1-4375 

244 

5 

i -75 

187 

6 

2 

146 


Sli ingle. 


Size. 

Length. 

No. per Lb 


Ins. 


5 d 

i -75 

1:78 

8 

2-5 

74 

9 

2-75 

60 

10 

3 

Tacks 

52 

1 oz. 

.125 

16 000 

i -5 

•1875 

10 666 

2 

•25 

8 000 

2-5 

•3125 

6 400 

3 

•375 

5 333 

4 

•4375 

4 000 

6 

•5625 

2 666 

8 

.625 

2 000 

10 

.6875 

1 600 

12 

•75 

1333 

14 

.8125 

1 i 43 

16 

•875 

1 000 

18 

•9375 

8S8 

20 

1 

Boat. 

800 


Size. 

No. per Lb. 

Ins. 


i -5 

206 


Spikes. 

19 
15 
13 

IO 

9 
7 


3 - 5 

4 

4 - 5 

5 

5 - 5 

6 


Railroad. Spikes. 
Number in a Keg of 150 lbs . 


Length. 

No. 

Length. 

No. 

Length. 

No. 

Length. 

No. 

No. 

3 X -375 
3-5 X .375 

4 X .375 

930 

890 

760 

Ins. 

3 - 5 X .4375 

4 X .4375 

4 - 5 X .4375 

675 

540 

510 

Ins. 

4 X .5 
4-5 X .5 

5 X .5. 

450 

400 

340 

Ins. 

5 X .5625 
5.5 X .5625 

300 

280 


5.5 X .5625 standard for a gauge of 4 feet 8.5 ins. 


Skip and Boat Spikes. 
Number in a Keg of 150 lbs . 


Length. 

No. 

Length. 

No. 

Length. 

No. 

Length. 

No. 

Ins. 


Ins. 


Ins. 


Ins. 


4 X.25 

1650 

5 X.3125 

930 

8X.375 

455 

10 x .4375 

270 

4-5 X.25 

1464 

6X.3125 

868 

9 X -375 

424 

8 x ^5 

256 

5 X.25 

1380 

7X.3125 

662 

ioX -375 

390 

9 X -5 

240 

6 x.25 

1292 

6 X -375 

570 

8 X. 4375 

384 

IOX -5 

222 

7 X.25 

Il6l 

7 X .375 

482 

9 X. 4375 

300 

IIX -5 

203 




































































VARIOUS METALS 


155 


Weigilit of "Varioias Metals. 

Per Cube Inch and Foot. 



Spec. 

W’ght 

Ins. 

Weight 


Specific 

W’ght 

Ins. 

Weight 

Metals. 

Gravi- 

in an 

in a 

in a 

Metals. 

Gravi- 

in an 

in a 

in a 

ty. 

Inch. 

Lb. 

Foot. 

ty- 

Inch. 

Lb. 

Foot. 

Wrought-iron 


Lb. 


Lbs. 



Lb. 


Lb. 

plates 

7734 

.2797 

3-57 

483.38 

Brass, rolled. 

8 217 

.2972 

3-37 

5 I 3.6 

“ wire. 

7774 

.2812 

3-55 

485.87 

“ cast... 

8 080 

.2922 

3 - 4 2 

505 

Cast iron.... 

7209 

.2607 

3-*4 

450-54 

Lead, rolled. 

11 340 

.4101 

2.44 

708.73 

Steel plates.. 

7804 

.2823 

3-54 

487.8 

Tin, cast. 

7 292 

.2673 

3-74 

462 

“ wire... 

7847 

.2838 

3 - 5 2 

490.45 

Zinc, rolled.. 

7188 

.26 

3-85 

449.28 

Copper, ( ... 
rolled (... 

8697 

8880 

.3146 
• 3 212 

• 3 i6 5 

3 -i 9 

3 -ii 

543-6 

555 

546.875 

Alumini- ) 
um, cast j 

2 560 

.0926 

10.8 

160 

Gun-metal,) 

8750 

3.16 

Silver. 

0 

00 

0 

H 

• 379 1 

2.64 

655 

cast.j 



English. (D. K. Clark.) 


Wrought iron 

7.698 

.278 

3-6 

480 

Tin. 

7.409 

.268 

3-74 

462 

Cast iron.... 

7- 2I 7 

.26 

3.84 

450 

Zinc. 

7.008 

•253 

3-95 

437 

Steel. 

7.852 

.283 

3-53 

489.6 

Lead. 

11.418 

.412 

2-43 

712 

Copper plates 

8.805 

.318 

3 -i 5 

549 

Brass, cast... 

8.099 

.292 

3-42 

505 

Gun-metal... 

8.404 

•304 

2.02 

5 2 4 

“ wire.. 

8.548 

■308 

3- 2 4 

533 


WROUGHT AND CAST IRON. 

To Compute Weight of' Wrought or Cast Iron. 

Rule. —Ascertain number of cube inches in piece; multiply sum by .2816* for 
wrought iron and .2607* for cast, and product will give weight in pounds. 

Or, for cast iron multiply weight of pattern, if of pine, by from 18 to 20, accord¬ 
ing to its degree of dryness. 

Example. —What is weight of a cube of wrought iron 10 inches square by 15 
inches in length? 

10 X 10 X 15 X .2816 = 422.4 lbs. 

COPPER. 

To Compute "Weight of* Copper. 

Rule. —Ascertain number of cube inches in piece; multiply sum by .32118,* 
and product will give weight in pounds. 

Sheathing and. Braziers’ Sheets. 

For dimensions and weights see Measures and Weights, pages 118-121, 131, 142. 


LEAD. 

To Compute Weight of Bead. 

Rule. —Ascertain number of cube inches in piece; multiply sum by .41015,* 
and product will give weight in pounds. 

Example. —What is weight of a leaden pipe 12 feet long, 3.75 inches in diameter, 
and 1 inch thick? 

By Rule in Mensuration of Surfaces, to ascertain Area of Cylindrical Rings. 
Area of (3.75 + 1 -f-1) = 25.967 
“ “ 3-75 — 11-044 

Difference, 14.923 (area of ring) X 144 (12 feet) = 2148.912 

X .41015 — 881.376 lbs. 

BRASS. 

To Compute "Weight of Ordinary Brass Castings. 
Rule.—A scertain number of cube inches in piece; multiply sum by .2922,* and 
product will give w r eight in pounds. 

* Weights of a cube inch as here given are for the ordinary metals ; when, however, the specific 
gravity of the metal under consideration is accurately known, the weight of a cube inch of it should 
be substituted for the units here given. 













































I 56 DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. 

TDiiTiensions and. NWeiglits of Wrought Iron Bolts 

and IN’m.ts. 

SQUARE AND HEXAGONAL HEADS AND NUTS. 

Ttougli, and from .25 Inch io 4 Inches in Diameter. 
Square Head and INnt. 


Diameter 
of Bolt. 

Wid 

Head. 

th. 

Nut. 

Diagonal. 
Head. 1 Nut. 

Depth. 

Head, j Nut. 

Weight. 

Head i Bolt 
and Nut. !per Inch. 

Threads 
per Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

No. 

•25 

•36 

•49 

•51 

.69 

•25 

•25 

.024 

.014 

20 

; 3 I2 5 

•45 

.58 

.64 

.82 

•3 

•3125 

•043 

.022 

18 

•375 

•54 

.67 

.76 

•95 

•34 

•375 

,c68 

.031 

l6 

•4375 

-63 

.76 

.89 

1.07 

•4 

•4375 

.104 

.O42 

14 

•5 

.72 

.84 

1.02 

1.19 

•44 

•5 

•145 

•055 

13 

.5625 

.82 

•94 

1.16 

i -33 

.48 

•5625 

.204 

•07 

12 „ 

.625 

.91 

1.03 

I.29 

1.46 

•53 

.625 

•273 

.086 

11 

.6875 

1 

1.12 

I.4I 

1.58 

•58 

.6875 

•356 

.IO4 

II 

•75 

1.09 

1.21 

i -54 

1.7 1 

•63 

•75 

•454 

.124 

IO 

.8125 

1.18 

I -3 

1.67 

1.84 

.67 

.8125 

•565 

•145 

IO 

•875 

1.27 

i -39 

1.8 

1.96 

•72 

•875 

.696 

.l6S 

9 

1 

i -45 

i -57 

2.05 

2.22 

.81 

1 

1-013 

.22 

8 

1-125 

1.63 

i -75 

2-3 

2.47 

•9 

1-125 

1.416 

.278 

7 

1.25 

1.81 

1.94 

2.56 

2-74 

1 

1.25 

1.923 

•344 

7 

1-375 

1.99 

2.12 

2.81 

3 

1.1 

I -375 

2-543 

.416 

6 

i -5 

2.17 

23 

3-07 

3-25 

1.18 

i -5 

3-234 

•495 

6 

1.625 

2.36 

2.48 

3-34 

3 - 5 i 

1.28 

1.625 

4.105 

.581 

5-5 

i -75 

2-54 

2.66 

3-59 

3 - 7 6 

i -37 

i -75 

5.087 

.674 

5 

1-875 

2.72 

2.84 

3-85 

4.02 

1.46 

i -875 

6.182 

•773 

5 

2 

2.9 

3.02 

4.1 

4.27 

1.56 

2 

7 - 49 1 

.88 

4-5 

2.125 

3.08 

3.21 

4-35 

4-54 

1.65 

2.125 

8.936 

•993 

4-5 

2.25 

3.26 

3-39 

4.61 

4-79 

i -75 

2.25 

10.543 

!•! 13 

4-5 

2-375 

3-44 

3-57 

4.86 

5-05 

1.84 

2-375 

12.335 

1.24 

4-375 

2.5 

3.62 

3-75 

5.12 

5-3 

1.94 

2-5 

14-359 

1-375 

4.25 

2.625 

3.81 

3-93 

5 49 

5-56 

2.03 

2.625 

16.549 

I- 5 I 5 

4 

2-75 

3-99 

4-ix 

5-64 

5 -Si 

2.12 

2-75 

18.897 

1.663 

4 

2.875 

4.17 

4.29 

5-9 

6.07 

2.22 

2-875 

21 545 

1.818 

3-75 

3 

4-35 

4 47 

6.15 

6.32 

2.31 

3 

24.464 

1.979 

3-5 

3-25 

4.71 

4.84 

6.66 

6.84 

2-5 

3 25 

30.922 

2.323 

3-5 

3-5 

5-07 

5-2 

7.17 

7-35 

2.68 

3 5 

38 - 39 1 

2.694 

3-25 

3-75 

5-44 

5-56 

7.69 

7.86 

2.87 

3-75 

47.168 

3 093 

3 

4 

5-8 

5 - 9 2 

8.2 

8-37 

3.06 

4 

56.882 

3 - 5 i 8 

3 


Finished. —Deduct .0625 from diameters of bolts and depths of all heads 
and nuts. 


Screws with square threads have but one half number of threads of those 
with triangular threads. 

Note.— The loss of tensile strength of a bolt by cutting of thread is, for one of 1.25 
ins. diameter, 8 per cent. The safe stress or capacity of a wrought iron bolt and nut 
may be taken at 5000 lbs. per square inch. 

Preceding width, depth, etc., arc for work to exact dimensions, whether 
forged or finished. 

To Compute "Weiglit of* a, Bolt and 1 N T ut. 

Operation .—Ascertain from table weight of head and nut for given di¬ 
ameter of bolt, and add thereto weight of bolt per inch of its length, multi¬ 
plied by full length of its body from inside of its head to end. 

Note.— Length of a bolt and nut for measurement , as such, is taken from inside 
of head to inside of nut, or its greatest capacity when in position. 
























DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS, 


157 


Illustration.— A wrought iron bolt and nut with a square head and nut is 1 inch 
in diameter and 10 inches in length; what is its weight? 

Weight of head and nut.1.013 lbs. 

“ holt per inch of length .22 X 10 = 2.2 “ 

3-213 “ 


Hexagonal Head and. Nut. 


Diameter 
of Bolt. 

Wit 

Head. 

th. 

Nut. 

Diag 

Head. 

>nal. 

Nut. 

D 

Head. 

ipth. 

Nut. 

Weig 
Head 
and Nut. 

ht. 

Bolt 

per Inch. 

Threads 
per Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

1 bs. 

Lbs. 

No. 

•25 

•375 

•5 

•43 

.58 

•25 

•25 

.022 

.014 

20 

•3125 

•4375 

•5625 

-5 

•65 

•3 

•3125 

•037 

.022 

18 

•375 

•5625 

.6875 

•65 

•79 

•34 

•375 

.062 

.031 

l6 

•4375 

.625 

•75 

•72 

,87 

•4 

•4375 

.094 

.O42 

H 

•5 

•75 

.875 

.87 

1 

•44 

•5 

.134 

•055 

13 

•5625 

.8125 

•9375 

•94 

1.08 

.48 

.5625 

.18 

.07 

12 

.625 

•9375 

1.0625 

1.08 

1.23 

•53 

.625 

•249 

.086 

II 

.6875 

1 

1-125 

1.16 

i -3 

.58 

•6875 

.318 

.104 

II 

•75 

1-125 

1.25 

i -3 

1.44 

•63 

•75 

•413 

.124 

IO 

.8125 

1.25 

i -375 

1.44 

i -59 

•67 

.8125 

.522 

• I 45 

IO 

•875 

1-3125 

1-4375 

1.52 

1.66 

.72 

•875 

•639 

.168 

9 

1 

i -5 

1.625 

i -73 

1.88 

.81 

1 

•931 

.22 

8 

1-125 

'1.6875 

1.8125 

i -95 

2.09 

•9 

1.125 

I.299 

.278 

7 

1.25 

i -875 

2 

2.17 

2.31 

1 

1.25 

i -759 

•344 

7 

1-375 

2 

2.1875 

2.31 

2-53 

1.1 

i -375 

2.263 

.416 

6 

i -5 

2.25 

2-375 

2.6 

2-74 

1.18 

i -5 

2.958 

•495 

6 

1.625 

2-4375 

2.5625 

2.81 

2.96 

1.28 

1.625 

3-741 

.581 

5-5 

i -75 

2.625 

2-75 

3-°3 

3 -i8 

i -37 

i -75 

4-654 

•674 

5 

i -875 

2.8125 

2 9375 

3-25 

3-39 

1.46 

i -875 

5-675 

•773 

5 

2 

3 

3-125 

3-46 

3.61 

1.56 

2 

6.854 

.88 

4-3 

2.125 

3- i8 75 

3-3125 

3.68 

3-83 

1.65 

2.125 

8.163 

•993 

4-5 

2.25 

3-375 

3-5 

3-9 

4.04 

i -75 

2.25 

9.658 

I-H 3 

4-5 

2-375 

3-5625 

3-6875 

4.1X 

4.26 

1.84 

2-375 

11.263 

1.24 

4-375 

2-5 

3-75 

3-875 

4-33 

4-47 

1.94 

2.5 

13-149 

i -375 

4-25 

2.625 

3-9375 

4.0625 

4-55 

4.69 

2.03 

2.625 

I 5 -I 5 

i- 5 i 5 

4 

2.75 

4.125 

4-25 

4-77 

4.91 

2.12 

2.75 

17.285 

1.663 

4 

2.875 

4-3125 

4-4375 

4.99 

5.12 

2.22 

2.875 

I9-75I 

1.S18 

3-75 

3 

4-5 

4.625 

5-2 

5-34 

2.31 

3 

22.378 

1.979 

3-5 

3-25 

4-875 

5 

5-63 

5-77 

2-5 

3-25 

28.258 

2.323 

3-5 

3-5 

5-25 

5-375 

6.06 

6.21 

2.68 

3-5 

35 -° 8 i 

2.694 

3-25 

3-75 

5-625 

5-75 

6-5 

6.64 

2.87 

3-75 

43 -I 78 

3-093 

3 

4 

6 

6.125 

6-93 

7.07 

3.06 

4 

51.942 

3 - 5 i 8 

3 


Finished. —Deduct .0625 from diameters of bolts and depths of all heads 
and nuts. 

For Wood or Carpentry, 

Head and Nut (Square), 1.75 diameter of bolt. Depth of Head, .75, and 
of Nut , .9. 

Washer. —Thickness, .35 to .4 of diameter of bolt, on Pine 3.5 diameter, 
and Oak 2.5. 

English. 

Molesworth gives following elements of Thread of Bolts: 

Angle of thread , 55 0 . Depth of thread = Pitch of screw. 

Number of threads per Inch. — Square. , half number of those in angular 
threads. 

Depth of thread. —.64 pitch for angular and .475 for square threads. 
























1^8 DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS 


French Standard. Bolts and USTnts. ( Armengaud’s.) 


HEXAGONAL HEADS AND NUTS. 


Equilateral Triangular Thread. 


Square Thread. 


Diamet< 

of Bolt. 

at Base of 
Thread. 

Threads 
per Inch. 

Thiel 

Head. 

mess. 

Nut. 

Breadth 

across Flats. 

Safe 

Tensile 

Stress. 

Diameter 
of Bolt. 

Depth of 

Thread. 

Threads 

per Inch. 

Thickness 

of Nut. 

Safe 

Tensile 

Stress. 

Mm. 

Ins. 

IllS. 

No. 

Ins. 

Ins. 

Ins. 

Lbs. 

Mm. 

Ins. 

Ins. 

No. 

Ins. 

Lbs. 

5 

.2 

•13 

18.1 

.24 

.2 

•55 

44 

20 

■19 

.072 

6.57 

1.82 

717 

7-5 

•3 

.22 

16 

•3 

•3 

.68 

99 

25 

.98 

.081 

5-97 

2.01 

I 142 

IO 

•39 

• 3 1 

14.I 

•38 

•39 

.88 

178 

30 

1.18 

•093 

5-4 

2.22 

1635 

12.5 

•49 

•39 

12.7 

•44 

•49 

I.O4 

277 

35 

1.38 

.1 

4-93 

2.4I 

2 218 

15 

•59 

.48 

n -5 

•52 

•59 

1.2 

4OO 

40 

i -57 

.106 

4-53 

2.63 

2 912 

17-5 

.69 

•58 

10.6 

•58 

.69 

1.4 

545 

45 

1.77 

.114 

4.2 

2.85 

3674 

20 

•79 

.66 

9.8 

.66 

■19 

i -5 

713 

5 ° 

I -97 

.128 

3 - 9 1 

3-07 

4547 

22.5 

.89 

.76 

9.1 

.72 

.89 

1.68 

902 

55 

2.17 

•13 

3-65 

3-3 

5288 

25 

.98 

.84 

8-5 

.8 

.98 

1.84 

I 120 

60 

2.36 

.14 

3 43 

3-5 

6540 

30 

1.18 

1.02 

7-5 

•94 

1.18 

2.16 

1635 

65 

2.56 

•15 

3-23 

3-7 

7 660 

35 

1.38 

1.2 

6.7 

1.08 

1.38 

2.48 

2 218 

70 

2.76 

•158 

3.06 

3 - 9 2 

8893 

40 

1.58 

1.4 

6 

1.22 

1.58 

2.8 

2 912 

75 

2-95 

.166 

2.92 

4 -i 3 

10 214 

45 

!-77 

1.56 

5-5 

1.36 

1.77 

3-2 

3674 

80 

3 -i 5 

.174 

2.76 

4-36 

11603 

50 

I, 97 

1.74 

5 -i 

i -5 

1.97 

3-44 

4547 

85 

3-35 

.183 

2.63 

4-58 

13 IOO 

55 

2.17 

I.92 

4-7 

1.64 

2.17 

3-76 

5288 

90 

3-54 

.192 

2.51 

4.78 

14794 

60 

2.36 

2.08 

4.4 

1.74 

2.36 

4.08 

6540 

95 

3-74 

.2 

2.41 

5 

16352 

65 

2.56 

2.26 

4.1 

1.92 

2.56 

4.4 

7 660 

IOO 

3-94 

.209 

2.31 

5.22 

18 144 

70 

2.76 

2.44 

3-8 

2.06 

2.76 

4-7 

8893 

105 

4 -i 3 

.22 

2.22 

5-43 

20 OOO 

75 

2-95 

2.6 

3-5 

2.2 

2-95 

5 

10214 

no 

4-33 

.226 

2.13 

5.66 

21950 

8o* 

3 -i 5 

2.78 

3-4 

2.34 

3 -i 5 

5-35 

n 468 

115 

4-53 

•23 

2.06 

5-87 

23990 


English. Bolts and IN"nts. ( Whitworth's.) 


Hexagonal Heads and Nixts, and. Triangular Threads. 


Diame 

Bolt. 

Base of r* 
Thread. 

Threads 
per Inch. 

D 

Head. 

eptli. 

Nut. 

Width 

of 

Head 

and 

Nut. 

Diam 

Bolt. 

eter. 

Base 

of 

Thread. 

Threads 
per Inch. 

Dej 

Head. 

)th. 

Nut. 

Width 

of 

Head 

and 

Nut. 

Ins. 

Inch. 

No. 

Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Ins. 

.125 

•093 

40 

.109 

.125 

•338 

1.25 

1.067 

7 

I.O94 

1.25 

2.048 

■1875 

•134 

24 

.164 

•1875 

•448 

i -375 

I.l6l 

6 

1.203 

i -375 

2.215 

.2187 

— 

24 

— 

— 

— 

i -5 

1.286 

6 

i- 3 12 

i -5 

2413 

•25 

,l86 

20 

.219 

•25 

•525 

1.625 

t -369 

5 

1.422 

1.625 

2.576 

•3125 

.241 

18 

•273 

• 3 I2 5 

.601 

i -75 

1.494 

5 

I - 53 I 

i -75 

2.758 

■315 

•295 

l6 

.328 

•375 

.709 

1-875 

i -59 

4-5 

I.64I 

i -875 

3.018 

•4375 

•346 

14 

•383 

■4375 

.82 

2 

i- 7 i 5 

4-5 

i -75 

2 

3-149 

•5 

•393 

12 

•437 

•5 

.919 

2.125 

1.84 

4-5 

1.859 

2.125 

3 337 

•5625 

•456 

12 

•492 

•5625 

I.OII 

2.25 

i -93 

4 

1.969 

2.25 

3 546 

.625 

.508 

II 

•547 

.625 

I.IOI 

2-375 

2-055 

4 

2.078 

2-375 

3-75 

.6875 

•571 

II 

.601 

•6875 

1.201 

2-5 

2.18 

4 

2.187 

2-5 

3-894 

•75 

.622 

IO 

.656 

•75 

1.301 

2.625 

2.305 

4 

2.297 

2.625 

4.049 

.8125 

.684 

IO 

.711 

•8125 

i -39 

2-75 

2.384 

3-5 

2.406 

2-75 

4.181 

.875 

•733 

9 

.766 

•875 

1.479 

2.875 

2.509 

3-5 

2.516 

2.875 

4-346 

•9375 

•795 

9 

.82 

•9375 

1-574 

3 

2.634 

3-5 

2.625 

3 

4-531 

I 

.84 

8 

•875 

I 

I.67 

3-25 

2.84 

3-25 

— 



1-125 

•942 

7 

.984 

1.125 

1.86 

3-5 

3.06 

3-25 

— 

, - 

— 

























































RETENTION" OF SPIKES AND NAILS. 


159 


Square Heads and. Nuts. (Whitworth’s.) 


Diameter. 

Threads 

Diameter. 


DiAmetp.r. 



Base of 
Thread. 


Base of 
Thread. 

Threads 



Threads. 

Bolt. 

per Inch. 

Bolt. 

per Inch. 

Bolt. 

Base of 
Thread. 

per Inch. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

No. 

3-75 

3- 2 5 

3 

4-5 

3-875 

2.875 

5-25 

4-4375 

2.625 

4 

3-5 

3 

4.75 

4.0625 

2.75 

5-5 

4.625 

2.625 

4-25 

3*75 

2.875 

5 

4-25 

2.75 

6 

4-875 

2-5 


"Weight oT Heads and IN"uts iiL 3L"bs. (Molesworth.) 

Hexagonal , 1.07 D 3 . Square , 1.35 3 D 3 . D representing diameter of bolt 
in inches. 


Retentiveness of YWrouglit Iron Spikes and. HSTails. 
Deduced from Experiments of Johnson and Bevan. 

SPIKES. 


Spike. 

Wood. 


.43 

og 

2*2 

- TJ 

00^ 

Remakes. 



I 

pL 


S J | 





& 

Q 

fi.3 

os" 
fe o 

££ > 




Ins. 

Ins. 

Ins. 

Lbs. 



Square. 

Hemloekf 

Chestnut 

•39 

•37 

.'I 

3-5 

3-5 

1297 

1873 

1.58 

2.16 

Seasoned in part. 
Unseasoned. 

U * 

.38 

44 * 

Yellow pine 

•375 

•375 

3-375 

2052 

2-37 

Seasoned. 

(4 

White oak 

•375 

•375 

3-375 

39IO 

4-52 

44 

44 

Locust 

.4 

.4 

3-5 

3-5 

5967 

2223 

6-33 

3-93 

44 

Flat narrow.. 

Chestnut 

•39 

•T 

.25 

Unseasoned. 

44 44 

White oak 

•39 

•25 

3-5 

399° 

7.05 

Seasoned. 

44 44 

Locust 

•39 

•25 

3-5 

5673 

9-32 

44 

u broad.. 

Chestnut 

•539 

.288 

3-5 

2394 

2.66 

Unseasoned. 

» u 

White oak 

•539 

.288 

3-5 

5330 

5-7i 

Seasoned. 

u u 

Locust 

•539 

.288 

3-5 

7040 

7.84 

44 

Square") > • 

Hemloekf 

•4 

•39 

3-5 

1638 

i-75 

Seasoned in part. 

“ L g « 

Chestnut f 

•4 

•39 

3-5 

1790 

1.81 

Unseasoned. 

“ J q«3 

Locust f 

•4 

•39 

3-5 

399° 

4.17 

Seasoned in part. 

Round and) 
grooved../ 

Ash 

Diam. .5 

3-5 

2052 

2.21 

Seasoned. 

44 

44 

46 

•5 

3-5 

2451 

2.41 

46 

44 

White oak 

44 

.48 

3-5 

3876 

3-2 

44 


* Burden’s patent. t Soaked Id water after the spikes were driven. 


NAILS. 




Depth of 
Insertion. 


Force required to draw it. 


Pressure required 

Nail. 

Length. 

Pine. 

Hemlock. 

Elm. 

Oak. 

Beech. 

to force them 
into Pine. 


Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sixpenny 

2 

I 

187 

3 12 

327 

507 

667 

235 

44 

2 

1-5 

327 

539 

571 

675 

889 

400 

44 

2 

2 

530 

857 

899 

1394 

1834 

6lO 


General Remarks. 

"With a given breadth of face, a decrease of depth will increase retention. 
In soft woods, a blunt-pointed spike forces the fibres downwards and 
backwards so as to leave the fibres longitudinally in contact with the faces 
of the spike. 

















































l60 ANGLES AND DISTANCES.-DISTANCES AND ANGLES, 


To obtain greatest effect, fibres of the wood should press faces of the spike 
in direction of their length; thus, a round blunt bolt, driven into a hole of 
a less diameter, has a retention equal to that of any other form, when wholly 
driven, as without boring. 

The retention of a spike, whether square or flat, in unseasoned chestnut, 
from two to four inches in length of insertion, is about 800 lbs. per square 
inch of the two surfaces which laterally compress the faces of the spike. 

When wood was soaked in water, after spikes were driven, order of their 
retentive power was Locust, White oak, Chestnut, Hemlock, and Yellow Pine. 


Gras IPipe Threads. 


Diameter in Inches. 
Threads per Inch .. 


1 - I2 5 

•25 

•375 

•5 I 

•75 

1 

1.25 

i-5 

i-75 

1 28 

19 

19 

14 1 

14 

11 

11 

11 

11 1 


2 

11 


ANGLES AND DISTANCES. 

Angles and. Distances corresponding to Opening 
Dale of r JC'v\^o Feet. 


of a 


Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

[ Distance. 

Angle. 

1 Distance. 

O 

Ins. 

O 

Ins. 

O 

Ins. 

O 

Ins. 

O 

Ins. 

I 

.2 

19 

3 - 9 6 

37 

7.61 

55 

II.08 

73 

14.28 

2 

.42 

20 

4.17 

38 

7.81 

56 

II.27 

74 

I4.44 

3 

•63 

21 

4-37 

39 

8.01 

57 

n -45 

75 

14.61 

4 

.84 

22 

4-58 

40 

8.2 

58 

II.64 

76 

14.78 

5 

1.05 

23 

4.78 

4 1 

8.4 

59 

11.82 

77 

14.94 

6 

1.26 

24 

4-99 

42 

8.6 

60 

12 

78 

I 5 -H 

7 

I.47 

25 

5 -i 9 

43 

8.8 

61 

I 2 .l 8 

79 

15-27 

8 

I.67 

26 

5-4 

44 

8.99 

62 

12.36 

80 

15-43 

9 

1.88 

27 

5-6 

45 

9.18 

63 

I2.54 

81 

15-59 

10 

2.09 

28 

5 . 8 i 

46 

9-38 

64 

12.72 

82 

15-75 

11 

2-3 

29 

6.01 

47 

9-57 

65 

I2.9 

83 

15-9 

12 

2.51 

30 

6.21 

48 

9.76 

66 

I3.O7 

84 

16.06 

13 

2.72 

31 

6.41 

49 

9-95 

67 

I 3-25 

85 

16.21 

14 

2.92 

32 

6.62 

50 

10.14 

68 

1342 

86 

16.37 

15 

3 -i 3 

33 

6.82 

5 i 

io -33 

69 

13-59 

87 

16.52 

16 

3-34 

34 

7.02 

52 

10.52 

70 

13-77 

88 

16.67 

17 

3-55 

35 

7.22 

53 

10.71 

7 i 

I 3-94 

89 

16.82 

18 

3-75 

36 

7.42 

54 

10.9 

72 

14.11 

90 

16.97 


Distances and Angles corresponding to Opening of a 
Rule of Two Feet. 


Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Distance. 

Angle. 

Ins. 

O 

Ins. 

O 

Ins. 

O 

Ins. 

O 

Ins. 

O 

•25 

1.12 

3 

14.22 

6 -5 

3:1.26 

IO 

49.14 

13-5 

68.28 

•375 

I.48 

3-25 

I 5-34 

6-75 

32-4 

IO.25 

50-34 

13-75 

69-54 

•5 

2.24 

3-5 

16.46 

7 

33-54 

IO.5 

5 i -54 

14 

71.22 

.625 

2-59 

3-75 

17-58 

7-25 

35-09 

IO.75 

53-14 

14.25 

72-5 

•75 

3-35 

4 

19.11 

7-5 

36.24 

II 

54-34 

14-5 

74.2 

•875 

4.12 

4-25 

20.24 

7-75 

37-4 

II.25 

55-54 

14-75 

75-5 

1 

4.48 

4-5 

21.37 

8 

38.56 

14.5 

57 -i 6 

15 

77.22 

1.25 

5-58 

d -75 

22.5 

8.25 

40.12 

n -75 

58.38 

15-25 

78-54 

i -5 

7 -i 

5 

24.4 

8-5 

41.28 

12 

60 

I 5-5 

80.28 

i -75 

8.22 

5-25 

25.16 

8-75 

42.46 

12.25 

61.23 

I 5-75 

82.2 

2 

9-34 

5-5 

26.3 

9 

44.2 

12.5 

62.46 

16 

83-36 

2.25 

10.46 

5-75 

27.44 

9-25 

45-2 

I2 -75 

64.1 

16.25 

85.14 

2.5 

11.58 

6 

28.58 

9-5 

46.38 

13 

65-36 

16.5 

86.52 

2-75 

i 3 -i 

6.25 

30.12 

9-75 

47-56 

I 3- 2 5 

67.02 

16.75 

88.32 
















































WIRE ROPE. 


161 


WIRE ROPE. 

Wire rope of same strength as new H$mp rope will run on sheaves 
of like diameter; but greater diameter of sheaves, less the wear. Short 
bends should be avoided, and wear increases with the speed. Adhesion 
is same as that of hemp rope. It should not be coiled, but should be 
wound as upon a reel. 

When substituting wire rope for hemp, it is well to allow for former 
same weight per foot which experience has approved of for latter. As 
a general rule, one wire rope will outlast three of hemp. To guard 
against rust, stationary rope should be coated once a year with linseed- 
oil, or well painted or tarred. Running rope in use does not require 
any protection. 

Where great pliability is required, centre or core of rope should be 
of hemp. 

Annealing wire, in rendering it more pliable than when unannealed, 
reduces its elasticity and consequent strength from 25 to 50 per cent. 

Running rope is made of finer wire than standing rope. 

For safe working load, deduct one fifth to one seventh of ultimate 
strength, according to speed and vibration. It is better to increase load 
than speed, as it increases wear. 

Standing rigging of a vessel of wire rope is one fourth less in weight 
than when of hemp. 

Rope of 19 wires to a strand is more pliable than one of 7 and 12 
wires, and hence it is better suited to operation over small drums, for 
hoisting, etc. 

Ultimate strength of iron ropes is 4480 lbs. for each pound in weight 
per fathom, and for galvanized steel 6720 lbs. 

Strength per square inch of section of a rope is about 53 per cent, of 
an equal section of solid metal of same tensile strength per square inch. 

Steel ropes may be one third less in weight than iron for same load. 
Their durability is much greater, especially when required to run rapidly 
over sheaves. Hemp should be one third heavier than iron. 

Steel wire Ho. 14 W. G. = .083 inch, weight 2 lbs. per yard, will bear a 
stress of 2000 lbs. 

The combined sectional area of the wires in a cable is to the area of the 
cable as 1 to 1.3. Hence, to ascertain areas of the wires in a cable multiply 
diameter by .77, and for areas of the voids, multiply area of cable by .23. 

In short transmissions, it is necessary to connect rope quite taut, and an 
additional diameter of two numbers of rope must be given to it. 

In long transmissions, when there is an insufficiency of height to admit 
of a proper deflection of rope, and it becomes necessary to connect it very 
taut, an additional diameter of one number of rope must be given to it. 

When distance exceeds 350 feet, transmission should be divided into two 
or more equal lengths by aid of intermediate wheels. 

Rope Nos. 7 and 8 ( Roebling's ) are made with Nos. 1 and 2 as strands, 
and twisting six of them around a hemp centre. 

Results of an Experiment with Galvanized Wire. 

A strand of 2-inch wire rope broke with a strain of 13 564 lbs., and a 
piece of a like rope, when galvanized, withstood a strain of 14 796 lbs. be¬ 
fore breaking. 


0* 


WIRE ROPES 


l62 


Elements of Running and Standing Wire Rope. 

J. A. Roebling's Sons Co. 

19 Wires in a Strand. 


Iron. 


Diam. 

Circum. 

Breaking 

Weight. 

Safe 

Load. 

Circum. 
of Hemp 
Rope. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Ins. 

•5 

i -5 

6 960 

I OOO 

3-5 

■5625 

1.625 

8 480 

1500 

4 

.625 

2 

10 260 

2 500 

4-5 

•75 

2.25 

17 280 

3500 

5 

•875 

2.75 

23 OOO 

5 OOO 

6 

I 

3-125 

32 OOO 

6 000 

7 

1-125 

3-5 

40 OOO 

8 000 

8 

1.25 

4 

54000 

II OOO 

9-5 

i -5 

4-375 

70 OOO 

14 OOO 

io -75 

1.625 

5 

88 000 

18 OOO 

12 

i -75 

5-5 

108 OOO 

22 OOO 

13 

2 

6 

130 OOO 

26 OOO 

14-5 

2 25 

6-75 

148 OOO 

30 OOO 

15-5 



<t 


Cast Steel. 


Weight 

per 

Foot. 

Breaking 

Weight. 

Safe 

Load. 

Circum. 
of Hemp 
Rope. 

Weight 

per 

Foot. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

•35 

II 

OOO 

2 OOO 

'4-5 

•35 

•44 

13 

OOO 

3 OOO 

4-75 

•44 

•7 

18 

OOO 

4 OOO 

5-25 

•65 

.88 

26 

OOO 

6 000 

6-5 

•83 

1.2 

40 

OOO 

8 000 

8 

1.14 

1.58 

48 

OOO 

10 OOO 

9- 2 5 

i-5 

2 

60 

OOO 

12 OOO 

IO 

i-95 

2-5 

78 

OOO 

16 OOO 

if-5 

2.44 

3-65 

no 

OOO 

22 OOO 

13 

3-i 

4.1 

128 

OOO 

26 000 

14-5 

4.1 

5-25 

156 

OOO 

34000 

15-75 

5.08 

6-3 

200 

OOO 

40 OOO 

19 

6.02 

8 

260 

OOO 

44 000 

22.5 

7.8 


Transmission and Standing Rope. 
7 Wires in a Strand. 


Diam. 

Circum. 

Irc 

Breaking 

Weight. 

)N. 

Safe 

Load. 

Circum. 
of Hemp 
Rope. 

Weight 

per 

Foot. 

Breaking 

Weight. 

Cast Si 

Safe 

Load. 

. 

EEL. 

Circum. 
of Hemp 
Rope. 

Weight 

per 

Foot. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

.28 

•875 

2 OOO 

500 

2 

.125 

4 200 

1050 

2-375 

.125 

•3125 

I 

2 760 

690 

2.25 

.16 

5 000 

1250 

3 

.16 

•375 

1-125 

330 0 

825 

2-5 

.19 

8 000 

2 OOO 

3-75 

.19 

■4375 

1.25 

4 260 

x 065 

2-75 

•23 

IO OOO 

2 4OO 

4.125 

•23 

•5 

!-375 

5 660 

1 4 r 5 

3-25 

•31 

II OOO 

2 500 

4-75 

•31 

•5625 

1.625 

8 200 

2050 

4 

.41 

12 OOO 

3500 

5 

.41 

.625 

i -875 

11 600 

2 9OO 

4-75 

• 5 * 

20 OOO 

4500 

5-5 

•51 

.6875 

2.125 

15 200 

3800 

5 

.68 

27 OOO 

6000 

6-5 

.68 

•75 

2-375 

17 600 

4 4OO 

5-25 

.86 

34000 

7 000 

7-25 

.86 

•875 

2.625 

24 600 

6 150 

6.25 

1.12 

44 OOO 

10 000 

8-5 

1.12 

I 

3 

32 OOO 

8 000 

7 

i -5 

60 OOO 

13 000 

IO 

i -5 

1-125 

3-375 

40 000 

10 OOO 

8 

1.82 

72 OOO 

16 000 

10.75 

1.82 

1.25 

3-75 

50000 

12 500 

9-25 

2.28 

90 OOO 

20 000 

12 

2.28 

i -375 

4-25 

60 000 

15 000 

IO 

2.77 

no 000 

25 000 

13 

2.77 

i -5 

4.625 

72 000 

18 OOO 

10.75 

3-37 

134 000 

32 000 

15 

3-37 


Note. —When made with wire centre instead of hemp, weight is 10 per cent. more. 


Gralvanized Charcoal Iron Wire. 
'V'essels 5 Ttigging and. Derrick Gray'S. 
12 Wires in a Strand. 


Circum. 

Circum. 
of Hemp 
Rope. 

Breaking 

Weight. 

Safe 

Load. 

Weight 

per 

Foot. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

3 

6 

24OOO 

6000 

i -33 

3-25 

6-5 

28000 

7 000 

1.58 

3-5 

7 

32 OOO 

8 000 

1.83 

3-75 

7-5 

40000 

IOOOO 

2 

4 

8 

46000 

II 500 

2.46 

4-25 

8-5 

52 OOO 

13000 

2.66 


Circum. 

Circum. 
of Hemp 
Rope. 

Breaking 
Weigh t. 

Safe 

Load. 

Weight 

per 

Foot. 

Ins. 

Ins* 

Lbs. 

Lbs. 

Lbs. 

4-5 

9 

60 000 

15 OOO 

3 

4-75 

9-5 

66000 

16 500 

3-46 

5 - 

IO 

70 000 

17500 

3 66 

5-25 

10.5 

80 OOO 

20 OOO 

4.12 

5-5 

II 

86000 

21 50O 

4.46 

6 

12 

100 OOO 

25 OOO 

4-83 








































































WIRE ROPES AND CABLES 


163 


Galvanized Charcoal Iron. 
■Vessels’ Digging and. Derrick Grays. 
(J. A. Roebling's Sons Co.) 

7 Wires in a Strand . 


Circura. 

Circum. 
of Hemp 
Rope. 

Breaking 
W eight. 

Safe 

Load. 

Weight 

per 

Foot. 

Circum. 

Circum. 
of Hemp 
Rope. 

Breaking 

Weight. 

Safe 

Load. 

Weight 

per 

Foot. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

IllS. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

I 

2 

4000 

I OOO 

.125 

3-5 

7 

32 OOO 

8 000 

1.79 

1.25 

2-5 

5000 

1250 

>25 

3-75 

7-5 

4OOOO 

IOOOO 

2 

i -5 

3 

7 000 

1750 

•334 

4 

8 

46 OOO 

II 500 

2.46 

i -75 

3-5 

10000 

2500 

.427 

4-25 

8-5 

52 poo 

13000 

2.67 

2 

4 

14000 

3500 

•583 

4-5 

9 

60000 

15 OOO 

3 

2.25 

4-5 

16000 

4OOO 

.708 

4-75 

9-5 

66000 

16 500 

342 

2.5 

5 

18 000 

4500 

•875 

5 

IO 

70000 

17500 

3.66 

2-75 

5-5 

20000 

5000 

Ia 7 

5-25 

10.5 

80 OOO 

20 000 

4.08 

3 

6 

24 000 

6 000 

i -33 

5-5 

II 

86000 

21 500 

4.41 

3-25 

6.5 

28000 

7 000 

1.58 

6 

12 

IOOOOO 

25 OOO 

4.81 


Grange, "Weight, and Length of Iron "Wire. 


Gauge. 

Diam. 

Weight 
per 100 
Feet. 

Weight 
of one 
Mile. 

63 lbs. 
Bundle. 

Area. 

Gauge. 

s 

C3 

5 

S 0 . 

H 0) 

<V U « 

£ 

Weight 
of one 
Mile. 

63 lbs. 
Bundle. 

Area. 

No. 

Inch. 

Lbs. 

Lbs. 

Feet. 

Sq. Inch. 

No. 

Inch. 

Lbs. 

Lbs. 

Feet. 

Sq.Inch. 

6/ 0 

.46 

56.1 

2962 

112 

.16619 

l6 

.063 

1.05 

55 

6000 

.003 117 

5/o 

•43 

49.01 

2588 

I29 

.145 22 

17 

•054 

•77 

41 

8 182 

.002 29 

4/0 

•393 

40.94 

2162 

154 

.121304 

18 

.047 

•58 

31 

10862 

.001734 

3/o 

.362 

34-73 

1834 

181 

.102 92I 

19 

.041 

•45 

24 

14000 

.OOI 32 

2/0 

•331 

29.O4 

1533 

217 

.086 049 

20 

•035 

•32 

17 

19687 

.000 962 

i/o 

•307 

27.66 

1460 

228 

.074023 

21 

•032 

.27 

14 

23 333 

.000 804 

I 

.283 

21.23 

1121 

296 

.062 9OI 

22 

.028 

.21 

II 

30 OOO 

.000615 

2 

.263 

18.34 

968 

343 

•054 325 

23 

.025 

•175 

9.24 

.36000 

.000 491 

3 

.244 

I 5-78 

833 

399 

.046 759 

24 

.023 

.14 

7-39 

45000 

.OOO415 

4 

.225 

13-39 

707 

470 

.039 76 

25 

.02 

.116 

6.124 

54310 

.OOO314 

5 

.207 

n-35 

599 

555 

•033653 

26 

.018 

•093 

4-9 1 

67 742 

.OOO 254 

6 

.192 

9-73 

514. 

647 

.028 952 

27 

.OI7 

.083 

4.382 

75 903 

.OOO 227 

7 

.177 

8.03 

439 

759 

.024 605 

28 

.016 

.074 

3-907 

85 135 

.OOO 201 

8 

.162 

6.96 

367 

9°5 

.020612 

29 

.015 

.061 

3.22 

103 278 

.000176 

9 

.148 

5.08 

306 

1086 

.017203 

30 

.OI4 

•054 

2.851 

116 666 

.OOO I54 

IO 

•135 

4-83 

255 

1304 

: -014313 

31 

•0135 

•05 

2.64 

126 000 

.000133 

II 

.12 

3.82 

202 

1649 

.011 309 

32 

.013 

.046 

2.428 

136 956 

.OOO I32 

12 

.105 

2.92 

154 

2158 

.008 659 

33 

.Oil 

•037 

i-953 

170 270 

.OOO O95 

!3 

.092 

2.24 

n8 

2813 

.006 647 

34 

.OI 

•03 

1.584 

210 OOO 

.000078 

14 

.08 

1.69 

89 

3728 

.005 026 

35 

•0095 

1 .025 

1.32 

252 OOO 

.OOOO7I 

15 

.072 

i-37 

72 

4598 

.OO407I 

36 

.009 

.021 

1.161 

286363 

.000 064 


Galvanized Steel Cables for Suspension Bridges. 


Diameter. 

Ultimate 

Strength. 

Weight 
per Foot. 

Diameter. 

Ultimate 

Strength. 

Weight 
per Foot. 

[Diameter. 

Ultimate 

Strength. 

Weight 
per Foot, 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

• Lbs. 

Lbs. 

i -5 

130 OOO 

3-7 

1-875 

200 OOO 

5-8 

2-375 

360 000 

IO 

1.625 

150 OOO 

4-35 

2 

220 OOO 

6-5 

2-5 

400 OOO 

JI -3 

i -75 

190 OOO 

5-6 

2.25 

310 OOO 

8.64 

2.625 

440 OOO 

13 
















































































164 IKON, STEEL, AND HEMP KOPE. 


'W'eigh.t and Strength, of Single Strand, and Cable 
laid. Fence "Wire. (F. Morton & Co.) 


Strands. 

No. 

Single Wire 
of equal 
Diameter. 

Len 
per 10 
Of a 
Strand. 

gth 

00 lbs. 

Of 

Rope. 

Strands. 

No. 

Single Wire 
of equal 
Diameter. 

Ler 
per 10 
Of a 
Strand. 

gth 

00 lbs. 
Of 

Rope. 

No. 


No. 

Inch. 

Feet. 

Feet. 

No. 


No. 

Inch. 

Feet. 

Feet. 

3 

2A 

8 

•159 

20 090 

15 270 

7 

OO 

4 

.229 

8300 

7366 

4 

2 

7 

.174 

14 730 

12 790 

7 

3/° 

3 

•25 

8036 

6228 

7 

I 

6 

.191 

13125 

10 580 

7 

4/0 

2 

.274 

7500 

515 6 

7 

O 

5 

.209 

10 446 

8 928 

7 

5/0 

I 

•3 

5090 

4286 


No. and diameter of wire is that of Ryland’s Bros., pp. 122-4. 

Hemp, Iron, and Steel. (K. S. Newall & Co.) 

ROUND. 


HEMP. 

Circumference. 

Weight 

per 

Foot. 

IRON. 

Circumference. 

Weight 

per 

Foot. 

STEEL. 

Circumference. 

Weight 

per 

Foot. 

Tensile 

Safe 

Load. 

Strength. 

Ultimate 

Strength. 

Ins. 

Lbs. 

Ins. 

Lbs. 

In 3 . 

Lbs. 

Lbs. 

Lbs. 

2.75 

•33 

I 

.16* 

— 

— 

672 

4 480 



i -5 

•25 

I 

.16 

1008 

6 720 

3-75 

.66 

1.625 

•33 

— 

— 

1344 

8 960 



i -75 

.42 

i -5 

•25 

1680 

II 200 

4-5 

.83 

1.875 

•5 

— 

— 

2 016 

13440 



2 

•58 

1.625 

•33 

2352 

15 680 

5-5 

1.16 

2.125 

.66 

i -75 

.42 

2688 

17 920 



2.25 

•75 

— 

— 

3024 

20 l6o 

6 

i -5 

2-375 

•83 

1-875 

•5 

3360 

22 4OO 


I 

2-5 

.92 

— 

— 

3696 

24 640 

6-5 

1.66 

2.625 

I 

2 

•58 

4032 

26 680 



2-75 

1.08 

2.125 

.66 

4368 

29 120 

7 

2 

2.875 

I.l6 

2.25 

•75 

4704 

3 1 360 



3 

1.25 

— 

— 

5040 

33600 

7-5 

^•33 

3-125 

i -33 

2-375 

.83 

5376 

36 840 



3-25 

1.41 

— 

— 

5672 

38 080 

8 

2.66 

3-375 

i -5 

2-5 

.92 

6048 

40320 



3-5 

1.66 

2.625 

I 

6 720 

44 800 

8-5 

3 

3-625 

1.83 

2-75 

1.. 08 

7 392 

49 280 



3-75 

2 

— 

— 

8 064 

53 760 

9-5 

3.66 

3-875 

2.16 

3-25 

i -33 

8736 

58 240 

IO 

4-33 

4 

2-33 

— 

— 

9408 

62 720 



4-25 

2-5 

3-375 

i -5 

10080 

67 200 

II 

5 

4-375 

2.66 

— 

— 

10752 

71680 



4-5 

3 

3-5 

1.66 

12 096 

80 640 

12 

5.66 

4.625 

3-33 

3-75 

2 

13440 

89600 


FLAT. 

Dimensions. Dimensions. Dimensions. 


4 X .5 

3-33 

2.25 

X -5 

1.85 

— 

— 

4928 

44 800 

5 X1.25 

4 

2-5 

X -5 

2.16 

— 

— 

5824 

51520 

5-5 X 1.375 

4-33 

2-75 

X.625 

2-5 

— 

— 

6 720 

60480 

5 - 75 X 1-5 

4.66 

3 

X.625 

2.66 

2 x .5 

1.66 

7 168 

62 720 

6 X 1.5 

5 

3-25 

X.625 

3 

2.25X-5 

H 

bo 

8064 

71680 

7 X 1.875 

6 

3-5 

X.625 

3-33 

2.25 X -5 

2 » 

8960 

80 640 

8.25X2.125 

• 6.66 

3-75 

X .6875 

3.66 

2-5 X -5 

2.16 

9850 

89 600 

8.5 X2.25 

7-5 

4 

X.6875 

4.16 

2-75 X .375 

2-5 

11200 

100 800 

9 X2.5 

8-33 

4-25 

x .75 

4.66 

3 X .3 75 

2.66 

12544 

112 000 

9-5 X 2.375 

9.16 

4-5 

X -75 

5-33 

3-25 X .375 

3 

14336 

125 440 

10 X2.5 

IO 

4.625 X -75 

5.66 

3-5 X.375 

3-23 

15232 

134400 

























































HOPES AND CHAINS 



From preceding tables following results are determined: 



Ultimate Strength 

Safe Load 


per Lb. Weight per 

per Lb. "Weight per 

per Square of Circurn- 


Foot. 

Foot. 

ference in Inches. 


Lbe. 

Lbs. 

Lbs. 

Hemp. 

15 OOO 

4550 

IOO 

Iron. 

22 OOO 

4500 

600 

Steel. 

f 30 OOO 

f 6000 

( I OOO 


(45 500 

( SOOO 

( 1300 


ROUND AND PLAT MINING ROPES. 
(MM. Harmegnies, Dumont <& Co., Amin, France.) 
For a Depth of 400 Metres or 440 Yards. 



Round. 




Flat. 



No. 

Diameter. 

W eight 
per Foot. 

Safe Load. 

No. of 
Strands. 

Width. 

Thick¬ 

ness. 

Weight 
per Foot. 

Safe Load. 


Ins. 

Lbs. 

Lbs. 


Ins. 

Ins. 

Lbs. 

Lbs. 

17 

•51 

2.l6 

560 

9 

2.4 

•55 

2 

3 3 6 0 

16 

•59 

1.66 

1120 

6 

2.8 

•59 

2.13 

4032 

15 

•63 

1.26 

1680 

6 

3-2 

•63 

2.66 

4 480 

14 

• 7 i 

1 

2240 

6 

3-2 

.67 

3 

5600 

13 

•83 

•S 3 

3360 

6 

3-5 

•79 

3-33 

6 720 

12 

.98 

.66 

4480 

8 

4-3 

.67 

3.66 

7840 

II 

1.1 

•5 

5600 

6 

3-9 

.83 

4 

8 960 

IO 

i -3 

•33 

6720 

8 

4-7 

•79 

4-33 

10 080 





8 

5 -i 

•87 

5-33 

11 200 


Ropes and. Chains of Eqnal Strength. 




CIRCUMFERENCE. 


WEIGHT PER FOOT. 


Diameter 

of 

Iron Chain. 

Hemp 

Rope. 

Crucible 

Steel 

Rope. 

Charcoal 

1 1011 
Rope. 

Steel 

Rope. 

Iron 

Rope. 

Hemp 

Rope. 

Iron 

Chain. 

Safe 

Load. 

Ins. 


Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Tons. 

.218 

75 

2-75 

— 

I 

— 

.14 

•34 

•5 

•3 

•25 


3 

— 

1.18 

— 

.21 

.46 

•65 

•4 

.281 

25 

3-5 

I 

i -39 

•17 

.28 

.67 

• .81 

•5 

• 3 12 

5 

4-25 

I.26 

i -57 

•25 

•33 

•75 

.96 

.6 

•375 


4-5 

i -45 

1.77 

•3 

•45 

•83 

I.38 

.8 

•437 

5 

5 

i -57 

1.97 

•35 

•57 

1.16 

I.76 

1 

.468 

75 

5-5 

t-77 

2.19 

•45 

•7 

1.2 

2.2 

*•3 

•5 


5-75 

1.96 

2.36 

•59 

•83 

1.6 

2.63 

i -5 

.625 


6-75 

2.36 

2-75 

•85 

1.08 

2 

4.21 

2-3 

.687 

5 

7-75 

2-75 

3- I 4 

1.1 

i -43 

2.65 

4-83 

3 - 1 

•75 


8-75 

2-95 

3-53 

1.28 

1.8 

3-35 

5-75 

3-8 

•875 


9-75 

3 -i 4 

3-93 

i -45 

2-3 

4.6 

7-5 

4.8 

•937 

5 

10.5 

3-53 

4-32 

1.83 

2.94 

4.92 

9-33 

5-9 

1.062 

5 

n -75 

3-93 

4.71 

2-33 

3-56 

5-83 

10.6 

7 

1.125 


12.75 

4-32 

5 -i 

2.98 

4 

6.2 

11.9 

8.2 

1.25 


14-75 

4.71 

5-5 

3-58 

4.8 

8.7 

14-5 

9-5 

i -375 


15-25 

4.81 

5-89 

3-65 

5-6 

9 

17.6 

11 

i -5 


15-75 

5 -i 

6.28 

4.04 

6-3 

10.1 

20 

12.5 

1.625 


17-75 

5-8 

7.07 

5-65 

7-95 

13-7 

22.3 

15-9 

I, 75 


i 9-5 

6-35 

7-85 

6-5 

9.81 

16.4 

24-3 

19.6 


By experiments of U. S. Navy, hemp rope of this circumference has a breaking 
weight of 71 309 lbs ., and a wire rope 0/5.34 bis. has equivalent strength. 



















































1 66 WEIGHT, STEESS, AND TENSION OF KOPES. 


Weight of Hemp and. "WTre Hope. ( Molesworth.) 


In Lbs. per Fathom. 


Circum¬ 

ference. 

He 

Common. 

MP. 

Good. 

Wire. 

Iron. | Steel. 

Circum¬ 

ference. 

He 

Common. 

MP. 

Good. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

I 

.18 

.24 

•87 

.89 

5 

4-5 

6 

i -5 

.41 

•54 

I.96 

2 

5-5 

5-45 

7.26 

i -75 

•55 

•74 

2.66 

2-73 

6 

6.48 

8.64 

2 

.72 

, .96 

3-48 

3-56 

6-5 

7.61 

10.14 

2.25 

.91 

1.22 

4.4 

4-51 

7 

8.82 

11.76 

2-5 

i-i 3 

i -5 

5-44 

5-56 

7-5 

10.13 

i 3-5 

2-75 

1.36 

1.82 

6.58 

6.73 

8 

11.52 

15-36 

3 

1.62 

2.16 

7-83 

8.01 

8-5 

13-05 

17-34 

3-25 

1.9 

2-54 

9.19 

9.4 

9 

14.58 

19.44 

3-5 

2.21 

2.94 

10.66 

10.9 

10 

18 

24 

3-75 

2-53 

3*38 

12.23 

12.52 

12 

26 

34-56 

4 

2.88 

3-84 

13.92 

14.24 

15 

40.52 

54 


To Compnte Stress upon a Hope set at an Inclination. 

Rule.— Multiply sine of angle of elevation by strain in lbs., add an allow¬ 
ance for rolling friction and weight of rope, and multiply by factor of safety. 

Factor of safety— For standing rope 4, for running 5, and for inclined 
planes from 5 to 7. 

Illustration. —Inclination of rope 92.5 feet in 100, velocity 1500 feet per minute, 
and strain 2000 lbs.; what should be diam. of iron rope, 7 wires to a strand? 

Angle of 92.5 feet in 100 = 43°, and sine °f 43° = .682. .682 X 2000= 1364, to 

which is to be added rolling friction and weight of rope, assumed to be n; hence, 
1364 + 11 = 1375. 

Factor of safety assumed at 6, consequently 1375 X 6 = 8250 lbs., capacity or break¬ 
ing weight or stress of rope. 

By table, page 162, 8200 lbs. is breaking weight of a wire rope of 7 strands, .625 
inch in diam. 


To Compute Tension of a Hope. 

jp 

~ — t. v representing velocity of rope in feet per minute , IP horses' power, 
and t tension in lbs. 

Illustration.— Assume wheel 7 feet in diameter, revolution 140 per minute, and 
IP as per preceding table, 29.6. 

Then 29-6X33000 ^976800 
7 X 3-141° X 14° 3°79 


To Compute Operative Deflection of a Hope. 


D 2 w 

10.7 t 


— d. D representing distance between centres of wheels or drums in 

feet, w weight of rope in feet per lb., t tension, or power required to produce 
required power or tension of rope when at rest, and d deflection in feet. 

Illustration.— Take elements of preceding case: diam. of wire rope of 7 strands 
= .5625 inch, and by table, page 162, ^ = .41 lb., and 0 = 300 feet. 

Then . 3 °° 3 X : 4 . = ,o, 8 7/m *. 

10.7 x 317-2 

Capacity .—At the Falls of the river Rhine there is a wire rope in operation 
that transmits the power of 600 horses for a distance exceeding one mile. 
















TRANSMISSION OF POWER AND EQUIVALENT BELT. 1 67 


Endless Ropes. 

Wire Ropes, when practicable and proper for application, can be used for 
transmission of power at a less cost than belting or shafting. 


Transmission of Power. 


I Diameter 
| of Wheel. 

Revolu¬ 
tions per 

1 Minute. 

Diameter 
of Rope. 

Horse 

Power. 

Diameter 

of Wheel. 

Revolu¬ 

tions per 
Minute. 

Diameter 

of Rope. 

j Horse 

Power. 

Diameter 

of Wheel. 

Revolu¬ 

tions per 
Minute. 

Diameter 

of Rope. 

Horse 

Power. 

Feet. 

4 

80 

Ins. 

•375 

3-3 

Feet. 

7 

IOO 

Ins. 

•5625 

21.1 

Feet. 

II 

I40 

Ins. 

.6875 

I32.I 

4 

IOO 

•375 

4.1 

7 

I40 

•5625 

29.6 

12 

80 

•75 

99-3 

4 

120 

•375 

5 

8 

80 

.625 

22 

12 

IOO 

•75 

124.1 

4 

I40 

•375 

5-8 

8 

IOO 

.625 

27-5 

12 

140 

•75 

173-7 

5 

80 

•4375 

6.9 

8 

140 

.625 

38.5 

13 

80 

•75 

122.6 

5 

IOO 

| -4375 

8.6 

9 

80 

.625 

4i-5 

13 

IOO 

•75 

153-2 

5 

120 

•4375 

10.3 

9 

IOO 

.625 

5i-9 

13 

120 

•75 

183.9 

5 

I40 

•4375 

12.1 

9 

140 

.625 

72.6 

14 

80 

•875 

148 

6 

80 

•5 

10.7 

10 

80 

.6875 

58-4 

14 

IOO 

•875 

176 

6 

IOO 

•5 

13-4 

10 

IOO 

.6875 

73 

14 

120 

.875 

222 

6 

120 

1-5 

16.1 

j 10 

140 

.6875 

102.2 

15 

80 

•875 

217 

6 

I40 

j -5 

18.7 

1 11 

80 

.6875 

75-5 

15 

IOO 

•875 

259 

7 

80 

! -5625 

1 16.9 

! 11 

IOO 

.6875 

94.4 

15 

120 

.875 

300 


Wire Rope and Equivalent Eelt. 

In substituting wire rope for an ordinary flat belt, the diameter is deter¬ 
mined by rule in practice for estimating power transmitted by a belt—viz., 

One horse power for every 70 square feet of running belt surface per 
minute. Thus, a belt 15 inches wide running at rate of 1400 feet per min¬ 
ute, its power would be equal to (1400 x 15) -4- (70 x 12) = 25 horses’ power. 

The same result is obtained by the use of a wire rope .5625 inch in diam¬ 
eter, running over a wheel 6 feet in diameter, making 130 revolutions per 
minute. 

Average life of iron wire rope with good care is from 3 to 5 years, and 
that of steel rope is greater. Wear increases rapidly with velocity. 

General Notes.—Hemp and. Wire Ropes. 

White Rope , 2 inches in circumference, of different manufactures, parted at 
a stress of from 4413 to 6160 lbs. 

Specimens of Italian, Russian, and French manufacture parted with an 
average stress of 5128 lbs. = 1633 lbs. per square inch of rope. 

Bearing capacity of a hemp rope is proportional to its thickness, number 
of its strands, slackness with which they are twisted, and quality of the 
hemp. 

Hemp and Wire Ropes — Ultimate Strength is 2240 lbs. per lb. per fathom 
for round hemp, 4480 lbs. for iron, and 6720 to 7840 lbs. for steel. 

Working Load is 336 lbs. per lb. weight per fathom for round hemp, 672 
lbs. for iron, and 1120 lbs. for steel. 

Or, .83 times square of circumference in inches for round hemp, 5 times 
square of circumference for iron, and 9 times square of circumference for 
steel. {D. K. Clark.) 

Steel Ropes may be one third less in weight than iron for like working 
load, and Hemp Ropes should be one third heavier than iron for like work¬ 
ing load. 





















168 


ROPES AND CHAINS 


IRON WIRE AND UNITED STATES NAYY HEMP ROPE. 


Wire 6 Strands , Hemp Core. Rope 4 Strands. 




WIRE. 




HEMP. 


Circumference. 

Wires. 

Breaking 

'* Circumference. 

Yarns. 

Breaking 

Actual. 

Nominal. 

Core. 

Weight. 

Actual. 

Nominal. 

Weight. 

Ins. 

IllS. 

Ins. 

No. 

Lbs. 

Ins. 

IlIS. 

No. 

Lbs. 

7 

7 

2-35 

108 

187 4OO 

12 

I 3-25 

Il68 

75966 

6 

6 

2.25 

108 

IO4 050 

II 

12.25 * 

IO36 

77 633 

4-937 

4.9 

i -57 

114 

65 409 

10-5 

II.875 

928 

76933 

4-375 

4-5 

i -57 

114 

55 316 

IO 

n -375 

876 

70 533 

3-5 

3-36 

1.27 

114 

34 480 

95 

10.5 

800 

58 766 

3 -i 87 

2.98 

1.17 

114 

28 606 

9 

10.312 

712 

56466 

2-75 

2.68 

.78 

114 

21 846 

8-5 

9 43 7 

640 

42 866 

2-5 

2-45 

.78 

114 

15692 

8 

8.812 

560 

38 500 

2-375 

2.4 

.78 

42 

15 718 

7-5 

8-437 

484 

40000 

2 

2.06 

•39 

II 4 

IO925 

7 

7.812 

436 

32 166 


eight and. Strength of Stnd-linli Chain Cable. 

(English.) 


D 

Diam. 
of each 
Side. 

IMENSTON 

Length 

of 

Link. 

S. 

Width 

of 

Link. 

Weight 

per 

Fathom. 

Admiral ty 
Proof-stress 
(adopted by 
Lloyds’). 

D 

Diam. 
of each 
Side. 

IMENSION 

Length 

of 

Link. 

s. 

Width 

of 

Link. 

Weight 

per 

Fathom. 

Admiralty 
Proof-stress 
(adopted by 
Lloyds’). 

Ins. 

IllS. 

Ins. 

Lbs. 

Tons. 

Ins. 

Ins. 

Ins. 

Lbs. ■ 

Tons. 

•4375 

2.625 

i -575 

11 -3 

3-5 

i -5 

9 

5-4 

121 

405 

•5 

3 

1.8 

13-4 

4-5 

1.625 

9-75 

5-85 

I 42 

47-5 

•5625 

3-375 

2.025 

17.2 

5-5 

i -75 

10.5 

6-3 

164.6 

55-125 

•625 

3-75 

2.25 

21 

7 

i -875 

11.25 

6.75 

189 

6325 

.6875 

4.125 

2-475 

25-4 

8-5 

2 

12 

7.2 

215 

72 

•75 

4-5 

2-7 

30.2 

10.125 

2.125 

12.75 

7-65 

242.8 

81.25 

•875 

5-25 

3 -i 5 

41.2 

13-75 

2.25 

13-5 

8.1 

276.2 

9 I - I2 5 

1 

6 

3-6 

53-8 

18 

2-375 

14.25 

8-55 

303-2 

101.5 

1-125 

6-75 

4 -oS 

69 

22.75 

2-5 

15 

9 

33 6 

112.5 

1.25 

1-375 

75 

8.25 

4-5 

4-95 

84 

101.6 

28.125 

34 

2-75 

16.5 

99 

406.6 

136.125 


Note i. — Safe Working-stress is taken at half Proof-stress, 3.82 tons per sq. inch 
of section. 

2. — Proof-stress and Safe Working - stress for close-link chains are respectively 
two-thirds of those of stud-link chains. 

3. — Proof-stress averages 72 per cent, ultimate strength, and Ultimate Strength 
averages 8 tons per square inch of section of rod or one side of a link. 

Weight of close-link chain is about three times weight of bar from which 
it is made, for equal lengths. 

Karl von Ott , comparing weight, cost, and strength of the three materials, 
hemp, iron wire, and chain iron, concludes that the proportion between cost 
of hemp rope, wire rope, and chain is as 2 : 1 : 3, and that, therefore, for 
equal resistances, wire rope is only half the cost of hemp rope, and a third 
of cost of chains. 


Safe Working Load, of drains. (Molesworth). 


Diameter 
ofIrop. 

Load. 

Diameter 
of Iron. 

Load. 

Diameter 
of Iron. 

Load. 

Diameter 
of Iron. 

Load. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs:* 

Ins. 

Lbs. 

•375 

2240 

•6875 

7 390 

•9375 

13 700 

I.1875 

22 4OO 

•5 

3800 

•75 

8 960 

1 

15 680 

I.25 

24 640 

•5625 

4900 

•8125 

10 280 

1.0625 

17 920 

I- 3 I 25 

26 680 

•625 

6270 

•875 

12 320 

1-125 

20 l6o 

i -375 

30 240 


























































ROPES AND CHAINS. 


169 


Breaking Strain, and. Proof of Chain Cables. 


Diam. 
of Chain. 

breaking 

Strain. 

Diam. 
of Chain. 

Breaking 

Strain. 

Diam. 
of Chain. 

Breaking 

Strain. 

Diam. 
of Chain. 

Breaking 

Strain. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

I 

67 7OO 

I.1875 

92 940 

i -5 

143 IOO 

2 

243 180 

I.0625 

75640 

I.25 

102 160 

1.625 

165 920 

2.125 

272 580 

I.125 

84 IOO 

1-375 

121 840 

i -75 

2l6 120 

2.25 

303 280 


Proof-stress is 50 per cent, of estimated strength of weakest link and 46 
per cent, of strongest. 

Comparison of Wire Ropes and Tarred Hemp Rope, 
Hawsers, and Cables. 


Diam¬ 

eter. 

Circum. 

COJ 

Safe 

Load. 

LRSE L 

Ro 

J- G 

A X 

M £ 
m 

AID. 

)es. 

u -3 

S3 S3 

O 33 
£ 

<n 

Haws’rs. 

JSg 

Cables. 

sf 

Js S 
Hi 
in 

Diam¬ 

eter. 

FI 

Safe 

Load. 

N'E LAI 
Ropes. 

03 

s- 'a 

If 

in 

D. 

Haws’rs. 

®J8 

£ S 

Three g, 
j Strands. “ 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

•25 

.78 

425 

X.25 

— 

— 

— 

•5 

1 875 

3.12 

2.87 

— 

•3125 

I 

690 

2-43 

2.25 

3-32 

— 

■5625 

2 42O 

3-56 

3-25 

4.87 

•375 

1-25 

825 

2.68 

2-375 

3-5 

— 

.625 

2 9OO 

3-93 

3.62 

5-25 

•5 

i -375 

1 600 

2.87 

2.62 

3-87 

— 

•75 

4320 

4.81 

4-37 

6-37 

-5625 

*•75 

2 800 

3.81 

3-5 

5-18 

— 

•875 

5 7 °° 

5-5 

5 

7-25 

.6875 

2.125 

3 800 

4-75 

4-25 

6.12 

— 

I 

8 200 

7-25 

6.25 

8-75 

•75 

2-375 

4 400 

5-25 

4.87 

7 

— 

1-125 

10 IOO 

8.18 

7 

9-5 

•875 

2.625 

6 150 

6.12 

5-75 

8 

8 

1.25 

13 600 

8.81 

8.06 

II 

I 

3 

8 400 

6.62 

6.12 

8.62 

8.62 

i -5 

17 500 

IO 

9-75 

12.5 

1.25 

3-75 

13400 

8.81 

8-5 

10.93 

10.93 

1.625 

21 800 

11.18 

10.93 

— 

i -375 

4-25 

16H00 

9.87 

9-56 

12.25 

12. 12 

i -75 

27 000 

12.5 

12. 12 

— 

i -5 

4.625 

20 160 

io -75 

10.5 

13 

13.12 

1 875 

32500 

— 

— 

— 

1.625 

5 

24600 

— 

11.87 

11 56 

n.75 

2 

37000 

— 

— 

— 


I11 above table, determination of circumference of rope, etc., is based upon 
Breaking Weight or Tensile resistance of wire being reduced by one fourth, 
and ultimate resistances of rope, etc., are reduced one third. 


Result of Experiments upon 'Wire Rope at XT. S. NTavy 
Yard, Washington. (J A. Roebling's Sons.) 


Circumference. 

13 'C 

O . 

Weight 

Breaking 

Circumference. 

JG -r-* 

2 2 c 

0 >> . 

.-°o 

S o> 


Breaking 

Actual. 

Nom¬ 

inal. 

> s 2 

Q > 

I oot. 

Weight. 

Actual. 

Nom¬ 

inal. 

> 

.2 jr ^ 

® P..° 

Weight. 

Ins. 

Ins. 

No. 

No. 

Lbs. 

Lbs. 

Ins. 

Ins. 

No. 

No. 

Lbs. 

Lbs. 

4-9375 

4.9 

J 9 

II 

3-14 

65 409 

2-375 

2.4 

7 

13 

.14 

I57i8 

4-375 

4-5 

J 9 

13 

2.15 

55 316 

2.1875 

2.12 

7 

14 

.11 

14478 

3-9375 

3 - 9 1 

19 

14 

2.0875 

4442° 

2 

2.06 

!9 

19 

,1 

IO925 

3-5 

3 - 3 6 

19 

14 

I-I 525 

34 840 

1-9375 

1.9 

7 

14 

.1 

IO Il8 

3-1875 

2.98 

19 

15 

I.09 

28606 

i -75 

I.85 

7 

17 

.07 

7 880 

2-75 

2.68 

19 

17 

I.O275 

21 846 

1-4375 

i -45 

19 

20 

.06 

5687 

2.6875 

2.56 

7 

13 

1.0225 

18 810 

1-3125 

i- 3 i 

7 

18 

•05 

4 428 

2-5 

2-45 

19 

18 

.14 

15692 

IO 

N 

H 

H 

1.11 

7 

19 

•035 

3 729 


To Compute Circumference of Wire Rope with Hemp 
Core, of Corresponding Strength to Hemp Rope, and 
of Hemp Rope to Circumference of "Wire Rope. 

Rule i. —Multiply square of circumference of hemp rope by .223 for iron 
wire and .12 for steel, and extract square root of product. 

2.—Multiply square of circumference of hemp-core wire rope by 4.5 for 
iron wire and 8.4 for steel wire. 

Example. —What are the circumferences of an iron and steel wire rope corre¬ 
sponding to one of hemp-core, having a circumference of 8 ins. ? 

V8 2 X .223 = 3.78 ins. iron , and V8 2 X -12 = 2.77 ins. steel. 

P 
































































170 


ROPES, HAWSERS, AND CABLES. 


HOPES, HAWSERS, AND CABLES. 

Ropes of hemp fibres are laid with three or four strands of twisted fibres, 
and are made up to a circumference of 12 ins., and those of four strands up 
to 8 ins. are fully 16 per cent, stronger than those of three strands. 

Hawsers are laid with three or four strands of rope. Cables are laid with 
but three strands of rope. Hawsers and Cables, from having a less propor¬ 
tionate number of fibres, and from the irregularity of the resistance of their 
fibres in consequence of the twisting of them, have less strength than ropes, 
difference varying from 35 to 45 per cent., being greatest with least circum¬ 
ference, and those of three strands up to 12 ins. are fully 10 per cent, strong¬ 
er than those having four strands. 

Tarred ropes, hawsers, etc., have 25 per cent, less strength than white 
ropes; this is in consequence of the injury fibres receive from the high tem¬ 
perature of the tar, viz. 290°. 

Tarred hemp and Manila ropes are of about equal strength, and have from 
25 to 30 per cent, less strength than white ropes. 

White ropes are more durable than tarred. 

The greater degree of twisting given to fibres of a rope, etc., less its 
strength, as exterior, alone resists greater portion of strain. 

Ultimate strength of ropes varies from 7000 to 12000 lbs. per square inch 
of section, according as they are wetted, tarred, or dry. One sixth of ulti¬ 
mate strength is a safe working load = 1166 to 2000 lbs. per square inch. 


Units for computing Safe Strain, that may he home hy 
New Ropes, Hawsers, and. Cables. (U . S. Navy.) 


Descrip¬ 

tion. 






Ropes. 


Hawsers. 

Cables. 

Circumference. 

White. 

Tarred. 

White. 

Tarred. 

White. 

Tarred 




3 strands. 

4 strands. 

3 str’ds. 

4 str’ds. 

3 str’ds. 

3 str’ds. 

3 str’ds. 

3 str’ds. 

Ins. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

2 -5 

to 

6 

II40 

1330 

— 

— 

600 

— 

— 

— 

6 

U 

8 

IO90 

1260 

— 

— 

570 

— 

510 

— 

8 


12 

1045 

880 

— 

— 

530 

— 

530 

— 

12 


18 

— 

— 

— 

— 

550 

— 

550 

— 

18 

U 

26 

— 

— 

— 

— 

— 

— 

560 

— 

2-5 

u 

5 

— 

— 


1005 

— 

460 

— 

— 

5 


8 

— 

— 

825 

940 

— 

480 

— 

— 

8 


12 

— 

— 

780 

820 

— 

505 

— 

505 

12 


18 

— 

— 

— 

— 

— 

— 

— 

525 

18 


26 

— 

— 

— 

— 

— 

— 

— 

550 

2-5 


6 

8lO 

950 

— 

— 

440 

— 

— 


6 


12 

760 

«35 

— 

— 

465 

— 

510 

— 

12 


18 

— 

— 

— 

— 

— 

— 

535 

— 

18 


26 

— 

— 

— 

— 

— 

— 

560 

— 


White 


Tarred 


u 

Manila 


Illustration.— What weight can be borne with safety by a Manila rope of 3 
strands, having a circumference of 6 inches ? (See Rule,page 167.) 

6 2 x 760 = 27360 lbs. 

When it is required to ascertain weight or strain that can be borne by 
ropes , etc., in general use , preceding Units should be reduced from one third 
to two thirds , in order to meet their condition or reduction of their strength 
by chafing and exposure to weather. Molesworth’s table is based upon a 
reduction of three fourths. 

Illustration.— What weight can be borne by a tarred hawser of 3 strands, 10 
inches in circumference, in general use? 


io 2 X (505 — 505 3) = 100 X 366.67 = 33 667 lbs. 















ROPES, HAWSERS, AND CABLES. 


I7I 


Destructive Strength, of Tarred. Hemp Ropes. 

(D. K. Clark.) 




Register. 



Register. 

Circum. 

Diam. 

Common 

Russian 

Circum. 

Diam. 

Common 

Russian 



Cold. 

Warm. 



Cold. 

Warm. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

3 

•95 

7 390 

8 620 

5-5 

i -75 

24800 

29 120 

3-5 

1.11 

II 200 

11 760 

6 

1.91 

28 985 

33150 

4 

1.27 

13 IOO 

15 340 

6-5 

2.07 

34 030 

40 550 

4-5 

i -43 

16330 

19440 

7 

2.24 

40320 

4704I 

5 

i -59 

19580 

23990 

8 

2-54 

52480 

6l 420 


Specimens f urnished by National Association of Rope and Twine Spinners , 
As tested by Mr. Kirkaldy. 


Rope. 

Circum¬ 

ference. 

Weight 
per Lb. 

Extreme 

Strength. 

Breaking 
Weight 
per lb.per 
Fathom. 

Extensic 
at Stres 
pe 

1O00 lbs. 

n in 50 in 
s per lb. 
r Fathom 
2000 lbs. 

s. Length 

Weight 

of 

3000 lbs. 


Ins. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Ins. 

Russian rope ... 48 thr’ds. 

5.26 

.926 

II 088 

1933 

5-29 

— 

— 

Machine yarn. .. 50 “ 

H 5-37 

.891 

II 514 

2152 

4-53 

6.56 

— 

Hand-spun yarn, 51 “ 

5-39 

1.006 

l8 278 

3024 

4.46 

5-91 

6.63 


Breaking Strength of Tarred Hemp Hopes. (Mr. Glynn.) 


Circum. 

Diam. 

Old M 

Common 

Hemp. 

ethod. 

Best 

Russian. 

By Re 

Cold. 

gister. 

Warm. 

Circum. 

s 

CS 

s 

Old M 

Common 

Hemp. 

ethod. 

Best 

Russian. 

By Re 

Cold. 

gister. 

Warm. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

3 

•95 

5 056 

6248 

7 392 

8624 

5-5 

i -75 

15 456 

18 414 

24 797 

29 120 

3-5 

1.11 

7 466 

8 668 

II 200 

II 760 

6 

1.91 

l8 I44 

21 6lO 

28 986 

33150 

4 

1.27 

8 780 

IO460 

13 I °4 

17 8lO 

6-5 

2.07 

20518 

23 6lO 

34630 

40 544 

4-5 

i -43 

10300 

12432 

16330 

19443 

7 

2.24 

22 938 

27 462 

40320 

47040 

5 

i -59 

i 3 3 2 8 

15 859 

20496 

23990 

8 

2-54 

26 680 

32 032 

52483 

6l 420 


To Compute Strain that may he home with safety- by 
new Ropes, Hawsers, and Cables. 

Deduced from experiments of Russian Government upon relative strength 
of different Circumferences of Ropes, Hawsers, etc. 

U. S. Navy test is 4200 lbs. for a White rope of three strands of best Riga 
hemp, of 1.75 inches in circumference (— 17 000 lbs. per square inch of fibre), 
but in preceding table (page 166) 14000 lbs. is taken as unit of strain that 
may be borne with safety. 

Rule. —Square circumference of rope, hawser, etc., and multiply it by 
Units in table. 

To Compute Circumference of a Rope, Hawser, or Cable 

for a Griven Strain. 

Rule.—D ivide strain in pounds by appropriate units in preceding table, 
and square root of product will give circumference of rope, etc., in ins. 

Example i.— Stress to be borne in safety is 165 550 lbs.; what should be circum¬ 
ference of a tarred cable to withstand it? 

165552-4-550 = 301, and ^301 = 17.35 ins - 

2._What should be circumference of a Manila cable to withstand a strain, in 

general use , of 149 336 lbs. ? 

Assuming circumference to exceed 18 ins., unit = 560. 

149 336 4- (560 — 560 4 - 3) = 400, a ^d y /400 = 20 ins. 


















































172 


ROPES, HAWSERS, AND CABLES. 


To Compute Weight of Ropes, Hawsers, and Oalales. 


Rule. —Square circumference, and multiply it by appropriate unit in 
following table, and product will give weight per foot in lbs.: 


HAWSERS. 

ROPES. CABLES. 

3-strand Hemp.032 .031 .031 

3-strand tarred Hemp, .042 .041 .041 

3-strand Manila.032 .031 .031 


, HAWSERS. 

ROPES. CABLES. 

4-strand Hemp.033 — — 

4-strand tarred Hemp, .048 — — 

4-strand Manila.035 .034 .034 


Units for Thread Ropes is same as that for Ropes of like material. 
Example.— What is weight of a coil of 10-inch Manila hawser of 4 strands of 120 


fathoms ? 


io 2 X -034 = 3-4, and 120 X 6 X 3.4 = 2448 lbs - 


Weight and Strength, of Hemp and. Wire Ropes. 

(Molesworth.) 

C 2 y = W; C 2 7 r = L; C 2 » = S; andyJ^ = C. 

C representing circumference in ins., W weight of rope in lbs. per fathom, 
L working load in tons , and 3 destructive stress in tons. 

VALUES OF y, X , AND k. 


ROPES. 

y 

X 

k 

ROPES. 

V 

X 

k 

Hawser, hemp. 

•131 

.117 

•235 

• 207 



Warm register, hemp 
Manila hawser. 

_ 

• 7 

. Il6 

u 



.177 

•155 

.87 

.89 

.27 
. IQ 

•045 

•033 

.29 

•45 

Tarred hawser, hemp. 

“ cable, “ . 

Cold register, “ . 

.22 

•15 

.6 

•037 

“ cable. 

Iron rope . 

1.8 

. I 

Steel “ . 

2.8 


To Compute Circumference of Hemp or Wire Rope 
for Fore or Alain Standing Rigging. (U. S. Navy.) 

Rule. —To length of mast between partners and deck, add half extreme 
breadth of beam of vessel and divide sum by half extreme breadth. Mul¬ 
tiply quotient by half square root of tonnage (OM) and extract square root 
of product. 

For Mizzen, take .74 of Fore and Main. 


Example. — Required circumference of hemp rope, for main-mast of a vessel 
having a breadth of beam of 45 feet and a burden of 3213 tons? 


Extreme length of mast.••94-4 feet. 

Depth of hold, or total bury of mast, 21.4 feet. 

Head. 15 “ 36.4 u 

Breadth of beam, 45 feet. 58 “ 

58 — -T- — = 3.58, and y/ (3.58 X ^ 32I3 N ) = V 101.46 = 10.n ins. 

22 \ 2 / 


Then if circumference for a wire rope is required, see table, page 164. 

Thus, a hemp rope 10 ins. in circumference has equivalent strength of an iron 
wire rope of 4 ins. and a steel rope of 3.25-f- ins. 


Galvanized Iron Wire .—Experiments at Navy Yard, Washington, gave for flex¬ 
ibility a mean loss of 30 per cent., and for tensile strength a like loss of 13.5 per 
cent. 

Relative Dimensions of Hemp Rope and Iron and Steel 
Wire Rope. (U. S. Navy.) 

Circumference in Inches. 

Hemp. 2.5 3.125 4 4-5 5-25 6.5 7.75 8.5 9.5 11 11.75 * 3-5 16.5 

Iron.. 1.25 1.625 2 2.125 2.5 3 3.5 4 4.5 5 5.5 6 7 

































ANCHORS, CABLES, ETC. 


173 


ANCHORS, CABLES, ETC. 

Anchors, Chains, etc., for a Griven Tonnage. 
(American Shipmasters Association.') 

SAILS. 


Tonnage 
computed as 
per Rule.* 

Bow 

With¬ 

out 

Stock. 

era . 

Admi¬ 

ralty 

Test. 

Anchors 

Inch 

Stream. 

iding Stc 

Kedge. 

ck. 

2d 

Kedge. 

Diameter. 

Ch.a 

to 

p 

IN Cabli 

Admi¬ 

ralty- 

Test. 

e.—Stub 
Weigh 

Stud. 

. 

t per Fa 

Short 

Link. 

thorn. 

Eng¬ 
lish.+ 


Lbs. 

Tons. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Faths. 

Tons. 

Lbs. 

Lbs. 


75 

616 

7 

168 

84 

— 

.8125 

90 

II 

40 

42 

35 

100 

728 

8 

196 

112 

— 

.875 

105 

!3 

44 

48 


125 

840 

9 

224 

112 

— 

•9375 

105 

15 

51 

55 

48 

150 

952 

10 

280 

I40 

— 

1 

120 

17-5 

59 

63 

54 

175 

1036 

11 

336 

l68 

— 

1.0625 

120 

20 

66 

70 


200 

1120 

12 

392 

196 

— 

1-125 

120 

22.5 

75 

79 

68 

250 

1288 

13 

448 

224 

112 

1.1875 

135 

25 

82 

88 

— 

300 

1456 

14 

504 

252 

126 

1.25 

135 

28 

9 i 

98 

84 

350 

1624 

i 5-5 

560 

280 

I40 

1-3125 

150 

31 

100 

106 


400 

1848 

17 

616 

3 ° s 

154 

1-3125 

150 

31 

100 

106 

— 

450 

I904 

18.5 

672 

33 6 

l68 

i -375 

165 

37 

115 

118 

102 

500 

2016 

20 

784 

392 

196 

1-4375 

165 

40 

120 

— 

— 

600 

2352 

22 

896 

448 

224 

i -5 

l8o 

44 

132 

— 

122 

700 

2688 

24 

IO08 

504 

252 

1-5625 

180 

47 

145 

— 

— 

800 

3024 

26 

1120 

560 

280 

1.625 

l8o 

5 i 

156 

— 

T 43 

900 

3248 

28 

1232 

616 

3 ° 8 

1.6875 

l8o 

55 

162 

— 


1000 

3584 

29-5 

1344 

672 

336 

i -75 

l8o 

59 

175 

— 

166 

1200 

3808 

3 i 

1456 

738 

364 

i-875 

180 

63 

189 

— 

191 

1400 

4032 

32.5 

1568 

784 

392 

1-9375 

180 

67 

205 

— 

— 

1600 

4256 

34 

1680 

840 

420 

2 

l8o 

72 

219 

— 

— 

1800 

4480 

35-5 

1792 

896 

448 

2 

180 

72 

240 

— 

217 

2000 

4704 

37 

1904 

952 

504 

2.0625 

l8o 

81 

— 

— 

— 

2500 

5040 

39 

2128 

1120 

560 

2.125 

l8o 

86 

— 

— 

244 

3000 

5376 

4 i 

2353 

1232 

616 

2.1875 

l8o 

96 

— 

— 

— 


t Brown, Lennox, & Co. 


To Compute Tonnage. 

Take dimensions as follows: Length. — From after-side of stem to for¬ 
ward-side of stern-post, measured on spar or upper deck in vessels having 
two decks and under, and on main deck in vessels having three or more 
decks. Breadth. —Extreme at widest point. Depth. —At forward coaming 
of main hatch, from top of ceiling at side of keelson to under side of deck. 

Then multiply these dimensions together, divide product by 100, and 
take .75 of quotient. 

All vessels to have 2 bowers and 1 each stream and kedge anchor, and for 
a tonnage exceeding 1400 a third bower is recommended. 

Hawsers and Warps to be 90 fathoms in length. 

Shrcmcls. 

Square-rigged. Hemp. —5.75 ins. in diameter for a tonnage of 75, in¬ 
creasing progressively up to 12.75 ins. for 3000 tons. 

Fore-and-aft rigged. From .25 to 1 inch in diameter progressively 
greater than for square-rigged. 

Wire. —One half diameter of hemp, increasing very slightly as tonnage 
increases. Thus, for 3000 tons, 12.75 ins. for hemp and 6.875 ins. for wire. 

p* 




















174 


ANCHORS, CABLES, ETC. 


{American Shipmasters' Association.) 
STEAM. 

Anchors. 


& 

Bowers. 

Including Stock. 





Weight per Fath. 


I • 


g 

© 

© 

Diam- 

d 

»- a> 

Diam. 


, . 


fi c 5 O 
O fa p- O) 
bSjft 

C5 ^ 

Witl 

out 

Stocl 

£ Af cc 
■« 

Cj 

© 

Ini 

VI 

bfl 

© 

\4 

•a J? 

eter. 

tx 

G 

© 

1-1 

h h 

<T* 

Stream. 

Stud. 

Shor 

Link 

a fo 


Lbs. 

Tons. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Faths. 

Tons. 

Ins. 

Lbs. 

Lbs. 


100 

336 

4.9 

112 

— 

— 

.6875 

105 

8.1 

•5 

— 

— 

25 

150 

448 

6.4 

I96 

— 

— 

.8125 

120 

11.9 

•5625 

40 

42 

35 

200 

616 

7.6 

224 

— 

— 

•875 

120 

13.8 

•5625 

44 

48 

— 

250 

672 

8.2 

280 

— 

— 

•9375 

120 

15-8 

.625 

5 i 

55 

48 

300 

8l2 

9-5 

308 

— 

— 

1 

120 

18 

.625 

59 

63 

54 

350 

924 

10.4 

336 

— 

— 

1.0625 

120 

20.3 

.6875 

66 

70 

— 

400 

1120 

12 

532 

252 

— 

1-125 

135 

22.8 

.6875 

75 

79 

68 

450 

1344 

13-9 

560 

280 

— 

1.1875 

135 

25-4 

•75 

82 

88 

— 

500 

15X2 

15.2 

672 

336 

— 

1.25 

150 

28.1 

•75 

9 i 

98 

84 

600 

1708 

16.7 

738 

364 

— 

I - 3 I2 5 

150 

3 i 

.8125 

100 

106 

— 

700 

1876 

18 

784 

392 

— 

i -375 

165 

34 

.8125 

115 

118 

104 

800 

2026 

19 

896 

448 

224 

1-4375 

165 

37-2 

•875 

120 

— 

— 

900 

235 2 

21.6 

IO08 

504 

252 

i -5 

l8o 

40.5 

.875 

132 

— 

122 

1000 

2632 

23-5 

1120 

560 

280 

1-5625 

l8o 

44 

•9375 

145 

— 

— 

1200 

2856 

25.2 

II76 

588 

308 

1.625 

l8o 

47-5 

•9375 

156 

— 

I 43 

1400 

3108 

26.9 

1232 

616 

308 

1.6875 

l8o 

51.2 

1 

162 

— 


1600 

3360 

28.6 

1344 

672 

336 

i -75 

l8o 

55 -i 

1 

175 

— 

166 

1800 

35*4 

30.1 

1456 

738 

364 

1.8125 

l8o 

59 -i 

1.0625 

189 

— 

— 

2000 

3808 

3 i -6 

1512 

766 

364 

i -875 

l8o 

63-3 

1.06251 

205 

— 

191 

2300 

4088 

33-4 

1568 

784 392 

1-9375 

180 

67.6 

1-125 

215 

— 


2600 

4256 

34-5 

1624 

812 

392 

2 

270 

72 

1-125 

240 

— 

217 

3000 

4480 

35-7 

l68o 

840 420 

2.0625 

270 

76.6 

1.1875 

— 

— 

— 

3500 

4592 

37 

1792 

896 476 

2.125 

270 

81.3 | 

1.1875 

— 

— 

244 

4000 

4816 

38 

i960 

952 

504 

2.1875 

270 

86.x 

1.25 

— 

— 


4500 

5040 

39 ‘ 2 

2128 

^64:532 

2.25 

270 

91.1 

1.25 

— 


— 

5000 

5264 

4 i 

2352 

1120 

560 

2.3125 

270 

96 

1-3125 1 

— 


— 


: Brown, Lennox, & Co. 


ANCHORS AND KEDGES. 

(U. S. Navy.) 

To Compute Weiglit of a, Bower A-nclior for a Vessel 
of a given Character and. Rate. 

Rule. —Multiply approximate displacement in tons, by unit in following 
table, and product will give weight in lbs., inclusive of stock. 


Units to determine Weights and T4umber of Anchors 

or Kedges. 


Displacement 
of Vessel in 
Tons. 

Unit. 

Bower. 

Sheet. 

Stream 

©* 

b£ 

r O 

© 

W 

Displacement 
of Vessel in 
Tons. 

Unit. 

Bower. 

Sheet. 

Kedge. 

Over 3700 

i -75 

2 

2 

I 

4 

Over 1500 . . 

2-5 

2 

2 

3 

“ 2400 

2 

2 

2 

I 

3 

“ 900 .. 

2-75 

2 

I 

3 

“ 1900 

2.25 

2 

2 

I 

3 

900 and under 

3 

2 

I 

2 


Example. —Tonnage of a bark-rigged steamer is 1500. 


1500 X 2.5 = 3750 lbs., weight of anchor. 

Bower and Sheet Anchors should be alike in weight. 

Stream Anchors and Kedges are proportional to weight of bowers. Thus, 
Stream. Anchor .25 weight. Kedges. — If 1, .125 weight; if 2, .16 and .1 
weight; if 3, .16, .125, and .1 weight. 























































ANCHORS, CABLES, ETC.-TONNAGE. 


175 


To Compute Diameter of a Chain. Oa"ble corresponding 
to a (Given Weight; of Anchor. 

(Z 7 . S. Navy.) 

Rule.— Cut off the two right-hand figures of the anchor’s weight in lbs., 
multiply square root of remainder by 4, and result will give diameter of 
chain in sixteenths of an inch. 

Example.— The weight of an anchor is 2500 lbs. 

V25.00 X 4 = 20 sixteenths — 1.25 ins. 

Note.— Diam. of a messenger should be .66 that of the cable to which it is applied. 
Lengths of Chain Cables for each Anchor. 
iff. S. Navy.) 


Weight of Anchor. 

Bower. 

Sheet. 

Stream. 

Weight of Anchor. 

Bower. 

Sheet. 

Stream. 

Lbs. 

Fathoms. 

Fathoms. 

Fathoms. 

Lbs. 

Fathoms. 

Fathoms. 

Fathoms. 

Under 800 

60 

60 

60 

Over 2000 

120 

120 

9 ° 

Over 800 

90 

90 

60 

“ 3000 

120 

120 

90 

“ 1200 

90 

90 

75 

“ 5000 

120 

120 

105 

“ 1600 

io 5 

105 

75 

“ 75 oo 

135 

135 

105 




ANCHORS. 





From Experiments of a Joint Committee of Representatives of Ship¬ 
owner's and Admiralty of Great Britain. 

An anchor of ordinary or Admiralty pattern, Trotman or Porter’s im¬ 
proved (pivot fluke), Honiball, Porter’s, Aylin’s, Rodgers’s, Mitcheson’s, and 
Lennox’s, each weighing, inclusive of stock, 27000 lbs., withstood without 
injury a proof strain of 45000 lbs. 

Breaking weights between a Porter and Admiralty anchor, as tested at 
Woolwich Dock-yard, were as 43 to 14. 

Comparative Resistance to Dragging. 

Trotman’s dragged Aylin’s, Honiball’s Mitcheson’s and Lennox’s ; Aylin’s 
and Mitcheson’s dragged Rodgers’s; and Rodgers’s and Lennox’s, dragged 
Admiralty’s. _ 

TONNAGE OF VESSELS. 

To Compute Tonnage of 'Vessels. 

For Laws of United States of America, with amendments of 1882 relative 
to Steam-vessels, see Mechanics’ Tables, with rule and illustrated diagrams, 
by Chas. H. Haswell, 3d edition, Harper & Bros., New York, 1878. 

English Registered. Tonnage. (New Measurement.) 

Divide length of upper deck between after-part of stem and fore part of stern- 
post into 6 equal parts, and note foremost, middle, and aftermost points of division. 
Measure depths at these three points in feet and tenths of a foot; also depths from 
under-side of upper deck to ceiling of limber-strake; or in case of a break in the 
upper deck, from a line stretched in continuation of the deck. For breadths, divide 
each depth into 5 equal parts, and measure the inside breadths at following points, 
viz. :—At .2 and .8 from upper deck of foremost and aftermost depths; and from 
.4 and .8 from upper deck of amidship depth. Take length at half amidship depth 
from after-part of stem to fore-part of stern-post. 

Then, to twice amidship depth add foremost and aftermost depths for sum of 
depths , and add together foremost upper and lower breadths, 3 times upper breadth 
with lower breadth at amidship, and upper and twice lower breadth at after division 
for sum of breadths. \ 

Multiply together sum of depths, sum of breadths, and length, and divide product 
by 3500, which will give number of tons. 

If the vessel has a poop or half-deck, or a break in upper deck, measure inside 
mean length, breadth, and height of such part thereof as may be included within 
the bulkhead; multiply these three measurements together, divide product by 92.4, 
and quotient will give number of tons to be added to result as above ascertained. 


















TONNAGE OF VESSELS 


176 


For Open Vessels. —Depths are to be taken from upper edge of upper strake. 

For Steam Vessels .—Tonnage due to engine-room is deducted from total tonnage 
computed by above rule. To determine this, measure inside of the engine-room 
from foremost to aftermost bulkhead; then multiply this length by amidship depth 
of vessel, and product by inside amidship breadth at .4 of depth from deck, and 
divide final product by 92.4. 

The volume of the poop, deck-houses, and other permanently enclosed spaces, 
available for cargo or passengers, is to be measured and included in the tonnage, 
but following deductions are allowed, the remainder being the Register tonnage. 

Deductions .—Houses for the shelter of passengers only; space allotted to crew 
(12 square feet in surface and 72 cube feet in volume for each person); and space 
occupied by propelling power. 

Approximate Fimle. 

Gross Register .—Tonnage of a vessel expresses her entire cubical volume in tons 
of 100 cube feet each, and is ascertained by following formula : 

^ ^ ^ t= Gross tonnage, and — - ^ ^ c = Register tonnage. L representing length 
100 100 

of keel between perpendiculars, B breadth of vessel, and D depth of hold, all in feet. 


Bia.ild.ers’ Measurement. 


(L-.6 B) X B X -5 B 
94 


Tonnage. 


Fore-perpendicular is taken at fore-part of stem at height of upper deck. 
Aft-perpendicular is taken at back of stern-post at height of upper deck. 

In three-deckers, middle deck is taken instead of upper deck. 

Breadth is taken as extreme breadth at height of the wales, subtracting differ¬ 
ence between thickness of wales and bottom plank. Deductions to be made for 
rake of stem and stern. 


18 /Girth -f- BreadthX 2 

Iron Vessels. --(- 1 - X length = Gross tonnage. 

10000 \ 2 / 

Length measured on upper deck, between outside of outer plank at stem and 
the after-side of stern-post and rabbet of stern-post, at point where counter-plank 
crosses it. Girth measured by a chain passed under bottom from upper deck at 
extreme breadth, on one side, to corresponding point on the other. 


Register tonnage 


L X B X D 

100 


X c. 


C representing a coefficient for vessels as 


follows: 


Ships of usual form.... 
Clippers and Steamers 


. 7 

2 decks... .65 

3 “ -68 


Yachts above 60 tons.... 

Small vessels .[ shar Pv- 
(very sharp 


•5 

45 

•4 


TJnits for Measurement and. Dead-weight Cargoes. 

(C. Mackrow, M. S. N. A.) 

To Compute Approximately for an Average Length of Voyage the Measure¬ 
ment Cargo, at 40 feet per 'Ton, which a Vessel can carry. 

# Rule.—M ultiply number of register tons by unit 1.875, and product will 
give approximate measurement cargo. 

To Compute Approximately Dead-weight Cargo in Tons which a Vessel can 
carry on an Average Length of Voyage. 

Rule.—M ultiply number of register tons by 1.5, and product will give 
approximate dead-weight cargo required. 

With regard to cargoes of coasters and colliers, as ascertained above, about 
10 per cent, may be added to said results, while about 10 per cent, may be 
deducted in cases of larger vessels on longer voyages. 












TONNAGE OF VESSELS. 


1 77 


In case of measurement cargoes of steam-vessels, spaces occupied by ma¬ 
chinery, fuel, and passenger cabins under the deck must be deducted from 
space or tonnage under deck before application of measurement unit thereto. 

In case of dead-weight cargoes, weight of machinery, water in boilers, and 
fuel must be deducted from whole dead weight, as ascertained above by 
application of dead-weight unit. 

The deductions necessary for provisions, stores, etc., are allowed for in 
selection of the two units. 

7 T o Ascertain Weight of Cargo for an Average Length of Voyage. (Moorsom.) 

Deduct tonnage of spaces of passenger accommodations from net register 
tonnage, and multiply remainder by 1.5. 

Average space for each ton weight of cargo on such a voyage 67 cube feet. 


F'reiglit Tonnage or Measurement Cargo. 

Freight Tonnage or Measurement Cargo is 40 cube feet of space for cargo, 
and it is about 1.875 times net register tonnage less that for passenger space. 


Royal Thames Yaclat Clxilo. 

Measure length of yacht in a straight line at deck from fore-part of stem to after¬ 
part of stern-post, from which deduct extreme breadth (measured from outside of 
outside planking), both in feet; remainder is length for tonnage. Multiply length 
for tonnage by extreme breadth, that product by half extreme breadth, divide re¬ 
sult by 94, and quotient will give tonnage. 

If any part of stem or stern-post projects beyond length as taken above, such 
projection or projections shall, for purpose of computing tonnage, be added to length 
taken as before mentioned. 

All fractional parts of a ton are to be considered as a ton. 

Measurements to be taken either above or below main wales. 


L —B X B X -5 B 
94 


= Tons. 


L representing length and B breadth , in feet. 


Ooriiathian and. IS’e'w Thames Yacht: Clnh. 


Measure length and breadth as in foregoing rule, and depth to top of covering 
board; multiply length, breadth, and depth together, divide result by 200, and quo¬ 
tient will give tonnage. 


L X B X D 


= Tons. 


200 


Suez Canal Tonnage. 

Gross Tonnage .—Spaces under tonnage deck, below tonnage and uppermost deck, 
all covered or closed - in spaces, such as poop, forecastle, officers’ cabins, galley, 
cook, deck, and wheel houses, and all inclosed or covered-in spaces for working the 
vessel. 

From which are to be deducted berthing accommodations for crew, not including 
spaces fo-r stewards and passengers’ servants; berthing accommodations for officers, 
except captain; galleys, cook-houses, etc., used exclusively for crew, and inclosed 
spaces above uppermost deck, designed for working the vessel. In none of these 
spaces can passengers be berthed or cargo carried, and total deduction under all of 
these spaces must not exceed 5 per cent, of gross tonnage. 

In steamers with standing coal-bunkers, English rule may bo followed, or owner 
may elect to have tonnage of his vessel computed by “Danube rule,” which is an 
allowance of 50 per cent, above space allowed to machinery in side-wheel steamers 
and 75 in screw steamers. 

In no case, however, except with tow-boats, must deduction for propelling power 
exceed 50 per cent, of gross tonnage. 




178 


WORKS OF MAGNITUDE. 


WORKS OF MAGNITUDE. 

YAm eric an. 

-A.qu.ecl'u.cts, Roads, and^ Railroads. 

Croton Aqueduct, N. Y. — Has a section of 53.34 square feet and capacity of 
100000000 to 118 000 000 gallons per day. and from Dam to Receiving Reservoir is 
38.134 miles in length. 

Aqueduct, Washington.— Cylinder of masonry 9 feet in diameter. Stone arch 
over Cabin John’s Creek, 220 feet span, 57.25 feet rise. 

National Road.— Over the Alleghany Mountains, Cumberland to Illinois Town, 
650.625 miles in length, and 80 feet in width. Macadamized for a width of 30 feet. 

Illinois Central Railroad.— Chicago to Cairo, length 365 miles, Centralia to Dun- 
leith 344 miles, total 709 miles. 


Bridges. 

Suspension Bridge, Niagara River.—Wire, Span 1042 feet 10 ins. 

Suspension Bridge, New York and Brooklyn. — Length of river span 1595 feet 6 
ins.; of each land span 930 feet; length of Brooklyn approach 971 feet; of N. Y. 
approach 1562 feet 6 ins.; total length of bridge 5989 feet; width 85 feet; number 
of cables 4; diameter of each cable 15.5 ins.; each consisting of 6300 parallel steel 
wires No. 7 gauge, closely laid and wrapped to a solid cylinder; ultimate strength 
of each- cable 11200 tons; depth of tower foundation below high water, Brooklyn, 
45 feet—New York 78 feet; towers at high-water line 140X59 feet; towers at roof 
course 136X53 feet; total height of towers above high water 277 feet; clear height 
of bridge in centre of river span above high water, at 50 0 , 135 feet; height of floor 
at towers above high water 119 feet 3 ins.; grade of roadway 3 feet in 100; anchor¬ 
ages, at base 129X119 feet, at top 117X104 feet; weight of each anchor-plate 23 tons. 

Iron Pipe Bridge over Rock Creek.—200 feet span, 20 feet rise. Arch of 2 lateral 
courses of cast-iron pipe, 4 feet internal diameter, and 1 inch thick. These pipes 
conveying the water not only sustain themselves over the great span, but support 
a street road and railway. 

Iron Bridge over Kentucky River near Shakers’ Ferry, Md.—3 spans, each 375 
feet, and 275.5 feet above low water. 

Bridge on line of New York, Erie, and Western Railroad across the Kinzua .— 
Of iron; length 2060 feet; central span 301 feet in height. 

Iron Truss .—Cincinnati and Southern Railway, over Ohio River, 519 feet. 

Foreign. 

Pyramids, Statues, etc. 

Pyramid of Cheops, Egypt.—Length of side at base 762 feet; height to present 
summit 453.3 feet; to original summit 485.2 feet; inclined length 568.25 feet; angle 
of side 51 0 51' 14"; area of each face = square of height; weight 5272600 tons; 
built 2170 years B.C. 

Peter the Great , St. Petersburg, Russia.—Bronze; height of horse 17 feet; of man 
11 feet; base of rock 42 feet at bottom, 36 at top, 21 wide, and 17 high, weighing 
1100 tons. 

Liberty, New York Harbor.—no feet in height from head to feet and 140 feet to 
flambeau; weight 150 tons. 

Daibutsu, of stone, Japan.—Sitting posture; height 44 feqt; circumference 87 feet; 
face 8.5 feet; circumference of thumb 3.5 feet. 

Bridge. 

Britannia Tubular Bridge. —Of iron, with a double line of Railway, 964 feet in 
length, with two approaches of 230 feet each. Weight 3658 tons. 


WORKS OF MAGNITUDE. 


179 


NEcmolitlis. 

Obelisk at Karnak, Egypt.—Of granite, 108 feet 10 ins.; pedestal 13 feet 2 ins.; 
freight 400 tons. 

Obelisk in Central Park, N. Y.—Of granite, 68 feet u ins.; weight 168 tons. 

U. S. Treasury , Washington.—Some stones of, are heavier than any in the Pyra¬ 
mids of Egypt. 

Steam Hammers. 

At workshops of Herr Krupp, at Essen, there is a steam hammer weighing 50 tons 
having a fall of 3 metres; and. at Creusot there is a hammer weighing between 75 
and 80 tons having a fall of 5 metres. 

Crane. 

At Creusot there is a steam crane having a capacity to lift and revolve with 150 
tons. 

Chimneys. 

J. Townsend’s chemical works, Glasgow, diameter at foundation 50 feet; at top 
12 feet 8 ins.; height from foundation 488 feet; from ground 474 feet. 

New York Steam Heating Co., 220 feet in height. 

IPillai’. 

At a gate near Delhi is a wrought-iron pillar having diameters of 16.4 ins. at 22 
feet in its height above ground and 12 ins. at its top. It is estimated from the re¬ 
sult of excavations at its base to be 60 feet in length or height and to weigh 17 
tons. Its period of structure is assigned to the 3d or 4th century A.D. 

Roofs. 

Midland Railway Station,London. 240 ft. j Union Railway Station, Glasgow. 195 ft. 
Imperial Riding-School, Moscow. 235 “ | Grand Central Station, N. Y 200“ 


Diameters of Domes. 


Domes. 

Feet. | 

Domes. 

Feet. 

Domes. 

Feet. 

Capitol, W ash ington 
Glasgow W. Railw’y 

124.75 

198 

St. Paul’s, London. 
St. Peter’s, Rome.. 

112 

139 

Midl’nd Rail’y, Lon. 
Great North’n, Eng. 

240 

210 


Dieiagtlis of Tunnels. 


Tunnels. 

Feet. 

Tunnels. 

Feet. 

Tunnels. 

Feet. 

Blaizy. 

Blue Ridge. 

Hoosac. 

13 455 

4 280 
25031 

Gunpowder, Md... 

Sutro.. 

Semtnering. 

36 500 
20 028 
5630 

Nerthe. 

Nochistongo.... 
Riquivel. 

15 153 
21 659 
18 623 


Thames and Medway, u 880 feet. Weehawken, 4000 feet. 

Mont Cenis 7.5 miles 242 yards, rises 1 in 45, and descends 1 in 2000. 

St. Gothard Tunnels and Roads 9 miles 477 yards; tunnels 116156.5 feet, and rises 
1 in 233 in whole length; 26.5 feet in width; 19 feet 10 inches in height. Maxi¬ 
mum grade 2.7 feet per 100. 

HYtiscellarLeons. 

Fortress Monroe, Old Point Comfort, Va.— Largest fortress. 

Telegraph IFire.—Span over river Kistnah between Bezorah and Sectanagran, 
6000 feet in length. 

Deer Park , Copenhagen.—4200 acres. 

Oxford College , England. —Largest University; said to have been founded by 
Alfred. 

Cathedral, St. Peter's, Rome.—Width of front 2x6 feet; of the cross 251 feet; total 
height 350 feet. 

Steamer Great Eastern.— Of iron, 680 feet in length; 83 feet width of beam; 60 
feet depth of hold; 22927 tons; built at Millwall, England, 1857. 

Chinese Wall.— 25 feet at base; 15 at top; height, with a parapet of 5 feet, 20 feet; 
length 1250 miles. 

Artesian Well , Perth.—3050 feet in depth; temperature of water 99 0 ; volume 
of discharge 18000 gallons per day. 
































ISO BELLS, CHURCHES, COLUMNS, TOWERS, ETC 


“WeigHts of IB ells. 


Bells. 

Lbs. 

Bells. 

Lbs. 

Bells. 

Lbs. 

Pekin. 

120 000 

Oxford, “Great 


St. Peter’s, Rome. 

18000 

Lewiston, Me. 

10233 

Tom,” Eng. 

\7 024 

Vienna. 

40 200 

Montreal, Can. 

28 560 

Olmutz, Bohemia. 

40320 

Westm’ster, “Big 


Moscow, Russia... 

443772 

Rouen, France... 

40 OOO * 

Ben,” England. 

35620 

Erfurt, Saxony.... 

30 800 

St. Paul’s, Eng... 

42 OOO 

York “ 

24 080 

Notre Dame, Paris 

28 670 

St. Ivan’s, Moscow 

127 830 

State House, Phila. 

13 OOO 


Rangoon, Burmali, 201 600 lbs. 


Capacity of Principal ClmrcLes ancl Opera Houses. 
Estimating a person to occupy an Area of 19.7 Ins. Square. 
Cliurches. 


St. Peter’s. 54000 

Milan Cathedral. 37000 

St. Paul’s, Rome. 32000 

St. Paul’s’ London. 25 600 

St. Petronio, Bologna. 24400 

Florence Cathedral. 24300 

Antwerp Cathedral. 24000 

St. Sophia’s, Constantinople. 23000 


Opera Houses 


St. John, Lateran. 22 900 

Notre Dame, Paris. 21 000 

P i sa .Cathed ral. 13 000 

St. Stephen’s, Vienna. 12 400 

St. Dominic’s, Bologna. 12000 

St. Peter’s, Bologna. n 400 

Cathedral of Sienna. 11 000 

St. Mark’s, Venice. 7 000 


and Theatres. 


Carlo Felice, Genoa. 2560 

Opera House, Munich.2370 

Alexander, St. Petersburg. 2332 

San Carlos, Naples. 2240 

Imperial, St. Petersburg. 2160 

La Scala, Milan. 2113 

Academy of Paris. 2092 


Heiglrts of Columns, 


Teatro del Liceo, Barcelona.4000 

Covent Garden, London. 2684 

Opera House, Berlin. 1636 

New York Academy. . 2526 

“ “ Windsor. 3400 

Philadelphia Academy. 3124 

Chicago “ 3000 


3, Domes, Spires, etc. 


Locations. 


Feet. 


Locations. 


Feet. 


CHIMNEYS. 


TOWERS AND DOMES. 


Townsend’s.Glasgow... 

St. Rollox. “ 

Musprat’s..Liverpool . 

Gas Works.Edinburgh 

New England Glass Co. Boston.... 
Steam Heating Co_New York. 


COLPMNS. 


Alexander. 

Bunker Hill. 


City. 

July. 

.. .Paris. 

Napoleon. 

4 4 

Nelson’s. 

...Dublin.... 

Nelson’s. 


Place Vendome.... 

.. .Paris. 

Pompey’s Pillar... 

...Egypt. 

Trajan. 


Washington. 


York. 



474 

455-5 

406 

34 i -5 

230 

220 


W 5 

221 

202 

157 

132 

1 34 

171 

136 

114 

i 45 

138 


TOWERS AND DOJIES. 

Babel. 

Balbec. 

Capitol.Wash’gton 

Cathedral...Antwerp .. 

“ .Cologne... 

“ Cremona.. 

“ .Escurial... 


680 

500 

287.5 

404.8 

524-9 

39 2 

200 


Cathedral. 


4 l 


44 


44 


Leaning . 


Porcelain. 


St. Mark’s. 


St. Nicholas... 


St. Paul’s. 


St. Stephen_ 


S trash lira - . 


Utrecht. 

Votive Church. 


SPIRES. 

Cathedral. 


44 


Grace Church.. 


Freibursr. 


Salisburv.. 

St. John’s. 


St. Paul’s. 

a 

St. Mary’s. 


St. Peter’s. ...*.. 


Trinity Church. 


Balustrade of 

Notre 

Dame. 



Towers of ditto. “ 

Hotel des Invalides... “. 


39°-5 

339-9 

438 

363 

188 

200 

328 

473 

355 -i 

443-8 

486 

464 

3I4-9 


325 

465-9 

2l6 

410 

450 

210 

200 

4°4 

469-5 

286 

216 

232.9 

344 


























































































































BRIDGES, CANALS, BREAKWATERS, ETC. l8l 


Areas of* Lakes in Europe, Asia, and. Africa. 


Lakes. 

Sq. 

Miles. 

Lakes. 

I sq- 
j Miles. 

Lakes. 

Sq. 

Miles. 

Geneva. 

Tchad, Africa. 

Bridges. 

*1 E 

CP M 

% . o\-^ 

_Ltrl 88 

Dembia, Abyssinia. 
Loch Lomond. 

engtlis of 13 

Bridges. 

13000 
! J 

27 

ridge 

Feet. 

Lough Neagh, lrel’d 
[Touting, China.... 

s. 

Bridges. 

80 

1200 

Feet. 

Avignon. 

Badajoz. 

Belfast. 

Blackfriars. 

Boston. 

London. 

Le 

Bridges. 

1710 

1874 

2500 

995 

3483 

950 

sngtL 

Feet. 

Lyons. 

Menai. 

N. Y. and Brook -1 
lyn spans and j 
approaches....) 
Pont St. Esprit... 

ls of Spans 

Bridges. 

1560 

1050 

59 8 9 

3060 

of 13 

Feet. 

Potomac. 

Riga. 

St. Law rence Riv’r 

Strasburg. 

Yauxhall. 

Westminster. 

ridges. 

Bridges. 

5300 

2600 

9 J 44 

339 ° 

860 

1223 

Feet. 

Britannia. 

Conway. 

Menai. 

460 

400 

580 

Niag’a at the Falls 
“ at Queens¬ 
town. 

1268 

1040 

Schuylkill. 

Southwark. 

Wheeling. 

340 

240 

IOIO 


Canals. 

Lengths. —Lake Erie to Albany 352 miles; Chesapeake and Ohio 307; Schuylkill 
108; Delaware and Hudson 109; Rideau 132; London to Liverpool 265; Caledonia 
25; Liverpool and Leeds 127.5; Rhone to Rhine 203. 

Capacity of Locks of Erie 240 tons, and of Welland 1500. 

Welland 26.77 miles. Lake Erie to Montreal via Canal 70.5; Lake and River 
375 miles. 

Montreal to Kingston.—Canal 120 miles; River 126.25. Suez, see page 183. 

Breakwaters. 

Delaware. —Average depth of water 29.4 feet below low-water level; range of tide 
6.66 feet; Outer slope 45 0 ; Inner slopes 1.5, 5, 3, and 1.3 to 1; length of base 172.12 
feet. 

Plymouth. —Outer slopes 1.75 to 1 from bottom to 7 feet 6 ins. below low-water 
line; 4 to 1 to low-water line; 16 to 1 to 4 feet. 6 ins. above low-water line; 5 to 1 
to high water; Inner slope 1.5 to 1 above low-water line; 2 to 1 below low-water line. 

Depth of water at high tide 46.5 feet; at low tide 30 feet. 

Body of breakwater cased with large squared stones cramped together. 

Portland. —Depth of high water 58 feet; of low water 51 feet; Outer slopes 1 to 1 
from bottom to 20 feet below low water; 2 to 1 to 12 feet below low water; 6 to 1 
to low-water line; 4 to 1 to high-water line; Inner slope 1.25 to 1. 

Body of breakwater, rubble, with crest wall of ashlar. 

Dover.— Depth of high-water line 61 feet; of low-water line 42 feet. 

Body of breakwater, concrete blocks faced with granite; batter 3 inches to the 
foot, stepped up in each course. 

Marseille §. —Depth of water 33 feet; Outer casing of beton 25.5 tons each; average 
thickness of casing from 14 to 20 feet; slope 1 to 1 from bottom to waiter line; 2.5 
to 1 above waiter-line; all other slopes .33 to 1; Inner casing of first-class rubble 
(of stones 2 to 5 tons weight), about 12 feet thick; Hearting, second-class rubble 
(of stones .5 to 2 tons w’eight), about 6 feet thick; Nucleus, of quarry rubbish. 

Algiers. —Depth of water 50 feet; rubble base carried up to 33 feet from surface of 
w r ater; the remainder composed of large beton blocks 25.5 tons each; slopes of rubble 
base 1 to 1; Outer slope of beton blocks 1.25 to 1; Inner slope of beton blocks 1 to 1. 

Port Said (Suez Canal).—Concrete blocks, 10 cubic metres each, composed of 1 
of hydraulic lime to 13 of sand, mixed with sea water; 4 days in the mold and dried 
for 4 months before being put in position. In some instances the composition of 
beton blocks is .33 lime or cement to .66 sand and broken stone, about the size of 
ballasting. 

Rubble or Block Filling. —Proportion of interstices to volume of breakwater fin¬ 
ished: First-class rubble of 2 to 5 tons, .25; second-class rubble of .5 to 2 tons, .2; 
third-class rubble, quarry chips, etc., .16; beton blocks, 15 to 25 tons, .33. 

Note. —For force of water, see Waves of the Sea, page 853. 

Q 




























































182 


LAKES, OCEANS, SEAS, MOUNTAINS, ETC 


Areas, Depths, and. I-Ieights of Great Northern Lakes 


Lakes. 

of 

Length. 

TJ nited 

Breadth. 

t^tGS* 

Mean Depth. 

Height 
above Sea. 

Area. 


Miles. 

Miles. 

Feet. 

Feet. 

Sq. Miles. 

Erie. 

250 

80 

200 

564 

6 000 

Huron... 

200 

160 

120 r 

574 

20 OOO 

Michigan. 

360 

109 

900 

587 

20 000 

Ontario. 

180 

65 

500 

234 

6 000 

Superior*. 

400 

160 

288 

635 

32 OOO 


* Greatest depth 5400 feet. 


Elevation .—Of Lake Erie above tide-water at Albany 323 feet. 

1 

NTearr Depths and Areas of the Oceans and Seas. 

(Herr Kriimmel.) 


! Fathoms. 

1 

Area 

Sq. Miles. 

Atlantic. 

2013 

487 

29514275 
3 046 600 
8800 

Archipelago. 

Azof.. 

Baltic Sea. 

36 

159690 
150 OOO 
864 555 

Black Sea. 

Behring’s Straits.... 
Caspian Sea. 

55o 

China (East) Sea.... 
Dead Sea. 

66 

472 210 

English Channel, etc. 

47 

78 416 



Fathoms. 

Area 

Sq. Miles. 

Gulf of Mexico. 

IOOI 

1 765 910 

“ St. Lawrence 

160 

101075 

Indian. 

1829 

28369595 

Japan . 

1200 

383 205 

Mediterranean. 

729 

I IO9 230 

North Sea. 

48 

210505 

North Ice Sea. 

845 

5 264 600 

Persian Gulf. 

20 

90 IOO 

Pacific. 

3887 

60 343 690 

Red Sea. 

243 

170 820 


Mean depth of Ocean surrounding land 1877 fathoms = 2.19 miles. 

In his subsequent computations he estimates ocean area at 143703000 square 
miles and determines area of land to water as 1 to 2.75, and that mean height of 
land = 1377 feet, or one eighth that of Ocean. 


Heights of IVLoiantaiius, Volcanoes, and. [passes 
above Level of Sea. 


Mountains. 


EUROPE. 

Azores Pico.j 

Barthelemy, France 

Ben Lomond. 

Ben Nevis. 

Elbrus, Caucasus... 
Guadarama, Spain.. 

Hecla. 

Ida. 

Jungfrau. Switz’d.. 

Mont Blanc. 

Cenis. 

Mont d’ Or, France. 
M ul abas sen, Gi'en’a. 
Nephin, Ireland.... 

Olympus. 

Parnassus. 

Plynlimmon, Wales. 
The Cylinder, Pyr.. 
Wetterhorn. 

ASIA. 

Ararat. 

Caucasus. 

Dhawalagheri. 

Geta, Java. 

Mount Lebanon... .j 


Feet. 


7 6 i 3 

7 365 
3240 
4380 

17776 

8 520 

5 i47 
4 960 

*3 7 2 5 
*5 797 

6 780 
6 510 

11 663 
2634 
6510 
6 000 
2463 

IO 030 

12 I54 


17 IOO 

16433 
28077 
8500 
12 000 


Mountains. 

Feet. 

Mountains. 

Mou n t Everest (FI i m - 


Mount Pitt. 

alaya, highest)... 

2 Q 003 

Mount Washington. 

Mount Libanus.... 

9523 

Nevado de Sorata.. 

Petcha.. . 

15 OOO 
7496 

Orizaba. 

Sinai. 

Potosi . 

AFRICA. 

Sierra Nevada. 

Tahiti. 

Atlas. 

10 400 

White Mountains.. 

Compass, Cape of 
Good Hope. 

10000 

VOLCANOES. 

Dianai Peak, St He- 


Cotopaxi. 

lena. 

2 7OO 
20 OOO 

Etna. 

Kilimanjaro. 

Hecla . 

Ruivo, Madeira.... 

5160 

Popocatapetl. 

Tenerifl'e Peak. 

12 3OO 

Sab am a. 

AMERICA. 

St. Helen’s, Oregon. 


Vesuvius. 7 .. . 

Aconcagua (highest 
in America). 

23 910 

PASSES. 

Blue Mount, Jam’a. 

8 000 

Cordilleras.j 

Catskill. 

3 804 

Chimborazo. 

21441 

16 036 

Mont, Cenis 

Correde, Potosi .... 

* “ Cervis. 

Crows’ Nest, High- 

Pont d’ Or. 

lands, N. Y. 

1370 

St. Bernard, Great.. 

Great Peak, New 

“ Little.. 

Mexico. 

W H 

00 VO 

0 OO 
0 CO 

St. Gothnrd 

Mauna Rauh,Owy’e 

Simplon. 


Feet. 


9 549 
6426 
25248 
18 879 
18 000 
15 700 
10895 
6 230 


18 887 
10 874 
5000 
*7 784 
22 350 
13320 
3 930 


13 525 
15225 
6 778 
11 100 

9843 

8 172 
7192 
6808 
6578 








































































































183 


CANAL LOCKS, ELEVATIONS, AND RIVERS. 


Dimensions of Canal XjocIcs.— (U. S.) 


Canal. 

43 

be 

S 

QJ 

Breadth' 

Depth. 

Length of 
Canal. 

Canal. 

Length. 

Breadth. 

Depth. 

Length 

of 

Canal. 



Feet. 

Ft. 

Ft. 


Miles. 


Feet. 

Feet. 

Feet. 

Miles. 

Albemarle and 




£ 



Champlain. 

no 

18 

5 

66.75 

Chesapeake.. 



4 ° 

0 


*4 

Cayuga and ) 





Black River, 






Seneca.j 

110 

10 

7 

2 4'75 

Crook’d L’ke, 





) 77 

Delaware and ( 





Chenango, 







Raritan.j 

220 

24 

7 

43 

Chemung, 


90 

15 

4 

-j 

97 

Dismal Swamp .'. 

9 ° 

17-5 

5-5 

44 

and Genesee 






33 

Erie. 

no 

18 

7 

35 2 

Valley. 






l II 3-75 

Falls of Ohio, Ky. 

350 

80 

2-60 


Chesapeake and 







Oswego. 

no 

18 

4 

38 

Delaware_ 


220 

2 4 

9 


J 4 

Welland,Canada.. 

270 

45 

14 

28 


Length of vessel that can be transported is somewhat less than lengths of locks. 

Suez Canal. —Width 196 to 328 feet at surface, 72 at bottom, and 26 deep. 
Length 99 miles. 


Heights of obtained. Elevations, and various IPlaces 
and [Points above the Sea. 


Locations. 

Feet. 

Locations. 

Feet. 

Locations. 

Feet. 

Aconcagua, Chili... 

2391° 

Geneva city. 

I 220 

Mont Rosa, Alps... 

15 155 

Antisana, highest 
established eleva- 

Geneva Lake. 

Gibraltar. 

1096 

1439 

Mount Adams. 

Mount Katahdin... 

5 930 
5 3 6c > 

tion (Farmhouse) . . 

I 3 434 

Humboldt’s highest 

Mount Pitt. 

9 549 

Balloon (Gay Lussac) 

22 9OO 

elevation. 

19 4OO 

Mount Washington. 

6 426 

“ (Green, 1837) 
“ (Glaisher and 

27 OOO 

Isthmus of Darien. 

645 

Paris, city. 

X15 

Jungfrau, Switz’d.. 

13725 

Pont d’ Oro, Pyr’s.. 

9843 

Coxwell). 

37000 

La Paz, Bolivia.... 

12 225 

Posthouse, Ap., Peru 

I 4 377 

Brazil, Quito, and ( 

6 000 

Laguna, TeneViffe... 

2 OOO 

Potosi, Bolivia. 

Quito. 

St. Bernard’s Mon’y 

Vegetation. 

White Mountain... 

13223 

Mexico plains.. ( 

Condor’s flight. 

Eagle’s “ . 

Everest, Himalaya. 

8 000 
29 500 
16 500 
29 003 

London, city. 

Madrid. 

Mexico, city of. 

Mont Blanc, Alps... 

64 

2 200 

7 5 2 5 
15797 

13 500 
8 040 
17 000 
6 230 


Lengths of [Rivers 


Rivers. 

Miles. 

Rivers. 

Miles. 



Ganges. 

1514 

EUROPE. 


Hoang Ho. 

3040 

Hnnulift 

l800 

Indus. 

1800 

Dnip.npr 

12AX 

Jordan . 

176 

Dmym . 

400 

Lena. 

2762 

Pwina 

IOQS 

Tigris. 

xi6o 

Elbe. 

780 

Yenesei and Se- 


Onrnrmp. 

442 

lenga. 

3580 

IjOirP . 


Yang-Tse. 

3314 

Po. 

420 


Rhine. 

760 

AFRICA. 



mo 

Gambia. 

700 

Sftj nA r . 

ako 

Niger. 

2400 

Shannon 

250 

Nile. 

4000 

Tagus. 

51° 


Thames. 

220 

NORTH AMERICA. 


Tiber 

190 

630 

Arkansas. 

2070 

Vi Rt.nl a. 

Colorado. 

1050 

Vnlcra Russia. 

2400 

Columbia. 

1200 


Connecticut. 

410 

ASIA. 


Delaware. 

420 

Amoor. 

2500 

Hudson and Mo- 


Euphrates. 

1786 

hawk. 

325 


Rivers. 


Miles. 


Kansas. 

La Platte. 

Mackenzie. 

Mississippi. 

Missouri. 

Ohio and Allegheny 

Potomac. 

Red. 

Rio Bravo. 

Rio Grande. 

St. Lawrence. 

Susquehanna. 

Tennessee. 


1400 

850 

2440 

1350 

3°3o 

1480 

420 

1520 

2300 

1800 

2172 

620 

790 


SOUTH AMERICA. 

Amazon. 

Essequibo . 

Magdalena. 

Orinoco. 

Platte. 

Rio Madeira. 

Rio Negro. 

Uruguay. 


4000 
520 
900 
1600 
2300 
2300 
1650 
1100 




























































































































SEA DEPTHS, BUILDING STONES, ETC. 


I84 


Large Trees in California. 

“ Keystone State. ■’Calavera Grove, is 325 feet in height. 

“ Father of the Forest.”— Felled, is 385 feet in length, and a man on horseback 
can ride erect go feet inside of its trunk. , 

“ Mother of the Forest.”— Is 315 feet in height, 84 feet in circumference (26.75 feet 
in diameter) inside of its bark, and is computed to contain 537000 feet of sound 1 
inch lumber. 

Sea Deptlis. 


Feet. 


Baltic Sea. 120 

Adriatic. 130 

English Channel... 300 

Straits of Gibraltar. 100 

Eastward of “ 3000 

Estimated depth of Atlantic 
“ “ Pacific.. 

250 miles off Cape Cod, no bottom at 7800 feet. 



Feet. 


Feet. 

Coast of Spain..... 

6 000 

Off Cape Canaveral. 

2400 

West of St. Helena. 

£7 000 

“ Charleston. 

4200 

Tortugas to Cuba .. 

4 200 

u Cape Hatteras.. 

3 I2 ° 

Gulf of Florida .... 

372° 

“ Cape Henry.... 

4200 

Off Cape Florida... 

1950 

“ Sandy Hook.... 

2400 


26 oco feet. 
29 000 


Location. 


Cascades and. "Waterfalls. 

Feet. || Location. Feet. Location. 


Arve, Savoy.. 
Cascade, Alps 


Cataracts of the Nile. 


Chachia, Asia.. 

Foyers, Scotland ... 

Garisha, India. 

Gavarny, Pyrenees . 


Genesee, N. Y. 

Lidford, England .. 
Lulea, Sweden 
Mohawk. 


1600 
2400 
3 ° 

34 
40 
362 
197 
1000 
1260 

Yosemite Valley. 


Missouri. 


Montmorenci ... 
Nant d’Apresias. 


100 

100 

600 

68 

(S° 

| 80 

(94 

250 

800 

.2600 


Niagara. 

Great Fall. 

Passaic. 

Potomac. 

Ribbon, Yosemite | 

Valley. ] 

Ruican, Norway ... 
Staubbach, Switz'd. 
Tendon, France.... 
feet. 


Feet. 

164 

15 * 

74 

74 

3300 

800 

798 

125 


Expansion and. Contraction of Building Stones for eacli 
Degree of Temperature. (Lieut. W. II. C Bartlett , U. S. E.) 


For One Inch. 

Granite. .000004 825 

Marble.000005 668 


For One Inch. 

Sandstone...000009532 

Whitepine.000002 55 


Ftesistan.ce of Stones, etc., to tlie Effects of Freezing. 

Various experiments show that the power of stones, etc., to resist effects of freez¬ 
ing is a fair exponent of that to resist compression. 

Magnetic Bearings of New York. 

The Avenues of the City of New York bear 28° 50' 30" East of North. 

Filters for Waterworks. 

1 square yard of filter for each 840 U. S. and 700 Imp’l gallons in 24 
hours; formed of 2.5 feet of fine sand or gravel and 6 inches of common 
sand or shells. 

Led off by perforated pipes laid in lowest stratum. 

Distances Between Yew York, Boston, Bliiladelpkia, 
Baltimore, and Western Cities of XT. S. 

Assuming Bostou as standard, New York averages 12 per cent, nearer to these 
cities, Philadelphia 18 per cent., and Baltimore 22 per cent. 

Between New York and Chicago the line of the Pennsylvania Railroad is 47 miles 
shorter than that by the Erie and its connections, 50 miles shorter than that by the 
N. Y. Central and Hudson River and its connections, and 114 miles shorter than that 
by the Baltimore and Ohio and its connections. 

For Distances between these and other cities of the U. S., see page 88. 





















































WEATHER-PLANTS, ANTIDOTES, ETC. 


185 


■Weather-foretelling 3?lants. ( Hanneman.) 

If Rain is imminent. —Chick weed,* Stellaria media; its flowers droop 
and do not open. Crowfoot anemone, Anemone ranunculoides; its blossoms 
close. Bladder Ketmia, Hibiscus trionum; its blossoms do not open. Thistle, 
Carduus acaulis; its flowers close. Clover, Trifolium pratense , and its allied 
kinds, and Whitlow grass, Draba verna; all droop their leaves. Nipple¬ 
wort, Lampsana communis; its blossoms will not close for the night. Yel¬ 
low Bedstraw, Galium verum; it swells, and exhales strongly; and Birch, 
Betula alba , exhales and scents the air. 

Indications of Rain. —Marigold, Calendula pluvialis; when its flowers do 
not open by 7 A. M. Hog Thistle, Sonchus arvensis and oleraceus; when its 
blossoms open. 

Rain of shoi't duration. —Chick weed, Stellaria media; if its leaves open 
but partially. 

If cloudy. — Wind-flower, or Wood Anemone, Anemone memorasa; its 
flowers droop. 

Termination of Rain. — Clover, Trifolium pratense; if it contracts its 
leaves. Birdweed and Pimpernel, Convolvulus and Anagallis arvensis; if 
they spread their leaves. 

Uniform Weather. —Marigold, Calendula pluvialis; if its flowers open early 
in the A. M. and remain open until 4 P. M. 

Clear Weather. —Wind-flower, or Wood Anemone, Anemone memorasa; 
if it bears its flowers erect. Hog Thistle, Sonchus arvensis and oleraceus; 
if the heads of its blossoms close at and remain closed during the night. 

Treatment and Antidotes to Severe Ordinary Poisons. 

Antidotes in very small doses. 

Chloroform and Ether.— Cold affusions on head and neck, and ammonia 
to nostrils. Antidote. —Camphor, petroleum, sulphur. 

Toadstools .—(Inedible mushroom). Antidote.—Same as for chloroform. 

Arsenic or Fly Poivder. —Emetic; after free vomiting give calcined mag¬ 
nesia freely. If poison has passed out of stomach, give castor oil. 

Antidote. —Camphor, nux vomica, ipecacuanha. 

Acetate of Lead (Sugar of lead). — Mustard emetic, followed by salts, 
Large draughts of milk with white of eggs. 

Antidote. —Alum, sulphuric acid alike to lemonade, belladonna, strychnine. 

Corrosive Sublimate (Bug poison). — White of eggs in 1 quart of cold 
water, give cupful every two minutes. Induce vomiting without aid of 
emetics. Soapsuds and wheat flour is a substitute for white of eggs. 

Antidote. —Nitric acid, camphor, opium, sulphate of zinc. 

Phosphorus Matches. — Rat Paste. —Two teaspoonfuls of calcined magne¬ 
sia, followed by mucilaginous drinks. Antidote.— Camphor, coffee, nux vomica. 

Carbonic Acid (Charcoal fumes), Chlorine , Nitrous Oxide, or Ordinary 
Gas. —Fresh air, artificial respiration, ammonia, ether, or vapor of hot water. 

Antidote.—Camphor, coffee, nux vomica. 

Belladonna (Nightshade). — Emetic and stomach pump, morphine and 
strong coffee. Antidote. —Camphor. 

Opium. —Stomach pump or emetic of sulphate of zinc, 20 or 30 grains, or 
mustard or salt. Keep patient in motion. Cold water to head and chest. 

Antidote. —Strong coffee freely and by injection, camphor, ether, and nux vomica. 

Strychnine (Nux vomica).—Stomach pump or emetic, chloroform, cam¬ 
phor, animal charcoal, lard, or fat. 

Antidote. —Wine, coffee, camphor, opium freely, and alcohol in small doses. 

Vegetable Poisons .— As a rule, an emetic of mustard and drink freely of 
warm water. 


* Spreads its leaves about 9 A. M., and they remain open until noon. 




VETERINARY, 


186 


"V eterinary. 

Horses.— Cathartic Ball. —Cape Aloes, 6 to io drs.; Castile Soap, i dr.; Spirit 
of Wine, i dr. ; Sirup to form a ball. If Calomel is required, add from 20 to 50 
grains. During its operation, feed upon mashes and give plenty of water. 

Cattle.— Cathartic.— Cape Aloes, 4 drs. to 1 oz.; Epsom Salts, 4 to 6 oz.; Gin¬ 
ger, 3 drs. Mix, and give in a quart of gruel. For Calves, one third will be sufficient. 

Horses ancl Cattle. — Tonic .— Sulphate of Copper, 1 oz. to 12 drs. ; Sugar, 
.5 oz. Mix, and divide into 8 powders, and give one or two daily in food. 

Cordial. —Opium, 1 dr.; Ginger, 2 drs.; Allspice, 3 drs., and Caraway Seeds, 4 
drs., all powdered. Make into a ball with sirup, or give as a drench in gruel. 

Cordial Astringent Drench, for Diarrhoea , Purging , or Scouring. —Tincture of 
Opium, .5 oz.; Allspice, 2.5 drs.; powdered Caraway, .5 oz.; Catechu Fowder, 2 drs.; 
strong Ale or Gruel, 1 pint. Give every morning till purging ceases. For Sheep 
.25 this quantity. 

Alterative.— Ethiop’s Mineral, .5 oz. ; Cream of Tartar, 1 oz.; Nitre, 2 drs. Divide 
into from 16 to 24 doses, one morning and evening in all cutaneous diseases. 

Diuretic Ball.— Hard Soap and Turpentine, each 4 drs.; Oil of Juniper, 20 drops; 
and powdered Resin to form a ball. 

For Dropsy , Water Farcy , Broken Wind, or Febrile Diseases, add to above, All¬ 
spice and Ginger, each 2 drs. Divide into 4 balls, and give one morning and evening. 

Alterative or Condition Powder. —Resin and Nitre, each 2 oz.; levigated Anti¬ 
mony, 1 oz. Mix for 8 or 10 doses, and give one morning and evening. When given 
to Cattle, add Glauber Salts. 1 lb. 

Fever Ball. —Cape Aloes, 2 oz.; Nitre, 4 oz.; Sirup to form a mass. Divide into 
12 balls, and give one morning and evening until bowels are relaxed; then give an 
Alterative Powder or Worm Ball. 

Hoof Ointment. —Tar and Tallow, each 1 lb.; Turpentine .5 lb. Melt and mix. 

Hogs. — Cathartic. —Cape Aloes, .5 dr. to 1 oz.; Calomel, 2 to 3 grs. ; Oil of 
Caraway, 6 drops; Sirup to form a ball. Repeat every 5 hours till it operates. 

Emetic. —2 to 4 grs. of Tartar Emetic in a meat ball, or a teaspoonful or two of 
common salt. Give twice a week if required. 

Distemper Powder. — Autimonial Powder, 2, 3, or 4 grs.; Nitre, 5, 10, or 15 grs.; 
powdered Ipecacuanha, 2, 3, or 4 grs. Make into a ball, and give two or three times 
a day. If there is much cough, add from .5 gr. to 1 gr. of Digitalis, and every 3 or 
4 days give an Emetic. 

Mange Ointment. —Powdered Aloes, 2 drs.; White Hellebore, 4 drs.; Sulphur, 4 
oz. ; Lard, 6 oz.— Red Mange, add 1 oz. of Mercurial Ointment, and apply a muzzle. 

Note.—P hysic, except in urgent cases, should be given in morning, and upon an 
empty stomach; and, if required to be repeated, there should be an interval of sev¬ 
eral days between each dose. 

Age of Horses. 

To Ascertain a. Horse’s Age. 

A foal of six months has six grinders in each jaw, three in each side, and also six 
nippers or front teeth, with a cavity in each. 

At age of one year, cavities in front teeth begin to decrease, and he has four 
grinders upon each side, one of permanent and remainder of milk set. 

At age of two years he loses the first milk grinders above and below, and front 
teeth have their cavities filled up alike to teeth of horses of eight years of age. 

At age of three years, or two and a half, he casts his two front uppers, and in a 
short time after the two next. 

At four, grinders are six upon each side; and, about four and a half, his nippers 
are permanent by replacing of remaining two corner teeth; tushes then appear 
and he is no longer a colt. 

At five, a horse has his tushes, and there is a black-colored cavity in centre of all 
his lower nippers. 

At six, this black cavity is obliterated in the two front lower nippers. 

At seven, cavities of next two are filled up, and tushes blunted; and at eight that 
of the two corner teeth. Horse may now be said to be aged. Cavities in°nippers 
of upper jaw are not obliterated till horse is about ten years old, after which time 
tushes become round, and nippers project and change their surface. 


DISTANCES, POPULATION, DROWNING, ETC. 1 87 


Distances Detween ^Principal Cities of* East and. West. 

In Miles. 


Cities. 

Boston. 

New 

York. 

Phila¬ 

delphia. 

Balti¬ 

more. 

Cities. 

Boston. 

New 

York. 

Phila¬ 

delphia. 

Balti¬ 

more. 

Burlington, la. 

I2l6 

1106 

1030 

995 

Louisville. 

1161 

870 

794 

706 

Chicago. 

IOO9 

900 

823 

802 

Memphis. 

MS 8 

1247 

it 71 

1083 

Cincinnati.... 

927 

743 

667 

576 

Milwaukee_ 

998 

947 

908 

887 

Cleveland. 

671 

580 

504 

483 

Omaha. 

1503 

1393 

I 3 W 

1294 

Columbus, 0 .. 
Detroit. 

807 

724 

623 

6 73 

547 

682 

512 

661 

St. Joseph. 

St. Louis. 

1478 

1212 

1356 

1050 

1280 

973 

1223 

Q17 

Indianapolis .. 

951 

810 

735 

700 

St. Paul. 

1418 

1308 

1232 

1211 

Kansas City... 

1487 

1324 

1248 

1192 

Toledo. 

784 

693 

617 

596 


Population of Principal Cities (1SS2). 


Loudon. 3832440 

Paris. 2225910 

Berlin. 1 222 500 

New York. 1206299 

Vienna. 1103 no 

St. Petersburg... 876 570 

Philadelphia.... 847 170 

Moscow. 611970 

Constantinople.. 600000 

Chicago. 583 185 

Brooklyn. 566 663 

Hamburg. 410120 

Naples. 403 no 

Lyons. 372 890 

Madrid. 367 280 

Boston. 362 839 

Buda-Pesth. 360580 


Marseilles. 357 53° 

St. Louis . 350518 

Warsaw. 339400 

Baltimore. 33 2 3 I 3 

Milan. 321440 

Amsterdam. 317 010 

Rome. 300470 

Lisbon. 246300 

Palermo. 244990 

Copenhagen. 234850 

San Francisco_ 233959 

Munich. 230200 

Cincinnati. 225 139 

Bucharest. 221 800 

Dresden. 220820 

New Orleans.216190 

Florence. 169000 


Stockholm. 168770 

Brussels. 161 820 

Cleveland. 160146 

Pittsburgh. 156389 

Buffalo. 155 134 

Antwerp. 150650 

Washington. 147293 

Cologne. 144770 

Frankfort. 136820 

Newark. 136508 

Venice. 132830 

Louisville. 123758 

Jersey City. 120722 

Detroit. 116340 

Milwaukee. 115587 

Providence. 104857 

Rouen. 104 010 


Treatment of Drowning Pei'sons. 

Practice adopted by Board of Healtli, New York 


Place patient face downward, with one of his wrists under his forehead. Cleanse 
his mouth. If he does not breathe, turn him on his back with shoulders raised on 
a support. Grasp tongue gently but firmly with fingers covered with end of a hand¬ 
kerchief or cloth, draw it out beyond lips, and retain it in this position. 

To Produce and Imitate Movements of Breathing. —Raise patient’s extended arms 
upward to sides of his head, pull them steadily, firmly, slowly, outwards. Turn 
down elbows by patient’s sides, and bring arms closely and firmly across pit of 
stomach, and press them and sides and front of chest gently but strongly for a mo¬ 
ment, then quickly begin to repeat first movement. 

Let these two movements be made very deliberately and without ceasing until 
patient breatjies, and let the two movements be repeated about twelve or fifteen 
times in a minute, but not more rapidly, bearing in mind that to thoroughly fill the 
lungs with air is the object of first or upward and outward movement, and to expel 
as much air as practicable is object of second or downward motion and pressure. 
This artificial respiration should be maintained for forty minutes or more, when the 
patient appears not to breathe; and after natural breathing begins, let same motion 
he very gently continued, and give proper stimulants in intervals. 


What Else is to he Done , and What is Not to he Done , while the Movements are 
Being Made. —If help and blankets are at hand, have body stripped, wrapped in 
blankets, but not allow movements to be stopped. Briskly rub feet and legs, press¬ 
ing them firmly and rubbing upward, while the movements of the arms and chest 
are in progress. Apply hartshorn, or like stimulus, or a feather within the nostrils 
occasionally, and sprinkle or lightly dash cold water upon face and neck. The 
legs and feet should be rubbed and wrapped in hot blankets, if blue or cold, or if 
weather is cold. 


What to Do when Patient Begins to Breathe. —Give stimulants by teaspoonful two 
or three times a minute, until beating of pulse can be felt at wrist, but be careful 
and not give more of stimulant than is necessary. Warmth should be kept up in 
feet and legs, and as soon as patient breathes naturally, let him be carefully removed 
to an enclosure, and placed in bed, under medical care. 
















































































188 


MISCELLANEOUS ELEMENTS. 


MISCELLANEOUS ELEMENTS. 

Earth- 

Polar diameter 7899.3 miles. Mean density or specific gravity of mass 5.672. Mass 
5 272 600 000 000 000 000 000 tons. Apparent diarfieter as seen from Sun 17 seconds. 

Sun. 

Heat of Sun equal to 322794 thermal units per minute for each sq. foot of pho¬ 
tosphere or solar surface. 

Diameter of Sun 882000 miles, tangential velocity 1.25 miles per second or 4.41 
times greater than that of the Earth. 

Distance from Earth 91.5 to 92 millions of miles. 

Mason and Dixon's Line. 


39 0 43' 26.3" N. mean latitude. 68.895 miles. 

Area and Population. (Behm and Wagner.) 


Divisions. 

Area. 

Population. 

Divisions. 

Area. 

Population. 

America. 

Sq. Miles. 
14 491 000 

3 760000 
16 313 000 
10 936 000 

95 495 500 
315 929000 
834 707 000 
205 679 000 

Oceanica. 

Greenland 1 
Iceland ) • • • 

Total. 

Sq. Miles. 

4 500 000 

4 031 000 

82 000 

Europe. 

Asia. 

Africa. 

50 000 000 

M 55 9 2 3 50 O 



Countries. 


38000000 


Austria ) 

Hungary) ' 

China.434 626 000 

France. . 37 000 000 

(United States... .50000000 
{Indians. 300000 


Germany.43 900 000 

Great Britain. .34000000 

J Russia.66 000 000 

{Territories 


India, British . .240298000 

Canada. 3 839 000 

Mexico. 9 485 000 

Brazil. n 106000 

) Turkey. 8866000 

in Asia.. 16 320 000 


.22 000 000 


t 


About one thirtieth of whole population are born every year, and nearly an equal 
number die in same time; making about one birth and one death per second. 
Earlier authority estimated population at 1 288000000, divided as follows: 


Caucasians.360 000 000 

Mongolians .... 552 000 000 

Ethiopians.190 000 000 

Asiatics. 60000000 


Malays and ) 

Indo Amer’s } W7 000 000 

Protestants.... 80000000 
Israelites. 5 000 000 


Moham medans. 190 000 000 

Pagans.300 000 000 

Catholics ) 

Rom. & Greek) 250000000 


Descent of Western Rivers. 

Slope of rivers flowing into Mississippi from East is about 3 inches per mile; 
and from West 6 inches. 

Mean descent of Ohio River from Pittsburgh to Mississippi, 975 miles, is about 5.2 
inches per mile; and that of Mississippi to Gulf of Mexico, 1180 miles, about 2.8 
inches. 

Transmission of Horse Rower. 


Largest, and perhaps most successful, wire rope transmission is one at Schaff- 
hausen, at Falls of the Rhine. Here, power of a number of turbines, amounting 
to over 600 FP, is conveyed across the stream, and thence a mile to a town, where it 
is distributed and utilized. 

At mines of Falun, Sweden, a power of over 100 horses is transmitted in like 
manner for a distance of three miles. 


A^cicls. 

Acetic Acid (Vinegar), acid of Malt beer , etc. Tartaric Acid , acid of Grape wine. 
Lactic Acid , acid of Milk , Millet beer , and Cider. 

Manures. 

Relative Fertilizing Properties of Various Manures. 

Peruvian Guano-1 I Horse.048 I Farm-yard.0298 

Human, mixed.069 | Swine.044 |£ow.0259 

Or, 1 lb. guano = 14.5 human, 21 horse, 22.5 swine, 33.5 farm-yard, and 38.5 cow. 
Relative Value, Covered and Uncovered , on an Acre of Ground. 

Covered .n tons 1665 lbs. potatoes, 61 lbs. wheat, 215 lbs. straw. 

Uncovered . 7 “ 1397 “ “ 61.5 “ “ 156 “ “ 












































MISCELLANEOUS ELEMENTS, 


I89 


Yield, of Oil of Several Seeds. 

PerCent. I Per Cent. I Per Cent. I PerCent.! PerCent. 

Poppy.. 56 to 63 I Castor .. 25 I Sunflower. 15 I Hemp. 14 to 25 I Linseed, n to 22 


Thickness of Walls of Buildings. (English.) (Molesworth.) 


Outer Walls. 


1st class dwelling 


3<1 

4th 


Party Walls. 

1st class dwelling 

2d “ “ 

3d “ “ 

4th “ “ 

If walls are more than 70 feet in length 
by half a brick. 

Warehouses 

1st Class. of Wall. 
For a height of 36 feet from ins. 

topmost ceiling. 17.5 

For a height of 40 feet lower.. 21.5 
“ “ 24 feet lower .. 26 

For footings.43.5 

3d Class. 

For a height of 28 feet below 

topmost ceiling.13 

For a height of 16 feet lower .. 17.5 
For footings.30.5 


Maximum 

Width 


Minimum Width of Walls. 


Height 

of 

Ground 

1 st 

2 d 

3 d 

4 th 

5 th 

6 th 

of V\ all. 

Footings. 

Floor. 

Floor. 

Floor. 

Floor. 

Floor. 

Floor. 

Floor 

Feet. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

85 

38.5 

2 I .5 

21.5 

17-5 

17-5 

17-5 

13 

13 

70 

30-5 

17-5 

I 7-5 

17-5 

13 

13 

13 


52 

30-5 

17-5 

13 

13 

13 

13 

— 

— 

38 

21.5 

13 

13 

8-5 

8-5 

—— 

— 

— 

8 S 

38.5 

21.5 

21.5 

17-5 

17-5 

17-5 

13 

13 

70 

30-5 

17-5 

i7-5 

17-5 

13 

13 

1.3 


52 

30-5 

17-5 

13 

LI 

13 

8-5 

— 

— 

38 

21.5 

13 

8-5 

8-5 

8-5 

— 

— 

— 


those of lower stories must be widened 

Warehouses 

2d Class. of Wall. 
For a height of 22 feet below ins. 

topmost ceiling.13 

For a height of 36 feet lower .. 17.5 
“ “ 8 feet lower.. 21.5 

For footings.34.5 

4th Class. 

For a height of 9 feet below 

topmost ceiling. 8.5 

For a height of 13 feet below .. 13 
For footings. 21.5 


Wooden Hoofs. (English.) 


Span 
in Feet. 

Princi 

Beat 

pal 

1 . 

Tie 

Beam. 

King 

Posts. 

Queen 

Posts. 

Small 

Queens. 

Straining 

Beam. 

Struts 


20 

4 

X 

4 

9 

X 

4 

4 X 4 

— 

— 

— 

3 

X 

3 

25 

5 

X 

4 

IO 

X 

5 

5X5 

— 

— 

— 

5 

X 

3 

30 

6 

X 

4 

II 

X 

6 

6X6 

— 

— 

— 

6 

X 

3 

35 

5 

X 

4 

II 

X 

4 

— 

4 X 4 

— 

7X4 

4 

X 

2 

45 

6 

X 

5 

13 

X 

6 

— 

6x6 

— 

7X6 

5 

X 

3 

50 

8 

X 

6 

13 

X 

8 

— 

8x8 

8 X 4 

9X6 

5 

X 

3 

55 

8 

X 

7 

14 

X 

9 

— 

9 x 8 

9X4 

10 x 6 

5-5 

X 

3 

60 

8 

X 

8 

15 

X 

10 

— 

10 X 8 

IO x 4 

11 X 6 

6 

X 

3 


^Mineral Constituents absorbed or removed from an 
Acre of Soil by" several Crops. (Johnson.) 


Crops. 

-*-» CO 

cS ■—> 

•S 

® O ® 

a> • 

Cm 00 

c 
s 0 

C 

5 \S 


£ % 

c 5 2 

« £ 

5 0 

H- 1 UI 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Potassa. 

29.6 

i 7-5 

47.1 

38.2 

Soda. 

3 

5-2 

8.2 

12 

Lime. 

12.9 

17 

29.9 

44-5 

Magnesia.... 

10.6 

9.2 

19.7 

7 - 1 

Oxide of Iron. 

2.6 

2. I 

7‘ 1 

.6 

Phosphoric 1 
Acid.j 

20.6 

25.8 

46-3 

1 5 -i 


Crops. 

Wheat, 

25 

bushels. 

Barley, 

4 ° 

bushels. 

Turnips, 
20 tons. 

c 

£\S 

tE vo 

M 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sulphuric\ 
Acid... j ' * 

10.6 

2.7 

13-3 

9.2 

Chlorine. 

2 

l6 

3.6 

A. I 

Silica. 

118.1 

129.5 

247.8 

78.2 


Alumina. 

_ 

2.4 

_ 






Total.... 

210 

213 

423 

209 





































































190 


MISCELLANEOUS ELEMENTS. 


Average Quantity of Tannin in Several Substances, 

(Morjit.) 


Catechu. 


Per Cent. 


Bombay. 55 

Bengal. 44 

Kino . 75 

Nutgalls. 

Aleppo. 65 

Chinese. 69 

Oak. 


Old, inner bark 


Oak. 


Per Cent. 


Young, inner b’k‘ 15.2 
“ entire b’k. 6 

“ spring- ) ' 

cut bark j 

“ root bark. 8.9 

Chestnut. 

Ainer. rose, bark 8 
14.2 Horse, “ 2 

21 Sassafras, root bark 58 

Alder hark ...'... 36 per cent. 


Sumac. 


Per Cent. 


Sicily and Malaga 16 

Virginia. 10 

Carolina. 5 

Willow. 

Inner bark. 16 

Weeping. 16 

Sycamore bark.... 16 

Tan shrub l - .... 13 

Cherry-tree . 24 


To Convert Chemical Formula; into a NLatliematical 

Expression. 


Rule. —Multiply together equivalent and exponent of each substance, and product 
will give proportion in compound by weight. Divide 1000 by sum of their products, 
and multiply this quotient by each of these products, and products will give re¬ 
spective proportion of each part by weight in 1000. 


Example. —Chemical formula for alcohol is C4 0 2 . Required their propor¬ 
tional parts by weight in 1000 ? 


C\ Carbon = 6.1 X 4 = 24.4 ) 
Hq Hydrogen — 1X6=6 [ 

0 2 Oxygen = 8 x 2 = 16 ) 
1000-7-46.4 



(525-82 

X 21.55 

{129.3 


( 344-8 

= 2 i -55 

999.92 


by weight. 


Elementary Bodies, with, their Symbols and. 
Equivalents. 


Body. 

Symb. 

Equiv. 

Body. 

Symb. 

Equiv. 

Body. 

Symb. 

Aluminium... 

A 1 

13-7 

Gold. 

Au 

196.6 

Platinum .... 

Pt 

Antimony.... 

Sb 

64.6 

Hydrogen.... 

H 

I 

Potassium... 

K 

Arsenic. 

As 

*27 .7 

Iodine. 

I 

126. < 

Rhodium.... 

R 

Barium. 

Ba 

68.6 

Iridium. 

Ir 

Q8.5 

Selenium.... 

Se 

Bismuth. 

Bi 

71 . nr 

Iron. 

Fe 

28 

Silicon. 

Si 

Boron. 

B 


Lead. 

Pb 


Silver. 

Ag 

Bromine. 

Br 

78.4 

Lithium. 

L 


Sodium. 

Na 

Cadmium.... 

Cd 

55-8 

Magnesium.. 

Mg 

12.7 

Strontium.... 

Sr 

Calcium. 

Ca 


Manganese. . . 

Mn 


Sulphur. 

s 

Carbon . 

c 

6.1 

Mercury. 

Hg 

200 

Tellurium.... 

Te 

Chlorine. 

Cl 

35-5 

Molybdenum. 

Mo 

47-9 

Tin . 

Sn 

Chromium... 

Cr 

26.2 

Nickel. 

Ni 

29-5 

Titanium.... 

Ti 

Cobalt. 

Co 

20. 

Nitrogen. 

N 


Tungsten .... 

W 

Columbium. . 

Ta 

184.8 

Osmium. 

Os 

99-7 

Uranium .... 

U 

Copper. 

Cu 

OI. 7 

Oxvgen . 

0 

' 8 

Yttrium . 

Y 

Fluorine . 

F 

18.7 

Palladium.. .. 

Pd 


Zinc . 

Zn 

Glucinum. ... 

G 

6.9 

Phosphorus. . 

P 

JJ 

i 5-9 

Zirconium ... 

Zr 


Equiv. 


98.8 

39 - 2 

52.2 

40 

22 

108.3 

23 - 5 
43-8 

16.1 

64.2 

58.9 

24 - 5 
92 
60 
32 
32-3 
34 


Analysis of certain Organic Substances by Weight. 


Body. 

Car¬ 

bon. 

Hydro¬ 

gen. 

Oxy¬ 

gen. 

Nitro¬ 

gen. 

Albumen. 

52-9 

7-5 

23-9 

I 5-7 

Alcohol. 

52-7 

12.9 

34-4 


Atmospheric air 

— 

— 

77 

23 

Camphor. 

73-4 

IO.7 

15.6 

•3 

Caoutchouc.... 

87.2 

12.8 

— 


Casein.. 

59 - 8 

7-4 

11.4 

21.4 

Fibrin. 

53-4 

7 

19.7 

x 9-9 

Gelatine. 

47-9 

7-9 

27.2 

17 

Gum. 

42.7 

6.4 

5°-9 

— 

Hordein. 

44.2 

6.4 

47.6 

1.8 

Lignin. 

52-5 

5-7 

41.8 

— 


Body. 

Car¬ 

bon. 

Hydro¬ 

gen. 

1 Oxy¬ 
gen. 

Nitro¬ 

gen. 

Morphine. 

72-3 

6.4 

16.3 

5 

Narcotine. 

65 

5-5 

27 

2-5 

Oil, Castor. 

74 

10.3 

I 5-7 


Linseed.... 

76 

11 .3 

12. 7 

_ 

Spermaceti. 

78 

11.8 

IO. 2 

— 

Quinine.. 

75-8 

7-5 

8.6 

8.1 

Starch . 

44.2 

6.7 

49.1 

— 

Strychnine. 

76.4 

6.7 

II. I 

5-8 

Sugar. 

42.2 

6.6 

51.2 


Tannin. 

52.6 

3-8 

43-6 

— 

Urea. 

18.9 

9-7 

26.2 

45-2 





































































































MISCELLANEOUS ELEMENTS. 


I 9 I 


Dilution Per Cent. Necessary to Reduce Spirituous 

Liquors. 

Water to be added to 100 volumes of spirit when of following strength: 


Strength 

Required. 

90 

85 

80 

75 

70 

65 

60 

55 

50 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

Per cent. 

85 

5-9 

— 

— 

— 

— 

— 

— 

— 

— 

80 

12.5 

6-3 

— 

— 

— 

— 

— 

— 

— 

75 

20 

P -3 

6.7 

— 

— 

— 

— 

— 

— 

70 

28.6 

21-4 

14-3 

7 -i 

— 

— 

— 

— 

— 

65 

38-5 

3 °. 8 

23.1 

i 5-4 

7-7 

— 

— 

— 

— 

60 

50 

4 i -7 

33-3 

25 

16.7 

8-3 

— 

— 

— 

55 

63.6 

54-5 

45-5 

36.4 

27.4 

18.2 

9 - 1 

— 

— 

50 

80 

70 

60 

5 ° 

40 

30 

20 

IO 

— 

40 

125 

112.5 

IOO 

87-5 

75 

62.5 

50 

37-5 

25 

30 

200 

183-3 

166.7 

150 

* 33-3 

116.7 

IOO 

83-3 

66.7 


Illustration. —100 volumes of spirituous liquor having 90 per cent, of spirit con¬ 
tains: alcohol 90, water io, = 100. 

To reduce it to 30 per cent, there is required 200 volumes of water. 

-r-r 1 . 9° 30 30 spirit, . 

Hence 200 + 10 = 210, and - z — = •—= , or 30 percent. 

210 70 70 water, 

Proportion of .Alcoliol Per Cent. 


Alcohol. 

In 100 

Specific 

Gravity. 

Parts of 

Alcohol. 

'Spirit, by 

Specific 

Gravity. 

Weight c 

Alcohol. 

r Volume , 

Specific 

Gravity. 

at 6o°. 

Alcohol. 

Specific 

Gravity. 

O 

5 

10 

Alcohol. 

I 

.991 

•984 

In 100 7 

Specific 

Gravity. 

20 

30 

40 

D arts of 

Alcohol. 

.972 

•958 

•94 

Alcohol am 

Specific 

Gravity. 

50 

60 

70 

l Water 

Alcohol. 

.918 

.896 

.872 

by Weight 

Specific 

Gravity. 

80 

90 

IOO 

at 6o°. 

Alcohol. 

.848 

.823 

•794 

Specific 

Gravity. 

0 

•53 

1.02 

I 

•999 

.998 

I -99 

3.02 

4.02 

.996 

•994 

•993 

5.01 

6.02 

7.02 

.991 

•99 ' 
.988 

7-99 

9-°5 

10.07 

.987 

.•985 

.984 


Tides of Atlantic and Pacific Oceans at Isthmus of 

Panama. [Totten.) 

Atlantic , Navy Bay .—Highest tide 1.5 feet; lowest .63 feet. 

Pacific , Panama Bay .—Highest tide 17.72 to 21.3 feet; lowest 9.7 feet. 


Areas of TJ. S. Coal Fields. 


State. 

Sq. Miles. 

State. 

Sq. Miles. 

State. 

Sq. Miles. 

Illinois 

44 000 
21 000 

15 437 
i 3 5 °° 

Ohio. 

II 900 

7 700 

6 000 
5000 

Tennessee. 

4300 

3400 

550 

150 

Virginia. 

Pennsylvania*... 
Kentucky. 

Indiana. 

Missouri!. 

Michigan!. 

Alabama. 

Maryland. 

Georgia. 


* Bituminous and Anthracite. t Anthracite. 


Extremes of Heat in Various Countries. 


England.. 

. 96° 

Denmark 1 


Greece... 

.. 105 0 

Egypt,... 


France... 


Sweden > 

• • 99 - 5 ° 

Italy. 


Africa.... 

•• 133-4° 

Holland 1 


Norway ) 


Spain.... 


Asia. 


Belgium | 

. 102^ 

Russia .... 


Tunis. 


Suez. 



Germany. 


Manilla... 

... 113.5 0 




Extremes of temperature upon the Earth 240 0 . 


Extremes of Cold in Various Countries. 


England...— 5 0 
Holland 1 
Belgium j 12 


Denmark 1 
Sweden / —67° 
Norway ) 


France_—24 0 

Russia ....—46° 
Germany..—32 0 


Italy. 

Fort Reliance, N. A.. 
Semipalatinsk, “ .. 

































































































192 


MISCELLANEOUS ELEMENTS. 


Nlean Temperatures of 'Various localities. 


London . 

•• 5 i° 

Rome. 

... 6o° 

Poles.—13 0 

Edinburgh .. 

.. 41 0 

Equator.... 

... 82° 

Torrid Zone. 75 0 


Polar Regions.. 36° 
Globe.50 0 


Bine of Perpetual Congelation, or Snow Line. 


Latitude. 

Height. 

Latitude. 

Height. 

Latitude. . 

Height. 

Latitude. 

Height. 

O 

Feet. 

O 

Feet. 

O 

Feet. 

O 

Feet. 

IO 

14764 

30 

11 484 

50 

6334 

70 

1278 

15 

14 760 

35 

10 287 

55 

5020 

75 

1016 

20 

13 47 8 

40 

9 000 

60 

3818 

80 

451 

25 

12557 

45 

7 670 

' 65 

2230 

85 

327 


At the Equator it is 15260 feet; at the Alps 8120 feet; and in Iceland 3084 feet 
At Polar Regions ice is constant at surface of the" Earth. 


Limits of 'Vegetation in Temperate Zone. 

The Vine ceases to grow at about 2300 feet above level of the sea, Indian Corn at 
2800, Oak at 3350, Walnut at 3600, Ash at 4800; Yellow Pine at 6200, and Fir at 6700. 


Periods of Grestation and Number of Young. 

Weeks. 

Elephant. 100 

Horse.... { 43 

15o 

Camel.... 45 

Ass. 43 


No. 


Weeks. 

No. 


Weeks. 

No. 


Weeks. 

I 

Cow.... 

• 4 1 

I 

Sheep.. 

. 21 

2 

d)cg- 

.. 9 


Buffalo. 

. 40 

I 

Goat... 

. 22 

2 

Fox.... 

• • 9 

I 

Stag.... 

• 36 

I 

Beaver. 

• w 

3 

Cat. 

.. 8 

I 

Bear... 

• 3 ° 

2 

Pig. 

• 17 

12 

Rat.... 

• • 5 

•I 

Deer ... 

. 24 

2 

Wolf... 

IO 

5 

Squirrel 

.. 4 


Rabbit. 


4 6 Guinea Pig. 3 


No. 

6 

5 

6 
8 
6 


Periods of Incubation of Birds. 

Swan. 42 days; Parrot, 40 days; Goose and Pheasant, 35 days; Duck, Turkey, and 
Peafowl, 28 days; Hens of all gallinaceous birds, 21 days; Pigeon and Canary, 14 
days. Temperature of incubation is 104 0 . 

AAges of Animals, etc. 


Whale, estimated 1000 years; Elephant, 400; Swan, 300; Camel, 100; Eagle, 100; 

Raven, 100; Tortoise, 100 to -; Lion, 70; Dolphin, 30; Horse, 30; Porpoise, 30; 

Bear, 20; Cow, 20; Deer, 20; Rhinoceros, 20; Swine, 20; Wolf, 20; Cat, 15; Fox, 15; 
Dog, 15; Sheep, 10; Hare, Rabbit, and Squirrel, 7. 


Relative Weiglats of Brain. 

Man, 154.33; Mammifers, 29.88; Birds, 26.22; Reptiles. 4.2; Fish, 1. 

Buoyancy of Casks. 

Buoyancy of a cask in fresh water in lbs. =11.97 times volume of it in U. S. gal¬ 
lons and 10 times in Imperial gallons, less weight of cask. 

Transportation of Horses and Cattle. 

Space required on board of a Marine Transport is: for Horses, 30 ins. by 9 feet; 
Beeves, 32 ins. by 9 feet. Provender required per diem is: for Horses, Hay, 15 lbs.; 
Oats, 6 quarts; Water, 4 gallons. Beeves, Hay, 18 lbs.; Water, 6 gallons. 

Rock and Earth Excavation and Embankment. 

Number of Cube Feet of various Earths in a Ton. 

Loose Earth. 24 | Clay. 18.6 I Clay with Gravel_ 14.4 

Coarse Sand. 18.6 | Earth with Gravel... 17.8 | Common Soil. 15.6 

The volume of Earth and Sand in embankment excaeds that in a primary ex¬ 
cavation in following proportions: 

Rock, large.1.5 I Rock, ballast.1.2 

“ medium.1.25 | Sand...143 

Clay and Earth will subside about .12. 


I Clay. hi 

I Gravel.. .oq 


























































MISCELLANEOUS ELEMENTS. 


193 


Hills or I? 1 ants in an Area of One A.cre. 
From 1 to 40 feet apart from centres. 


Feet apart. 

No. 

Feet apart. 

No. 

Feet apart. 

No. 

Feet apart. 

No. 

I 

43560 

5 

1742 

9 

538 

16 

171 

i-5 

19 360 

5-5 

1440 

9-5 

482 

W 

I5i 

2 

10890 

6 

I2IO 

IO 

435 

18 

135 

2.5 

6969 

6-5 

1031 

10.5 

361 

20 

108 

3 

4 840 

7 

889 

12 

302 

25 

69 

3-5 

3 556 

7-5 

775 

13 

258 

3° 

48 

4 

2 722 

8 

680 

14 

223 

35 

35 

4-5 

2151 

8-5 

692 

15 

i 93 

4 ° 

27 


Number of several Seeds in a 33 nsh.el, and Number per 


Square Foot per Acre. 



No. 

Sq. Foot. 


No. 

Sq. Foot 

Timothy. 

41 823 360 

960 

Rye. 

888 390 

20.4 

Clover. 

16 400 960 

376 

Wheat. 

556 290 

12.8 



Volumes. 




Permanent gases, as air, etc., are diminished in their volume in a ratio direct 
with that of pressure applied to them. With vapor,’ as steam, etc., this rule is 
varied in consequence of presence of the temperature of vaporization. 


Minerals. 

Relative Hardness of some Nlinerals. 


Talc. 

I 

Barytes. 

3-5 

Opal. 

.. 6 

Emerald.... 

.. 8 

Gypsum. 

2 

Fluor-spar... 

4 

Quartz. 

•• 7 

Topaz. 

.. 8 

Mica. 

2-5 

Feldspar .... 

6 

Tourmalin .. 

•• 7 

Ruby...... 

.. 9 

Carbonate of lime. 

3 

Lapis Lazuli 

6 

Garnet. 

• 7-5 

Diamond.... 



Weiglit of Diamonds. 
Carats. Carats. 

Mattam . . 367 i Regent or Pitt.136.75 

Grand Mogul*.279.9 | Star of the Southf.. 125 

Orloflf..194-25 I Koh-i-Noorf.106.06 

Florentine, brilliant . 139.5 i Piggott. 82.25 

Crown of Portugal... 138.5 | Napac. 78.625 

* Rough 900. t Rough 254.5. 


Carats. 


Dresden.76.5 

Sancy.53.5 

Eugenie, brilliant. 51 

Hope (blue)...48.5 

Polar Star.40.25 


X Originally 793. 


Heat of tlie Sun. 

Sir Isaac Newton. 3 138740 0 I Waterston. 16000000° 

Capt. John Ericsson. 4 909 860° | Soret. 10 443 323° 

Sundry others ranging from 2520° to 183600°. 

Moon. —Distance of Moon from Earth 237000 miles. 


Frigorific HVIixtore. 

Lowest temperature yet procured. Faraday obtained 166° by evaporation of a 
mixture of solid carbonic acid and sulphuric ether. 


Current of Rivers. 

A fall of. 1 of an inch in a mile will produce a current in rivers. 


Sandstones. 

Structures of sandstone erected in England in 12th century are yet in good 
condition. 

Canal Transportation. 

Erie Canal and Hudson River.— From Buffalo to New York, 495 miles, cost of 
transportation 2.46 mills per ton (inclusive of tolls) per mile. Transportation of 
wheat costs when it reaches New York 4.72 cents per bushel, and .61 cents per 
bushel for elevating and trimming. 

Towing.—Erie Canal.— Four mules will tow 230 tons of freight down and 100 
tons back, involving a period of 30 days, at a cost of 8 cents per mile for a course 
of 690 miles. 

R 



































































i 9 4 


MISCELLANEOUS ELEMENTS. 


Matter. 

Unit of the Physicist is a molecule, and a mass of matter is composed of them, 
having same physical properties as parent mass. 

It exists in three forms, known as solid, liquid, and gaseous. Solids have indi¬ 
viduality of form, and they press downward alone. Liquids have not individuality 
of form, except in spherical form of a drop, and they press downward and sideward. 
Gases are wholly deficient in form, expanding in all directions, and consequently 
they press upward, downward, and sideward. 

Liquids are compressible to a very moderate degree. Water has been forced 
through pores of silver, and it may be compressed by a pressure of one pound per 
square inch to the 3 300 oooth part of its volume. » 

Gases may be liquefied by pressure or by reduction of their temperature. 

Combustible matter (as coal) may be burned, a structure (as a house) may be 
destroyed as such, and the fluid (of an ink) may be evaporated, yet the matter of 
which coal and house, were composed, although dissipated, exists, and the water 
and coloring matter of the ink are yet in existenoe. 

Spaces between the particles of a body are termed pores. 

All matter is porous. Polished marble will absorb moisture, as evidenced in its 
discoloration by presence of a colored fluid, as ink, etc. 

Silica is the base of the mineral world, and Carbon of the organized. 

NLiniateness of HVtatter. 

A piece of metal, stone, or earth, divided to a powder, a particle of it, however 
minute, is yet a piece of the original material from which it was separated, retain¬ 
ing its identity, and is termed a molecule. 

It is estimated there are 120000000 corpuscles in a drop of blood of the musk-deer. 

Thread of a spider’s web is of a cable form, is but one sixth diameter of a fibre of 
silk, and 4 miles of it is estimated to have a weight of but 1 grain. 

One imperial gallon (277.24 cube ins.) of water will be colored by mixture therein 
of a grain of carmine or indigo. 

A grain of platinum can be drawn out the length of a mile. \ 

Film of a soap-and-water bubble is estimated to be but the 300000th part of an 
inch in thickness. 

It is computed that it would require 12000 of the insect known as the twilight 
monad to fill up a line one inch in length. 

A drop of water, or a minute volume of gas, however much expanded—even to 
the volume of the Earth—would present distinct molecules. 

Gold leaf is the 280000th part of an inch in thickness. 

A thread of silk is 2500th of an inch in diameter. 

A cube inch of chalk in some places in vicinity of Paris contains 100000 of shells 
of the foraminifera. 

Ihere are animalcules so small that it requires 75 000 000 of them to weigh a grain. 

Velocity, Weight, and Volume of Molecules. 

Velocity. —Collisions among the particles of Hydrogen are estimated to occur at 
the rate of 17 million-million-million per second, and in Oxygen less than half this 
number. 


Weight. A million-million-million-million molecules of Hydrogen are estimated 
to weigh but 60 grains. 

Volume. —19 million-million million molecules of Hydrogen have a volume of.061 
cube ins. Diameter. —Five millions in a line would measure but .1 inch. 


Charcoal, Alcohol, 

Charcoal as yet has not been liquefied, nor has Alcohol been solidified. 

NLetals. 

dull^ S ^ aVC ^ VG de £ rees °f lustre— splendent, shining,'glistening, glimmering, and 

All metals can be vaporized, or exist as a gas, by application to them of their ap¬ 
propriate temperature of conversion. p 

Repeated hammering of a metal renders it brittle; reheating it restores its tenacity. 
sti-on"er te<i me tlDS of iron renders harder, and up to twelfth time it becomes 

Platinum is the most ductile of all metals. 


MISCELLANEOUS ELEMENTS. 


195 


Impenetrability-. 

Impenetrability expresses the inability of two or more bodies to occupy same 
space at same time. 

A mixture of two or more fluids may compose a less volume than that due to sum 
of their original volume, in consequence of a denser or closer occupation of their 
molecules. This is evident in the mixture of alcohol and water in the proportion 
of 16.5 volumes of former to 25 of latter, when there is a loss of one volume. 

Elasticity. 

Elasticity is the term for the capacity of a body to recover its former volume, 
after being subjected to compression by percussion or deflection. 

Glass, ivory, and steel are the most elastic of all bodies, and clay and putty are 
illustrations of bodies almost devoid of elasticity. Caoutchouc (India rubber) is but 
moderately elastic; it possesses contractility, however, in a great degree. 

Momentum. 

Momentum is quantity of motion, and is product of mass and its velocity. Thus, 
the momentum of a cannon ball is product of its velocity in feet per second and its 
weight, and is denominated foot-pounds. 

A foot-pound is the power that will raise one pound one foot. 

Sound, 

Velocity of sound is proportionate to its volume; thus, report of a blast with 2000 
lbs. of powder passed 967 feet in one second, and one of 1200 lbs. 1210 feet. It passes 
in water with a velocity of 4708 feet per second. Conversation in a low tone has 
been maintained through cast-iron water pipes for a distance of 3120 feet, and its 
velocity is from 4 to 16 times greater in metals and wood than air. 

Eight. 

Sun’s rays have a velocity of 185000 miles per second, equal to 7.5 times around 
the Earth. 

Color Blindness 

Is absence of elementary sensation corresponding to red. 

Luminous Point. 

To produce a visual circle, a luminous point must have a velocity of 10 feet in a 
second, the diameter not exceeding 15 ins. 

All solid bodies become luminous at 800 degrees of heat. 

Mirage. 

When air near to surface of Earth becomes so highly heated, as upon a sandy 
plain, that its density within a defined distance from it increases upwards, a line 
of vision directed obliquely downwards will be rendered by refraction, gradually 
increasing, more and more nearly horizontal as it advances, until its direction is so 
great as to produce a total reflection, and the reflected ray then, by successive re¬ 
fractions, is gradually elevated until it meets the eye of the observer. 

Looming is inverted mirage, frequently seen over calm water, and is effect of 
lower or surface stratum of air being colder than that above it. 

Snow Flakes. 

96 forms of snow flakes have been observed. 

Nlelted Snow- 

Produces from .25 to .125 of its bulk in water. 

Strength, of Ice. 

Two inches thick will support men in single file on planks 6 feet apart; 4 inches 
will support cavalry, light guns, and carts; and 6 inches wagons drawn by horses. 

Temperature. 

Sulphuric acid and water produce a much greater proportionate contraction than 
alcohol and water. Both of these mixtures, however low their temperature, pro¬ 
duce heat which is in a direct proportion to their diminution in volume. 

At the depth of 45 feet, the temperature of the Earth is uniform throughout the 
year. 

Temperature of Earth increases about x° for every 50 to 60 feet of depth, and its 
crust is estimated at 30 miles. 

, A body at Equator weighs two hundred and eighty-nine parts less than at the Poles. 


MISCELLANEOUS ELEMENTS, 


I96 


Colors for Drawings. 


Material. 

COLOES. 

Material. 

COLOES. 

Brass .... 
Brick .... 
Cast Iron. 
Clay. 

Gamboge. 

Carmine. 

Neutral tint. 

Burned umber. 

“ “ light. 

Sepia with dark spots. 
Lake and Bur’d Sienna. 

Granite. 

Lead. 

Indian Ink, light. 

• “ “ and Prussian blue. 

Light blue and Lake. 

Cobalt or Verdigris. 

(Burned Sienna, deep and light, 
{ for dark and light wood. 
Prussian Blue, light. 

Steel. 

Water. 

Earth.... 
Concrete . 
Copper... 

Woods. 

Wr’ght Iron. 


Birds aird Insects. *-(d/. De Lacy.) 

Elements of Flight. —Resistance of air to a body in motion is in ratio of surface 
of body and as square of its velocity. 

Wing Surface. —Extent or area of winged surface is in an inverse ratio to 'weight 
of bird or insect. 

A Stag-beetle weighs 460 times more than a Gnat, and has but one fourteenth of 
its wing surface; 150 times more than a Lady Bird (bug), and has but one fifth. 
An Australian Crane weighs 339 times more than a sparrow, and has but one sev¬ 
enth; 3000000 times more than a Gnat, and has but one hundred and fortieth. A 
Stork weighs eight times more than a Pigeon, and has but one half. A Pigeon 
■weighs ten times more than a Sparrow, and has but one half; 97 000 times more than 
a Gnat, and has but one fortieth. 

A resisting surface of 30 sq. yards will enable a man of ordinary weight to descend 
safely from a great elevation. 

Strength of Insects. —Insects are relatively strongest of all animals. A Cricket 
can leap 80 times its length, and a Flea 200 times. 

-Application fox* Stiixgs and Burns. 

Sting of Insects. —Ammonia, or Soda moistened with water, and applied as a paste. 

Burns. —Hot alcohol or turpentine, and afterwards bathed with lime water and 
sweet oil. Cold water not to be applied. 

To Preserve INI eat. 

Meat of any kind may be preserved in a temperature of from 8 o° to ioo°, for a 
period of ten days, after it has been soaked in a solution of 1 pint of salt dissolved 
in 4 gallons of cold water and .5 gallon of a solution of bisulphate of calcium. 

By repeating this process, preservation maybe extended by addition of a solution 
of gelatin or white of an egg to the salt and water. 

To Detect Starclx iix Ntilli. 

Add a few drops of acetic acid to a small quantity of milk; boil it, and 
after it has cooled filter the w T hey. If starch is present, a drop of iodine 
solution will produce a blue tint. 

This process is so delicate that it will show the presence of a milligram of starch 
in a cube centimeter of wiiey (1 grain of starch in 2.16 fluid-ounces). 

Betainixxg Walls of” Iroxx Biles. 

Sheet Piles. —7 feet from centres, 18 ins. in width and 2 ins. in thickness, strength¬ 
ened with 2 ribs 8 ins. in depth. 

Plates. —7 feet in length by 5 feet in width and 1 inch in thickness, with one 
diagonal feather 1 by 6 ins. 

Tie-rods 2 ins. in diameter. 

Stone Sawing.' 

Diamond Stone Sawing. — (Emerson.) Alabama marble 6 feet X 2.5 feet in 22 min¬ 
utes — 41 sq. feet per hour. 

Wood Sawing. 

7722 feet of poplar, board measure, from 9 round logs in 1 hour. Engine 12 ins. 
diameter by 24 ins. stroke. 
















MISCELLANEOUS ELEMENTS:. 


197 


Cost of Dredging. 

Actual cost, if on an extended tvork , inclusive of Delivery, if dredging into or on a 
vessel alongside of dredger. — (Trautwine.) 

Labor at $ 1 per day and Repairs of Plant included. 


Depth. 

Cents. 

Depth. 

Cen ts. 

Depth. 

Cents. 

Depth. 

Cents. 

Feet. 

10 

IS 

Cube Yards. 

6 

7 

Feet. 

20 

22 

Cube Yards. 

8 

9 

Feet. 

25 

30 

Cube Yards. 
10 

13 

Feet. 

35 

40 

Cube Yards. 
18 

25 


Discharge of Scows or Camels. —Towing .25 mile 4 cents per cube yard, .5 mile 6 
cents, .75 mile 8 cents, and 1 mile 10 cents. 

Note. —A Scow is a flat-bottomed vessel or boat. A Camel is a shallow, flat- 
bottomed and decked vessel, designed for the transportation of heavy freight or the 
sustaining of attached bodies, as a vessel, by its buoyancy. 

Dredging. 

A steam dredge will raise 6 cube yards, or 8.5 tons, per hour per IP. 

NLetal Doring and Turning. 

Boring. — Cast iron. —Divide 25 by the diameter of the cylinder in inches for the 
revolutions per minute. 

Wrought iron. —The speed is one fifth greater than for cast iron. 

Brass. —The speed is one half that for cast iron. 

Turning.— Cast iron. — The speed is twice that of boring. 

Wrought iron,— The speed is one fifth greater than that for cast iron. 

Brass. —The speed is twice that of boring. 

Vertical boring.— The speed may be twice that of horizontal boring. 

The feed depends upon the stability of the machine and depth of the cut. 

"Well Boring. 

At Coventry, Eng., 750000 galls, of water per day are obtained by two borings of 
6 and 8 ins., at depths of 200 and 300 feet. 

At Liverpool, Eng., 3000000 galls, of waiter per day are obtained by a bore 6 ins. 
in diameter and 161 feet in depth. 

This large yield is ascribed to the existence of a fault near to it, and extending to 
a depth of 484 feet. 

At Kentish Town, Eng., a well is bored to the depth of 1302 feet. 

At Passy, France, a well with a bore of 1 meter in diameter is sunk to a depth of 
1804 feet, and for a diameter of 2 feet 4 ins. it is further sunk to a depth of 109 feet 
10 ins., or 1903 feet 10 ins., from which a yield of 5 582 000 galls, of water are obtained 
per day. 

Tempering Boring Instruments. 

Heat the tool to a blood-red heat; hammer it until it is nearly cold; reheat it to 
a blood-red heat, and plunge it into a mixture of 2 oz. each of vitriol, soda, sal-am¬ 
moniac, and spirits of nitre, 1 oz. of oil of vitriol, .5 oz. of saltpetre, and 3 galls, of 
water, retaining it there until it is cool. 

Circular Saws. 

Revolutions per Minute. —8 ins. 4500, 10 ins. 3600, and 36 ins. 1000. 

Masonry. 

Concrete or Beton should be tlirow'n, or let fall from a height of at least 10 feet, 
or well beaten down. 

The average weight of brickw r ork in mortar is about 102 lbs. per cube foot. 

Plastering. 

In measuring Plasterers’ work all openings, as doors, windows, etc., are com¬ 
puted at one half of their areas, and cornices are measured upon their extreme 
edges, including that cut off by mitring. 

Gflazing. 

In Glaziers’ work, oval and round windows are measured as squares. 

R* 



















igS 


MISCELLANEOUS ELEMENTS, 


Corn Nleasnre. 

Two cube feet of corn in ear will make a bushel of corn when shelled. 

Tenacity of Iron Bolts in Woods. 

Diameter 1.125 ins. and 12 ins. in length required for Hemlock 8 tons, and for 
Pine 6 tons to withdraw them. 

Length of Gun Barrels. (C. T. Coathupe.) 

The length of the barrel of a gun, to shoot well, measured from vent-hole, should 
not be less than 44 times diameter of its bore, nor more than 47. 

I 

Bay and. Straw. 

Hay, loose. 5 lbs. per cube foot. Ordinarily pressed, as in a stack or mow, 8 lbs. 
Close pressed, as in a bale, 12 to 14 lbs. 

Ordinarily pressed, as in a wagon load, 450 to 500 cube feet will weigh a ton. 

Straw in a bale 10 to 12 lbs. per cube foot. 

Natural Powers. 

Sun .—The power or work performed by the Sun’s evaporation is estimated at 
90 000 000 000 BP. 

Niagara .—Volume of water discharged over the falls is estimated at 33000000 
tons per hour, and the entire fall from Lake Erie at Buffalo to Lake Ontario is 323.35 
feet. 

Velocity of Stars. 

According to computation of Mr. Trautwine a Star passes a range in 3' 55.91" less 
time each day. 

Service Train of a Quartermaster. 

Quartermaster’s train of an army averages 1 w r agon to every 24 men; and a well- 
equipped army in the field, with artillery, cavalry, and trains, requires 1 horse or 
mule, upon the average, to every 2 men. 

Tides. 

The difference in time between high water averages about 49 minutes each day. 

Atlantic and Pacific Oceans .—Kise and fall of tide in Atlantic at Aspinwall 2 feet, 
in Pacific at Panama 24 feet. 

Dimensions of Drawings for Patents. 

United States, 8.5 X 12 inches. 

Batitnde. 

One minute of latitude, mean level of Sea, nearly 6076 feet = 1.1508 Statute miles. 

Artesian Well. 

White Plains, Nev., Depth 2500 feet. 

Foundation Biles. 

A pile, if driven to a fair refusal by a ram of 1 ton, falling 30 feet, will bear 1 ton 
weight for each sq. foot of its external or frictional surface, or a safe load of 750 lbs. 
per sq. foot of surface. 

Earth. 

Density of its mass 5.67. 

TripolitH. 

A new building material, compounded of Coke, Sulphate of Lime, and Oxide of 
Iron. It has increased tensile strength after exposure to the air, being much in 
excess of that of lime and cement. 

Gf as and Electric XJigLt. 

Gas light of 16 candle power costs 5 cent per hour; Electric, 4.15 cents. 

1 ST iagara. 

Discovered, 1678. Falls have receded 76 feet in 175 years. Height, American 
Falls, 164 feet; Horseshoe, 158 feet. 


BRIDGES.-U. S. ENSIGNS, PENNANTS, AND FLAGS. 1 99 


Suspension Bridges. 
Lengths of Spans in Feet. 


You-Mau, China. 

Schuylkill (P'nila.). 

Hammersmith, Eng. 

Pesth (Danube). 

New York and Brooklyn 
water at qo°, 135 feet. 


330 

342 

422 

660 


Niagara . 

Lewistown and Queenstown 

Cincinnati. 

Niagara Falls... 


822 

1040 

1057 

1280 


, 93 °i * 595 - 5 > an( i 930; clear height of Bridge above high 


IT. S. ENSIGN, PENNANTS, AND FLAGS. 

Ensign. —Head (Depth, or Hoist). — Ten nineteenths of its length. 

Field.' —Thirteen horizontal stripes of equal breadth, alternately red and 
white, beginning with red. 

Union .—A blue field in upper quarter, next the head, .4 of length of field, 
and seven stripes in depth, with white stars ranged in equidistant, horizon¬ 
tal, and vertical lines, equal in number to number of States of the Union. 

Pennants (Narrow).— Head. —6.24 ins. to a length of 70 feet; 5.04 ins. to a 
length of 40 feet; 4.2 ins. to a length of 35 feet. Night , 3.6 ins. to a length of 20 
feet, and 3 ins. to a length of 9 feet.— Boat , 2.52 ins. to a length of 6 feet. 

Union. —A blue field at head, one fourth the length, with 13 white stars in a hori¬ 
zontal line. Field. —A red and white stripe uniformly tapered to a point, red up¬ 
permost. Night and Boat Pennants. —Union to have but 7 stars. 

Union Jack.—Alike to the Union of an Ensign in dimensions and stars. 

Elags.— ^President. —Rectangle, with arms of the U. S. in centre of 
a blue field. 

Secretary of Navy.— Rectangle, with a vertical white foul anchor 
in centre of a blue field. 

Admiral.— Rectangle, with 4 white stars in centre of a blue field, set as 
a square. 

Vice-Admiral. —Same as Admiral’s, with 3 white stars set as an equi¬ 
lateral triangle. 

Rear-Admiral.— Same as Admiral’s, with two white stars set vertical. 

If two or more rear-admirals in command afloat should meet, their seniority is to 
be indicated respectively by a Blue flag, a Red with White stars, and a White with 
Blue stars, and another or all others, a White flag with Blue stars. 

Commodore’s ( Broad Pennant). —Blue, Red, or White, according to 
rank, with one star in centre of field, being white in blue and red pennants, 
and blue in white. 

Swallow-tailed, angle at tail, bisected by a line drawn at a right angle from centre 
of depth or hoist, and at a distance from head of three fifths of length of pennant; 
the lower side rectangular with head or hoist; upper side tapered, running the width 
of pennant at the tails .1 the hoist. Head.—. 6 length. Fly 1.66 hoist. 

Divisional NLarlrs. — Triangle, 1st Blue, 2d Red, 3d White. Blue 
vertical. Reserve Division. —Yellow Red vertical. Division mark is worn 
by Commander of a division of a squadron at mizzen, when not authorized 
to wear Broad Pennant of a Commander or Flag of an Admiral. Fly .8 hoist. 

Signal Numbers.— Fly 1.25 hoist. Signal Pennants, Fly 4.6 hoist. 
Repeaters 1.89 hoist. 

International , Signal Number , Square , Signal Pennants. Fly .3 hoist. 












200 


ANIMAL FOOD. 


-AJirtieiatary Principles. 

Primary division of Food is into Organic lyid Inorganic. 

Organic is subdivided into Nitrogenous and Non-Nitrogenous; Inorganic 
is composed of water and various saline principles. The former elements 
are destined for growth and maintenance of the body, and are termed “ plas¬ 
tic elements of nutrition.” The latter are designed for undergoing oxidation, 
and thus become source of heat, and are termed “ elements of respiration,” or 
“ Calorihcient.” 

Although Fat is non-nitrogenous, it is so mixed with nitrogenous matter that it 
becomes a nutrient as well as a calorihcient. 

Alimentary Principles. —i. Water; 2. Sugar; 3. Gum; 4. Starch; 5. Pectine; 
6. Acetic Acid; 7. Alcohol; 8. Oil or Fat. Vegetable and Animal. —9. Albumen; 
10. Fibrine; n. Caseine; 12. Gluten; 13. Gelatine; 14. Chloride of Sodium. 

These alimentary principles, by their mixture or union, form our ordinary foods, 
which, by way of distinction, may be denominated compound aliments; thus, meat 
is composed of fibrine, albumen, gelatine, fat, etc.; wheat consists of starch, gluten, 
sugar, gum, etc. 


./Analysis of Aleats, Fisli, 'Vegetables, etc. 


Food. 

Water. 

Nitro¬ 

genous 

Matter. 

Fat. 

[ Saline 
Matter. 

hon-Nitro- 

genous 

Matter. 

Sugar. 

Cellu¬ 

lose. 

Ash, etc 

Arrowroot. 

18 

_ 


_ 

82 


_ 


Barley Meal. 

15 

6 -3 

2.4 

2 

69.4 

4.9 

_ 

_ 

Beans, White. 

9.9 

25-5 

2.8 

— 

55-7 


2.9 

3-2 

Beef, roast. 

54 

27.6 

15-45 

2.95 


— 



fat. 

5 i 

14.8 

29.8 

4-4 

— 

— 

— 

— 

lean. 

72 

19-3 

3-6 

S-i 

— 

— 

— 

— 

salt. 

49.1 

29.6 

.2 

21.1 

— 

— 

_ 

_ 

Beer and Porter.... 

91 

. I 

— 

.2 

8.7 

_ 

_ 

_ 

Buckwheat. 

13 

T 3 - 1 

3 

•4 

64-5 

— 

3-5 

2.5 

Butter and Fats.... 

15 


83 

2 


_ 



Cabbage. 

9 1 

2 

•5 

• 7 

5-8 

— 

_ 

_ 

Carrots. 

83 

i -3 

.2 

I 

7-4 

6.1 

_ 

I 

Cheese. 

36.8 

33-5 

24-3 

5-4 


— 

_ 

_ 

Corn Meal. 

14 

II. I 

8.1 

i -7 

57 - 6 

•4 

5-9 

1.2 

Cream. 

66 

2.7 

26.7 

1.8 


2.8 


_ 

Egg. 

74 

14 

10.5 

i -5 

— 

— 

_ 

_ 

volk. 

5 ? 

l6 

3°-7 

i-3 

— 

— 

_ 

_ 

Fish, white hesh... 

78 

18.1 

2.9 

I 

— 

— 

_ 

_ . 

Eels. 

75 

9.9 

13.8 

i -3 

— 

— 

_ 

_ 

Lobster, hesh. 

76.6 

I 9- I 7 

1.17 

1.8 

1.26 

— 

_ 

_ 

Oysters. 

80.39 

14.01 

1.52 

2.7 

1-38 

— 

_ 

_ 

Liver, Calf’s. 

7 2 -33 

20.55 

5-58 

i -54 


— 

_ 

_ 

Milk, Cow’s. 

86 

4.1 

3-9 

.8 

— 

5-2 

_ 

1 • 

Mutton, fat. 

53 

12.4 

3 1 - 1 

3-5 

— 


_ 

- 

Oatmeal. 

15 

12.6 

5-6 

3 

58-4 

5-4 

— 

_ 

Oats. 

21 

14.4 

5-5 


48.2 


7.6 

3.3 

Parsnips . 

82 

I.I 

•5 

I 

9.6 

5-8 



Peas. 

15 

23 

2. I 

2-5 

50.2 

2 

3 - 1 

2.1 

Pork, fat. 

39 

9.8 

48.9 

2-3 

— 

_ 



Bacon, dry... 

15 

8.8 

73-3 

2.9 

_ 

_ 



Potatoes. 

75 

2. I 

.2 

•7 

16.8 

3-2 

I 

I 

Poultry. 

74 

21 

3-8 

1.2 

— 




Rice. 

13 

6-3 

•7 

• 5 

78.1 

•4 

_ 

I 

Rye Meal. 

i 5 

8 

2 

x.8 

6 9-5 

3-7 

_ 


Sugar. 

5 

— 

— 

— 

95 


_ 

_ 

Tripe. 

68 

13.2 

16.4 

2.4 


_ 


_ - 

Turnips. 

9 1 

1.2 

— 

.6 

4-3 

2. I 

, 

.8 

Veal. 

63 

16.5 

15-8 

4-7 


_ 

_ 


Wheat Flour. 

15 

10.8 

2 

i -7 

6l.I 

4.2 

3.5 

1.7 

Bread*. 

37 

8.1 

1.6 

2-3 

45-4 

3-6 


2 

Bran. 

13 

18 

6 


60 


— 

3 


* Water absorbed by flour varies from 40 to 60 per cent, of weight of flour, the best quality absorb¬ 
ing most. 100 lbs. flour yield 130 lbs. bread. 





























































ANIMAL FOOD 


201 


Analysis of” Different Foods 

In their Natural Condition. 



Ni¬ 

trates. 

Carbon¬ 

ates. 

Phos¬ 

phates. 

Water. 


Ni¬ 

trates. 

Carbon¬ 

ates. 

Phos¬ 

phates. 

Water 

Apples. 

5 

IO 

I 

84 

Milk of cow.. 

5 

8 

I 

86 

Barley. 

17 

69-5 

3-5 

IO 

Mutton. 

12.5 

40 

4-5 

43 

Beans . 

24 

57-7 

3-5 

14.8 

Oats. 

17 

66.4 

3 

13.6 

Beef. 

15 

3 ° 

5 

50 

Parsnips. 

9.2 

7 

I 

82.8 

Buckwheat .. 

8.6 

75-4 

1.8 

14.2 

Pork. 

IO 

5 o 

i -5 

38-5 

Cabbage . 

4 

5 

I 

9 ° 

Potatoes . 

2.4 

22.5 

•9 

74.2 

Chicken. 

19 

3-5 

4-5 

73 

“ sweet 

i -5 

28.4 

2.6 

67.5 

Corn,North’n 

12 

73 

I 

14 

Rice. 

6-5 

79-5 

.5 

13.5 

“ South’n 

35 

48 

0 

D 

*4 

Turnips. 

5 

4 

• 5 

90.5 

Cucumbers... 

i -5 

I 

•5 

97 

Veal. 

l6 

16.5 

4-5 

63 

Lamb. 

II 

35-5 

3-5 

50 

Wheat. 

i5 

69.2 

1.6 

14.2 


Nitrates —Are that class which supplies waste of muscle. 

Carbonates —Are that class which supplies lungs with fuel, and thus furnishes heat 
to the system, and supplies fat or adipose substances. 

Phosphates —Are that class which supplies bones, brains, and nerves, and gives 
vital power, both muscular and mental. 

From above it appears, that Southern corn produces most muscle and least fat, 
and contains enough of phosphates to give vital power to brain, and make bones 
strong. Mutton is the meat which should be eaten with Southern corn. 

The nitrates in all the fine bread which a man can eat will not sustain life beyond 
fifty days; but others, fed on unbolted flour bread, would continue to thrive for an 
indefinite period. It is immaterial whether the general quantity of food be reduced 
too low, or whether either of the muscle-making or heat-producing principles be 
withdrawn while the other is fully supplied. In either case the effect will be the 
same. A man will become weak, dwindle away and die, sooner or later, according to 
the deficiency; and if food is eaten which is deficient in either principle, the appe¬ 
tite will demand it in quantity till the deficient element is supplied. All food, be¬ 
yond the amount necessary to supply the principle that is not deficient, is not only 
wasted, but burdens the system with efforts to dispose of it. 


Analysis of Fruits. 


Fruit. 

Water. 

Sugar. 

Acid. 

Albumi¬ 
nous sub¬ 
stances. 

Insoluble 

matter. 

Pectous 

sub¬ 

stances. 

Ash. 

Apple, white. 

85 

7.6 

I 

.22 

1.83 

3.88 

•47 

Apricot, average. 

83-5 

1.8 

I. I 

•51 

4-7 

7-55 

.84 

Blackberry. 

86.4 

4.44 

1.19 

•51 

5.26 

I.72 

.48 

Cherry, red. 

75-4 

13 -1 

•35 

•9 

5-83 

3 ’ 73 

.69 

sour. 

80.5 

8.77 

.1.28 

•83 

5 - 9 1 

2.07 

.64 

black. 

79-7 

IO.7 

•56 

I 

6.04 

i -33 

.67 

Currant, red. 

85-4 

5-6 

i -7 

•36 

3-74 

2.4 

.8 

Gooseberry, red. 

85-6 

8 

i -35 

•44 

2.92 

1.26 

•43 

yellow.... 

85-4 

7 

1.2 

.46 

3 -i 7 

2.4 

•37 

Grape, white. 

80 

I 3-78 

I 

•83 

2.48 

1.44 

•47 

Peach. Dutch. 

85 

1.58 

.6l 

.46 

5-49 

6.4 

.46 

Pear, red. 

83-5 

7-5 

.07 

•25 

3-54 

4.8 

•34 

Plum, yellow gage.... 

80.8 

2.96 

.96 

.48 

3 - 9 8 

10.48 

•34 

large “ .... 

79-7 

3-4 

.87 

•4 

3 - 9 1 

n -3 

.42 

black blue. 

88.7 

2 

I.27 

•4 

6.86 

•23 

•54 

“ red. 

85-3 

2.25 

i -33 

•43 

4-23 

5-85 

.6l 

Italian, sweet... 

81.3 

6-73 

.84 

•83 

4.01 

5-63 

.66 

Raspberry, wild. 

83-9 

3-6 

2 

•55 

8-37 

1.28 

•4 

Strawberry, “ . 

87 

4 

i -5 

.6 

5-5 

•4 

1 

Banana. 

73-9 

Sugar, Pectin, 

Salt, Acid, etc., 26 

I. 

« 


Sugar and. Water in Various Products not Included in 

tlie Ta“ble. (Per Cent.) 


Sugar. 

Sugar, crude.95 

Molasses. 77 

Buttermilk.. 6.4 


Water. 


Molasses. 23 

Lean beef.. 72 

Buttermilk. 88 


Water. 


Cabbage. 91 

Ale and Beer. 91 

Coffee and Tea. 100 















































































202 


ANIMAL FOOD 


Relative Values of* Roods or Assimilating Quality- to 
make an Equal Quantity of Flesh, in Cattle or Sheep. 

[Ewart.) 


Article. 

Cattle. 

Sheep. 

Turnips . 

800 

400 

Carrots. 

630 

— 

Beets. 

600 

3 °° 

Parsnips and Swedes ... 

600 

200 

Meadow grass in bloom. 

400 

— 

Vetches, pods open. 

360 

90 

Potatoes at maturity.... 

280 

200 

Oat straw, cut green .... 

125 

— 

Bean or Vetch straw.... 

— 

200 

Meadow hay. 

IOO 

IOO 

Vetch “ . 

90 

— 

Linseed cake. 

5 ° 

— 


Note. — When these values express w 
about 4 to 5 lbs. beef or mutton. 


Article. 

Cattle. 

Sheep 

Wheat bran. 

45 

i °5 

Corn and Barley meal.. 

35 

— 

Oatmeal. 

34 

— 

Beanmeal. 

33 

— 

Peameal. 

32 

— 

Cabbagq . 


500 

Pea straw. 

— 

200 

Rye bran. 

— 

109 

Oats . 

— 

70 

Buckwheat. 

— 

65 

Barley. 

— 

60 

Pease or Beans. 

— 

■54 


eight in lbs., then such food will produce 


Nutritive Constituents and Values of Food in Grams 

per Found. 


Food. 

Carbon. 

Nitrogen. 

Bakers’ Bread. 

1975 

88 

Barley Meal. 

2563 

68 

Beef.. 

1854 

184 

Beer and Porter. 

274 

I 

Bullock’s Liver. 

934 

204 

Buttermilk. 

387 

44 

Carrots. 

508 

14 

Cheddar Cheese. 

3344 

306 

Cocoa. 

3934 

140 

Dry Bacon. 

5987 

95 

Fat Pork. 

4 H 3 

106 

Flour, Seconds. 

2700 

Il6 

Fresh Butter. 

6456 

— 

Green Bacon. 

5426 

76 

Green Vegetables. 

420 

14 

Indian Meal. 

3016 

120 

Lard. 

4819 

— 

Molasses. 

2395 

— 


Food. 

Carbon. 

Nitrogen 

Mutton. 

1900 

189 

New Milk. 

599 

44 

Oatmeal. 

2831 

136 

Parsnips. 

554 

12 

Pearl Barley. 

2660 

9 1 

Potatoes. 

769 

22 

Red Herrings. 

1435 

217 

Rice. 

2732 

68 

Rye Meal. 

2693 

86 

Salt Butter. 

4585 

— 

Skim Cheese. 

1947 

483 

Skimmed Milk. 

438 

43 

Split Pease. 

2698 

248 

Suet. 

4710 

— 

Sugar. 

2955 

— 

Turnips. 

263 

13 

Whey. 

i 54 

13 

Whitefish. 

871 

I 95 


The Full Daily Diet of a man is held to be 12 oz. bread, 8 oz. potatoes, 
6 oz. meat, 4 oz. boiled rice with milk, .375 pint of broth or pea soup, 1 pint 
milk, and 1 pint of beer. 


Nutritive Values and Constituents of IVIilk.— [Payen.) 


Animal. 

Nitrogenous 
Matter and 
insoluble 
Sal ts. 

Butter. 

Lactic 

and 

soluble 

Salts. 

Water. 

Animal. 

Nitrogenous 
Matter and 
insoluble 
Salts. 

Butter. 

Lactic 

and 

soluble 

Salts. 

Water. 

Goat.... 

4-5 

4.1 

5-8 

85-6 

Ass. 

i -7 

1.4 

6.4 

9°-5 

Cow.... 

4-55 

3-7 

5-35 

86.4 

Mare... 

1.62 

.2 

8-75 

89-43 

Woman. 

3-35 

3-34 

3-77 

89-54 

Ewe.... 

4.68 

4.2 

5-5 

85.62 


VVeight of some Different Foods, required to furnish 
1220 Grains of Nitrogenous Matter. 


Cheese. 

Lbs. 

. .4 

Meat, fat. 

Lbs. 

i -3 

Bacon, fat. 

Lbs. 
- 1.8 

Bariev Meal. 

Pease. 

• -7 

Oatmeal. 

i -5 

Bread. 


Milk. 

Meat, lean... 

• -9 

Corn Meal. 

1.6 

Rye Meal.. 


Potatoes. 

Fish, White. 

. I 

Wheat Flour.. 

i -7 

Rice. 


Parsnips. 


Turnips, 15.9 lbs. 

; Beer or Porter, 

158.6 lbs. 


Lbs. 

2.9 

4.2 

8-3 

15-9 

















































































































ANIMAL FOOD 


203 


Proportion of Sugar and. Acid in Various Fruits. 

(Fresenius.) 


Fruit. 

Sugar. 

Acid. 

Apple. 

Per Cent. 
8.4 

Per Cent. 
.8 

Apricot. 

1.8 

I. I 

Blackberry. 

4.4 

1.2 

Currants. 

6.1 

2 

Gooseberry. 

7.2 

1.5 

Grape. 

14.9 

.7 

Mulberry. 

9.2 

1.9 

Peach . 

1.6 

•7 


Fruit. 

Sugar. 

Acid. 

Plum. 

Per Cent. 

Per Cent. 

2. I 

1.3 

Prune. 

6.3 

• Q 

Raspberry. 

4 

1.5 

Red Pear... 

7-5 

. I 

Sour Cherry... 

8.8 

1.3 

Strawberry. 

5-7 

1.3 

Sweet Cherry. 

10.8 

.6 

Whortleberry. 

5-8 

i -3 


Proportion of Oil in Various Air-dry Seeds. ( Berjot .) 


Beeclinut. 

Hemp. 

Watermelon ... 


24 Mustard 
28 Flax.... 
36 Peanut . 


3c Almond 

Colza... 

3 8 


40 Orange. 40 

iTs P0IW . {to 


Analysis of different Articles of Food, with Reference 
only to their Properties for giving Heat and Strength.. 

(Payen.) In 100 Parts. 


Substances. 

Car¬ 

bon. 

Nitro¬ 

gen. 

Substances. 

Car¬ 

bon. 

Alcohol. 



Coffee. 


Barley. 

40 

I Q 

Corn. 


Beans. 

42 

4-5 

Eels. 

30.05 



Beef, meat.... 

II 

n 

J 

Eggs. 

13-5 

Beer, strong.. 

4-5 

.08 

Figs, dried.... 

34 

Bread, stale... 

28 

I.O 7 

Herring, salt- 


Buckwheat... 

4 2 -S 

2.2 

ed. 

23 

Butter. 

83 

.64 

Liver, Calf’s.. 

15.68 

Carrots. 

<. t; 

. QI 

Lobster . 


Caviare . 

J <-7 

27.41 

4- 4Q 

Mackerel. 


Cheese,Chestf 

41.04 

4- I 3 

Milk, Cow’s... 

8 

Chocolate .... 

58 

1.52 

Nuts. 

10.65 

Cod-fish, salt’d 

16 

5.02 

Oatmeal. 

44 


Nitro¬ 

gen. 

Substances. 

Car¬ 

bon. 

Nitro 

gen. 

I. I 

Oil, Olive. 

98 

_ 

!-7 

Oysters. 

7.18 

2.13 

2 

Pease . 

44 

3.66 

1.9 

Potatoes. 

II 

■33 

.92 

Rice. 

4 1 

1.8 


Rye Flour.... 

4 1 

i -75 

3 -11 

Salmon. 

l6 

2.0Q 

3-93 

Sardines. 

29 

6 ' 

2 -93 

Tea. 

2. I 

.2 

3-74 

Truffes. 

9-45 

i -35 

.66 

Wheat. 

41 

3 

1.4 

“ Flour.. 

38.5 

1.64 

i -95 

Wine. 

4 

.015 


Note. —Multiply figures representing nitrogen by 6.5, and equivalent amount of 
nitrogenous matter is obtained. 


Human and Animal Sustenance. 

Least Quantity of Food required to Sustain Life. (E. Smith , M. D.) 


Carbon. Hydrogen. 

Grs. Grs. 

Adult Man, 4300) 200 

Adult Woman, 3900} Mean ’ 4IO °' 180 


| Mean, 190. 


An adult man, for his daify sustenance, requires about 1220 grs. nitrog¬ 
enous matter or 200 of nitrogen, and bread contains 8.1 per cent, of it. 


1220 

Hence,-= 15 062 grains which A- 7000 in alb. =2 lbs. 2.43 oz. of bread. 

.081 

These quantities and proportions are also contained in about 16 lbs. of 
turnips. 

Thus, by table of nutritive values, page 202, turnips have 263 grains of carbon and 
13 of nitrogen. 

Hence, and — — 16.35 lbs. for the necessary carbon and 15.4 lbs. for the 
263 13 

nitrogen. 


Relative Value of Foods compared with IOO lbs. of 


Lbs. 

Clover, green.. 400 
Corn, green ... 275 
Wheat straw .. 374 


Grood 
■ Lbs. 

Bye straw.442 

Oat straw.195 

Cornstalks .... 400 


Hay. 

Lbs. 


Carrots.276 

Barley.. 54 

Oats. 57 


Lbs. 

Corn. 59 

Linseed cake .. 69 
Wheat bran.... 105 































































































204 


ANIMAL FOOD, 


Weight of* .Articles of* Food required. to be consumed in 
the human system, to develop a, power equal to rais¬ 
ing IdrO lbs. to a height of lO OOO feet. (Frankland.) 


Substances. 

Weight. 

Cod-liver oil. 

Lbs. 

*553 

•555 

.67 

•693 

. 707 

Beef, fat. 

Bacon. 

Butter. 

Cocoa . 

Fat of Pork. 

* /y/ 

•97 

1.156 
1.281 

Cheese. 

Oatmeal. 

Arrowroot.. 

1.287 

13 11 

Wheat flour. 


Substances. 

Weight. 

Rice. 

Lbs. 

I - 34 I 

1-379 

I-505 

2.062 

Isinglass. 

Sugar, lump. 

Cream.. 

Egg, boiled. 

2. 209 
2 - 345 
2.826 

Bread . 

Salt Pork. 

Ham, lean, boiled.. 
Mackerel. 

3.001 

3.124 

3.461 

Ale, bottled. 


Substances. 


Salt Beef.... 
Veal, lean.... 

Porter. 

Potatoes. 

Fish. 

Apples. 

Milk. 

Egg, white of. 

Carrots. 

Cabbage. 


Weight. 


Lbs. 

3 - 6 5 

4 - 3 
4.615 
5.068 
6.516 

7- 8i5 

8.021 

8 - 745 
9.685 

12.02 


Relative Value of Various Foods as Productive of Rorce 


when Oxidized in the Body. 


Cabbage. 1 

Carrots. 1.2 

Skimmed Milk. 1.2 
White of Egg.. 1.4 

Milk. 1.5 

Apples. 1.5 

Ale. 1.8 

Fish. 1.9 

Potatoes. 2.4 


Porter. 2.6 

Veal, lean. 2.8 

Salt Beef. 3.3 

Poultry. 3.3 

Lean Beef._3.4 

Mackerel. 3.8 

Ham, lean. 4 

Salt Pork. 4.3 


Bread, crumb.. 5.1 


Egg,hard boil’d 5.4 

Cream. 5.9 

Egg, yolk. 7.9 

Sugar. 8 

Isinglass. 8.7 

Rice. 8.9 

Pea Meal.9. 

Wheat Flour .. 9.1 
Arrowroot. 9.3 


Oatmeal. 9.3 

Cheese. 10.4 

Fat of Pork. 12.4 

Cocoa. 16.3 

Pemmican.. 16.9 

Butter. 17.3 

Bacon. 17-94 

Fat of Beef.. 21.0 
Cod-liver Oil. 21.7 


Nutritious Properties of different Vegetables and Oil- 
calse, compared with each other in Quantities. 


Oil-cake. 1 

PeaseandBeans 1.5 
Wheat, flour... 2 
“ grain.. 2.5 
Oats. 2.5 


Rye. 2.5 

Bran, wheat! 2 ' 7 
’ (3 

Corn. 3 

Barley. 3 


Clover hay. 4 

Hay. 5 

Potatoes. 14 

“ old.... 20 

Carrots. 17.5 


Cabbage. 18 

Wheat straw.. 26 
Barley “ 26 

Oat “ 27.5 

Turnips. 30 


Illustration.— 1 lb. of oil-cake is equal to 18 lbs. of cabbage. 


Volume of Oxygen required to Oxidize 100 parts of folloioing Foods as con¬ 
sumed in the Body. 

Grape Sugar .. 106 | Starch. 120 | Albumen. 150 | Fat. 293 

Hence, assuming capacity for oxidation as a measure, albumen has half value of 
fat as a food-producing element, and a greater value than either starch or sugar. 


Proportion of .Alcohol in IOO Parts? of follow: 

{Brandt.) 


Small Beer... 1 and 1.08 
Porter....... 3.5 and 5.26 

Cider.5.2 and 9.8 

Brown Stout. 5.5 and 6.8 

Ale.6.87 and 10 

Rhenish. 7.58 

Moselle. 8.7 

Johannisberger. 8.71 

Elder Wine. 8.79 

Claret ordinaire. 8.99 

Tokay. 9.33 

Rudesheimer.10.72 

Marcobrunner.11.6 

Gooseberry Wine ... 11.84 

Hockheimer.12.03 

Vin de Grave.12.08 


Hermitage, red ..... 12.32 

Champagne.12.61 

Amontillado.12.63 

Frontignac.12.89 

Barsac. .13-86 

Sauterne.14.22 

Champagne Burg’dy, 14.57 

White Port.15 

Bordeaux.15.1 

Malmsey.16.4 

Sherry. 17.17 

Malaga.17.2 

Alba Flora.17.26 

Hermitage, white... 17.43 

Cape Muscat.18.25 

Constantia, red.18.92 


ing Licpuors. 


Lisbon.18.94 

Lachryma.19.7 

Teneriffe.19.79 

Currant Wine.20.55 

Madeira.22.27 

Port..23 

Sherry, old..23.86 

Marsala.25.09 

Raisin Wine.25.12 

Madeira, Sercial _27.4 

Cape Madeira.20.^1 

Gin.51.6 

Brandy.53.39 

Rum.. . 53-63 

Irish Whiskey.53.9 

Scotch Whiskey_54.32 






































































































































ANIMAL FOOD, 


205 


Proportion of* Food. Appropriated and Expended "by 

following Animals. 


Proportion appropriated 
“ in manure... 

“ respired. 


Oxen. 

Sheep. 

Swine. 

6.2 

8 

17.6 

36-5 

3 i -9 

16.9 

57-3 

60.1 

65-5 

IOO 

IOO 

IOO 


Specific GJ-ravity of* Atilli and Percentage of Cream, etc. 


Milk. 

Specific 

Gravity. 

Volume 

of 

Cream. 

Volume 

of 

Curd. 

Specific 
Gravity when, 
skimmed. 

Milk, pure*. 

0 T 

m n cs 

0 0 C 

H M H* 

12 

6 .1 


“ 10 per cent, water. 

10.5 

8-5 

6 

5-6 

A, Q 

1029 

“ 20 “ “ “ . 

1026 

“ 30 “ .. 

1021 

1 y 

4.2 

1023 


* For a method of testing the purity of milk, see Pavy on Food (Philadelphia, 1874), page 196. 


Note.—T he average proportion of cream is io, or io per cent. 


Proportion Per cent, of Starch in sundry "Vegetables. 

Arrowroot_82 Wheat flour... 66.3 Oatmeal.58.4 I Potatoes.18.8 

Rice.79.1 Corn meal.... 64.7 Pease.55.4 | Turnips. 5.1 


Composition of Clieese of Different Countries.— (Payen.) 



Fat. 

Nitrogen. 

Salt. 

Water. 


Fat. 

Nitrogen. 

Salt. 

Water. 

Neufchatel.. 

18.74 

2.28 

4-25 

61.87 

Chester. 

25.41 

5-56 

4.78 

30-39 

Parmesan .. 

21.68 

5-48 

7.09 

30 - 3 I 

Gruy6res ... 

28.4 

5-4 

4.29 

32.05 

Brie. 

24.83 

2-39 

5-63 

53-99 

Marolles.... 

28.73 

3-73 

5-93 

40.07 

Holland .... 

25.06 

4.1 

6.21 

41.41 

Roquefort... 

32 - 3 I 

5-07 

4-45 

26.53 


Nutritive Equivalents. Computed from Amount of Ni¬ 
trogen in Substances when Dried. Human ZVIilli at 1. 


Rice. 

.81 

Potatoes. 

.84 

Corn. 

I 

Rye.. 

1.06 

Wheat. 

1.19 

Barley. 

1.25 

Oats. 

1.38 


Bread, White. 

1.42 

Cheese. 

Milk, Cows’ .. 

2-37 

Eel. 

Pease. 

2. 

Mussel. 

Lentils. 

2.76 

Liver, Ox .... 

Egg, Yolk.... 

3-05 

Pigeon. 

Oysters. 

3-05 

3-2 

Mutton. 

Beans. 

Salmon. 


Herring, 9.14. 


3- 3i 

4 - 34 
5.28 

5 - 7 
7 - 56 
7-73 
7.76 


Lamb. 

Egg, White. 

Lobster_ 

Veal. 

Beef. 

Pork. 

Ham. 


8 -33 

8-45 

8-59 

8-73 

8.8 

8-93 

9.1 


Thermometric Power and IVleclianical Energy of IO 
G-rains of "Various Substances in tlieir Natural Con¬ 
dition, wlien Oxidized in tire Animal Body into Oar— 
bonio .Acid, Water, and Urea.— (Frankland.) 


Substance. 

Water 

raised 

i° 

Lifted 

1 foot 
high. 


I,b 3 . 

Lbs. 

Ale, Bass’s .. 

1.99 

i -54 

Apples. 

1.48 

1.29 

Arrowroot... 

10.06 

7-77 

Beef, lean ... 

3.66 

2.83 

Bread. 

5-52 

4.26 

Butter. 

18.68 

14.42 

Cabbage. 

1.08 

•83 

Carrots. 

i -33 

1.03 


Substance. 

Water 

raised 

i°. 

Lifted 
x foot 
high. 


Lbs. 

Lbs. 

Cheese. 

II .2 

8.65 

Cocoa-nibs .. 

17 

7-3 

Cod-liver oil. 

II 

18.12 

Egg, h’d boil. 

5-86 

4-53 

“ yolk.... 

8-5 

6.56 

“ white... 

1.48 

1.14 

Flour, wheat. 

9.87 

7.62 

Ham, boiled. 

4-3 

3 - 3 2 


s 


Substance. 

Water 

raised 

x°. 

Lifted 

1 foot 
high. 


Lbs. 

Lbs. 

Mackerel.... 

4.14 

3-2 

Milk. 

1.64 

1.25 

Oatmeal. 

IO. I 

7.8 

Pea meal.... 

9-57 

7-49 

Potatoes .... 

2.56 

1.99 

Porter. 

2.77 

2. 19 

Rice, ground. 

9-52 

7-45 

Sugar, grape. 

8.42 

6.51 












































































































20 6 


ANIMAL FOOD, 


Digestion. 

Time required, fbr Digestion of several Articles of Food. 

(Beaumont, M.D.) 

Time. 


Food. 


Apple, sweet and mellow .... 

sour and mellow. 

sour and hard. 

Barley, boiled. 

Bean, boiled. 

Bean and Green Corn, boiled. 

Beef, roasted rare. 

roasted dry. 

Steak, broiled. 

boiled. 

boiled, with mustard, etc. 

Tendon, boiled. 

“ fried. 

old salted, boiled. 

Beet, boiled. 

Bread, Corn, baked. 

Wheat, baked, fresh ... 

Butter, melted. 

Cabbage, crude. 

crude, vinegar. 


crude, vin’r, boiled - 


Carrot, boiled. 

Cartilage, boiled. 

Cheese, old and strong. 
Chicken, fricasseed. .. 
Custard, baked. 

Duck, roasted. 


Dumpling, Apple, boiled. 

Egg. 

whipped. 

boiled hard. 

“ soft. 

fried. 

Fish, Cod or Flounder, fried .. 

Cod, cured, boiled. 

Salmon, salt’d and boil’d 
Trout, boiled or fried. . . 

Fowl, boiled or roasted. 

Goose, roasted. 

Gelatine, boiled. 


h. m. 

1 50 

2 

2 50 
2 

30 

45 


2 

3 
3 
3 
3 

2 

3 
5 

4 
4 
3 
3 
3 

3 
2 

2 

4 
4 

3 

4 

3 
2 

2 

4 
4 

3 

2 

1 

3 
3 
3 

3 

2 

4 

1 30 
4 

3 

2 30 


30 

45 

30 

30 

15 

45 

15 

30 

30 

30 


Food. 

Time. 

Heart, Animal, fried . 

h . m . 

4 

Lamb, boiled . 

2 30 

Liver, Beefs, boiled . 

Meat and Vegetables, hashed . 

Milk, boiled or fresh . j- 

Mutton, roasted . 

2 

2 30 

2 

2 15 

3 15 

broiled or boiled .... 

3 

Oyster. 

2 55 

3 i 5 

roasted . 

stewed . 

3 30 

Parsnip, boiled . 

2 3 ° 

Pig, sucking, roasted. 

2 30 

Feet, soured, boiled . 

1 

Pork, fat and lean, roasted .. . 

5 i 5 

recently salted, boiled .. 

4 30 

“ “ fried. . . 

4 15 

“ “ broiled . 

3 15 

“ “ raw. ... 

3 

Potato, boiled . 

3 30 

baked . 

3 20 

roasted . 

2 30 

Rice, boiled . 

1 

Sago, boiled . 

1 45 

Sausage, Pork, broiled . 

3 20 

Soup, Barley . 

1 30 

Beef and Vegetables . .. 

4 

Chicken . 

3 

Mutton or Oyster . 

3 30 

Sponge-cake, baked . 

2 30 

Suet, Beef, boiled . 

5 3 ° 

Mutton, boiled . 

4 3 ° 

Tapioca, boiled . 

2 

Tripe, soured . 

1 

Turkey, roasted J Dome ^ tic __ 

2 18 

2 30 

boiled. 

2 25 

Turnip, boiled. 

3 30 

Veal, roasted. 

4 

fried. 

4 50 

Brain, boiled. 

1 45 

Venison Steak, broiled. 

1 35 


Gi-eneral NTotes. 

The per-centage of loss in the cooking of meats i§as follows: Boiling 23; Baking 
31; Roasting 34. 

Potatoes possess anti-scorbutic power in a greater degree than any other of the 
succulent vegetables. 

The average yearly consumption of wheat and wheat flour in Great Britain is 5.5 
bushels per capita of its population. 

The daily ration of an Esquimaux is 20 lbs. of flesh and blubber .—{Sir John Ross.) 














































































ANIMAL FOOD. 


207 


An adult healthy man, according to Dr. Edward Smith, requires daily of 

Phosphoric acid from .. 32 to 79 grains. Potash-27 to 107 grains. 

(Chlorine.51 “ 175 “ Soda.80 “ 171 “ 

(Or of common salt_85 “ 291 “ Limes. 2.3 “ 6.3 “ 

and of Magnesia 2.5 to 3 grains. 

A common fowl’s egg contains 120 grains of Carbon and 17.75 of Nitrogen. 
An ordinary working-man requires for his daily sustenance 


Oxygen . 

Albuminous matter 
Fat. 


x.47 

• 3°5 

.22 


.Starch 
Salts.. 
Water. 


= 7.23 lbs. avoirdupois. 


.66 

.04 

4-535 


Milk .—If the milk of an animal is taken at three immediately successive periods, 
that which is first received will not be as rich in milk-fat as the last. 

In a Devon cow', milked in this manner, the first milk gave but 1.166 per cent, of 
fat, and the last, or that known as “strippings,” 5.81 per cent. 

Relative Rielaness of NLilli of Several ykniinals. 

Human Milk = 1. 



Milk-fat. 

Casein. 

Sugar. 


Milk-fat. 

Casein. Sugar. 

Cow. 


1.38 

.69 

Ass. 

. 5 

.38 .94 

Mare.... 

.I-I 9 

•75 

•94 

Sheep.. . 


2.1 .72 

Goat.... 


1.04 

.69 

Camel... 

.i -4 

— .96 


The condensation of milk reduces it to about one third of its original volume. 

A Farm of second-rate quality, properly cultivated, will sustain 100 head of cattle 
per 100 acres, besides laboring stock (employed in cultivation of farm), and swine. 
— {Ewart.) 

Thus, calves 25; do. 1 year 25; do. 2 years 25; cows 25. 

Cane Sugar (Saccharose)—Is insoluble in absolute alcohol, and in diluted alcohol 
it is soluble only in proportion to its weakness. Loaf sugar, as a rule, is chemically 
pure. 

Beet Root Sugar— Contains 85 to 96 per cent, of cane sugar, 1.6 to 5.1 of organic 
matter, aud 2 to 4.3 of water. 

Honey —Contains 32 per cent, of sugar (levulose), 25.5 of water, 27.9 of dextrine, 
and 14.6 of other matter, as mannite, wax, pollen, and insoluble matter. 

Molasses —Contains 47 per cent, of cane sugar, 20.4 of fruit sugar, 2.6 of salts, 2.7 
extractive and coloring matter, and 27.3 of water. 

Flour. —Tests of flour, see A. W. Blyth, London, 1882, page 152. 

Bread. —Wheat loses of water after 1 day 7.71 per cent., 3 days 8.86, and 7 days 
14.05 per cent. 

Sago. —2.5 lbs. per day will support a healthy man. 

Fig —Contains nearly as much gluten as wheat bread (as 6 to 7), and in starch and 
sugar it is 16 per cent, richer. 

Gooseberry (dry)—Is as nutritious as wheat bread. 

Watermelon , Vegetable marrow , and Cucumber —Contain 94, 95, and 97 per cent, 
of water respectively. 

Onion (dry)—Contains 25 to 30 per cent, of gluten. Potato containing but 5. 

Cabbage , Cauliflower , Broccoli , and Leaves are generally rich in gluten, while the 
potato is poor. 


Ratio of FTesL.-formers of Tubers. 

Per Cent. 


Tubers. 

Flesh- 

formers. 

Starch, 

etc. 

Ratio to 
Heat-giv’rs. 

Tubers. 

Flesli- 

formers. 

Starch, 

etc. 

Ratio to 
Heat-giv’rs. 

Beet root. 

Turnip. 

•4 

•5 

J 3-4 

4 

1:30 

1:8 

Parsnip. 

Onion. 

1.2 

1-5 

8.7 

4.8 

i: 10 

i* 3-5 

Carrot. 

•5 

5 

i: 10 

Sweet Potato. 

1-5 

20.2 

1:13 

Potato. 

1.2 

18 

1:16 

Yam. 

2.2 

16.3 

1: 7-5 










































208 GEAYITY OF BODIES.-GEAYITY AND WEIGHT. 


GRAVITY OF BODIES. 

Gravity acts equally on all bodies at equal distances from Eaith s 
centre; its force diminishes as distance increases, and increases as dis¬ 
tance diminishes. 

Gravitating forces of bodies are to each other, 

1. Directly as their masses. 

2. Inversely as squares of their distances. 

Gravity of a body, or its weight above Earth’s surface, decreases as 
square of its distance from Earth’s centre in semi-diameters of Earth. 

Illustration i.—I f a body weighs goo lbs. at surface of the Earth, what will n 
weigh 2000 miles above surface?—Earth’s semi-diameter is 3963 miles (say 4000). 

900 7 . ■ 

Then 2000 -f 4000 = 6000 = 1.5 semi-diam\s , and 900 -f 1.5 = ^7 = 4 00 UjS - 

Inversely, If a body weighs 400 lbs. at 2000 miles above Earth’s surface, what will 
it weigh at surface ? 

400 X 1.5 2 = 900 lbs. 

2. —A body at Earth’s surface weighs 360 lbs.; how high must it be elevated to 
weigh 40 lbs.? 

2-22 — 9 semi-diameters, if gravity acted directly; but as it is inversely as square 

40 * 

of the distance, then f 9 — 3 semi-diameters = 3 X 4000 = 12 000 miles. 

3. —To what height must a body be raised to lose half its weight? 

As y/i : s /2 ;; 4000 : 5656 = as square root of one semi-diameter is to square root 
of two semi-diameters, so is one semi-diameter to distance required. 

Hence 5656 — 4000 == 1656 = distance from Earth's surface. 

Diameters of two Globes being equal , and their densities different , weight 
of a body on their surfaces will be as their densities. 

Their densities being equal and their diameters different , weight of them 
will be as their diameters. 

Diameters and densities being different , weight will be as their product. 

Illustration. —If a body weighs 10 lbs. at surface of Earth, what will it weigh at 
surface of Sun, densities being 392 and 100, and diameters 8000 and 883000 miles? 

883 000 x 100 4 - 8000 x 392 = 28.157 = quotient of product of diameter of Sun and 
its density, and product of diameter of Earth and its density. 

Then 28.157 X 10 = 281.57 lbs. 

Note. —Gravity of a body is .003 46 less at Equator than at Poles. 


SPECIFIC GRAVITY AND WEIGHT. 

Specific Gravity or Weight of a body is the proportion it bears to the 
weight of another body of known density or of equal volume, and which is 
adopted as a standard. 

If a body float on a fluid, the part immersed is to whole body as specific 
gravity of body is to specific gravity of fluid. 

When a body is immersed in a fluid, it loses such a portion of its own 
weight as is equal to that of the fluid it displaces. 

An immersed body, ascending or descending in a fluid, has a force equal 
to difference between its own weight and weight of its bulk of the fluid, less 
resistance of the fluid to its passage. 

Water is well adapted for standard of gravity; and as a cube foot of it 
at 62° F. weighs 997.68 ounces avoirdupois, its weight is taken as the unit, 
or approximately 1000. 





SPECIFIC GRAVITY AND WEIGHT. 


209 


French standard temperature for comparison of density of solid bodies 
and determination of their specific gravities, is that of maximum density of 
water, at 4 0 C. or 39. i° F., and for gases and vapors under one atmosphere or 
.76 centimeters of mercury is 32 0 F. or o° C., and specific gravity of a body 
is expressed by weight in kilogrammes of a cube decimeter of that body. 

Densities of metals vary greatly. 

Potassium, Sodium, Barium, and Lithium are lighter than water. Mercury 
is heaviest liquid and Platinum heaviest metal. Volcanic scoriae is lighter 
than water. 

Pomegranate and Lignum-vitse are heaviest of woods. Pearl is heaviest 
of animal substances, and Flax and Cotton are heaviest of vegetable sub¬ 
stances, former weighing nearly twice as much as water. 

Zircon is heaviest of precious stones, being 4.5 times heavier than water. 
Garnet is 4 times heavier, Diamond 3.5 times, and Opal, lightest of all, is but 
twice as heavy as water. 


To Ascertain Specific Grravity of a Solid Body- Heavier 

tlian Water. 

Rule. —Weigh it both in and out of water, and note difference ; then, as 
weight lost in water is to whole weight, so is 1000 to specific gravity of body. 
"W x 1000 

Or,— - == G, W and 10 representing weights out and in water, and G 

specific gravity. 

Example. — What is specific gravity of a stone which weighs in air 15 lbs., in 
water 10 lbs.? 

15 —10 ==5; then 5 : 15 : 1000 : 3000 Spec. Grav. 

To Ascertain Specific Grravity of a Body ligliter tlian 

Water. 


Rule. —Annex to lighter body one that is heavier than water, or fluid 
used ; weigh piece added and compound mass separately, both in and out of 
water, or fluid; ascertain how much each loses, by subtracting its weight 
from its weight in air, and subtract less of these differences from greater. 

Then, as last remainder is to weight of light body in air, so is 1000 to 
specific gravity of body. 

Example.— What is specific gravity of a piece of wood that weighs 20 lbs. in air; 
annexed to it is a piece of metal that weighs 24 lbs. in air and 21 lbs. in water, and 
the two pieces in water weigh 8 lbs.? 

20 -p 24_8 — 44 — 8 = 361= loss of compound mass in water ; 

24_21 — 3 = loss of heavy body in water. 

33 : 20 1000 ; 606 = 24 Spec. Grav. 


To Ascertain. Specific Grravity of a Fluid. 

Rule.—T ake a body of known specific gravity, Weigh it in and out of 
the fluid; then, as weight of body is to loss of weight, so is specific gravity 
of body to that of fluid. 

Example. — What is specific gravity of a fluid in which a piece of copper [spec, 
grav. — good) weighs 70 lbs. in, and 80 lbs. out of it? 

80 : 80 — 70 = 10 '.: 9000 : 1125 Spec. Grav. 


To Ascertain Specific Grravity of a Solid Body wliicli 
is soluble in Water. 

Rule. —Weigh it in a liquid in which it is not soluble, divide its weight 
out of the liquid by loss of its weight in the liquid, and multiply quotient 
by specific gravity of liquid; the product is specific gravity. 

Example.— What is specific gravity of a piece of clay, which weighs 15 lbs. in air 
and 5 lbs. in a liquid of a specific gravity of 1500, in which it is insoluble? 

15-rioX 1500 = 2250 Spec. Grav. 



210 


SPECIFIC GEAYITY AND WEIGHT, 


Substances. 


NEetals. 


Aluminum, cast. 

“ wrought. 

“ Bronze.. 


Antimony. 

Arsenic. 

Barium. 

Bismuth. 

Boron. 

Brass. 

Sheet, cop. 75, zinc 25. 
Yellow “ 66, “ 34. 

Muntz “ 60, “ 40. 

Plate . 

Cast. 

Wire. 

Bromine. 

Bronze, gun metal. 

“ ordinary mean . 

“ cop. 84, tin 16 .. 

“ “ 81, “ 19 .. 

“ small bells, cop. 

35, tin 65.... 
“ cop. 21, tin 74 .. 

Cadmium. 

Calcium. 

Chromium. 

Cinnabar. 

Cobalt,... 

Columbium. 

Copper, cast. 

“ plates. 

“ wire and bolts.. 

“ ordinary mean. 

Gold, pure, cast. 

“ hammered. 

“ 22 carats fine. 


Iridium. 

“ hammered. 

Iron, Cast, gun metal... 

“ minimum. 

“ maximum. 

“ ordinary mean.... 

“ mean, Eng. 

u cast, hot blast. 

“ “ cold “ . 

“ Wrought bars. 

“ “ wire. 

“ “ rolled plates 

“ “ average.... 

“ “ Eng. rails.. 

“ “ Lowmoor... 

“ “ pure. 

“ ordinary mean.... 

Lead, cast. 

“ rolled. 

Lithium. 

Magnesium. 

Manganese. 

Mercury —40 0 . 

“ +3 2 °. 


SOLIDS. 


Specific 

Gravity, 

Weight 
of a Cube 
Inch. 

'Substances. 

Specific 

Gravity 

Weight 
of a Cube 
Inch. 


Lb. 

IVIet&ls. 


Lb. 

2560 

.0926 

Mercury 6o°. 

13 569 

.4908 

2 670 

.0906 

“ 212°. 

13 37 ° 

.4836 

7 700 

.2785 

Molybdenum. 

8 600 

•3m 

6 712 

.2428 

Nickel. 

8 800 

•3183 

5 763 

.2084 

“ cast. 

8 279 

.2994 

470 

.017 

Osmium. 

10 OOO 

•3613 

9823 

•3553 

Palladium. 

II350 

.4105 

2 OOO 

.0723 

Platinum, hammered... 

20337 

•7356 



u native. 

16 OOO 

•5787 

8450 

.3056 

“ rolled. 

22 069 

.7982 

8300 

.2997 

Potassium, 59 0 . 

865 

•0313 

8 200 

.2966 

Red lead. 

8 940 

.324 

8380 

. 3026 

Rhodium. 

10 650 

.3852 

8 100 

.2930 

Rubidium. 

1 520 

•055 

8 214 

. 2972 

Ruthenium. 

8 600 

•3m 

3 000 

• 1085 

Selenium. 

4500 

.1627 

8750 

•3165 

Silver, pure, cast. 

10474 

.3788 

8 217 

.2972 

“ “ hammered. 

10511 

.3802 

8832 

•3194 

Sodium. 

970 

•0351 

8 700 

.2929 

Steel, minimum. 

7 700 

.2785 



“ maximum. 

7 900 

.2857 

8 060 

.291 

“ plates, mean. 

7 806 

.2823 

7 39 ° 

.2668 

“ soft. 

7833 

•2833 

8650 

3129 

“ temper’d andhard- 



1 580 

•057 

ened. 

7 818 

.2828 

5 9 °° 

.2134 

“ wire... 

7847 

.2838 

8 098 

• 2920 

“ blistered. 

7823 

.283 

8 600 

. 3I I I 

“ crucible. 

7842 

.2836 

6 000 

. 217 

“ cast. 

7848 

.2839 

8788 

3 W 9 

“ Bessemer. 

7852 

.284 

8698 

.3146 

“ ordinary mean.... 

7834 

.2916 

8880 

.3212 

Strontium. 

2540 

.0918 

8880 

.3212 

Tellurium. 

6 no 

.221 

19258 

.6965 

Tlialium. 

ii 850 

.4286 

19361 

.7003 

Tin, Cornish, hammered. 

7 39 ° 

.2673 

17 486 

•6325 

“ “ pure ...... 

7291 

• 2637 

15 7°9 

.5682 

Titanium. 

5 300 

. I917 

18 680 

.6756 

Tungsten. 

17 OOO 

.6149 

23 OOO 

8319 

Uranium. 

18330 

.6629 

7308 

.264 

Wolfram. 

7 119 

• 2575 

6 900 

.2491 

Zinc, cast. 

6861 

.2482 

7 500 

.2707 

“ rolled. 

7 191 

.26 

7207, 

.2607 




7217 

.2609 

Woods [ Dry ). 


Cube 

7065 

•2555 



Foot. 

7 218 

.2611 

Alder. 

800 

50 

7 788 

.2817 

Apple. 

793 

49-562 

7 774 

.2811 

Ash.| 

845 

52.812 

7 704 

.2787 


690 

43-125 

7 698 

.2779 

Bamboo. 

400 

25 

7 ^ 4 ° 

. 2722 

Baytree. 



7 808 

.2819 

Reeeh. j 

852 

u A * C> / D 

53 -25 

8 140 

.2938 


690 

43-125 

7 744 

.2801 

Birch... 1.| 

567 

35-437 

11 352 

.4106 


720 

45 

11 388 

•4119 

Blackwood, India. 

898 

56.125 

59 ° 

.0213 

Boxwood, Brazil. 

1 031 

64-437 

1 7 So 

•0633 

“ France. 

1 328 

83 

8 000 

.2894 

“ Holland. 

912 

57 

i 5 6 3 2 

.5661 

Bullet-wood. 

928 

58 

13 59 8 

.4918 

Butternut. 

376 

23-5 













































































































SPECIFIC GRAVITY AND WEIGHT, 


21 I 


Substances. 


Woods (Dry). 

Campeachy. 

Cedar. 

“ Indian.. 

Charcoal, pine. 

“ fresh burned.. 

‘ ‘ oak.. 

“ soft wood .... 

“ triturated.... 

Cherry... 

Chestnut, sweet. 

Citron. ;.... 

Cocoa . 

Cork. 

Cypress, Spanish. 

Dog-wood..... 

Ebony, American. 

“ Indian. 

Elder. 


Elm.J 

“ rock. 

Erroul, India. 

Filbert. 

Fir, Norway Spruce.... 

“ Dantzic.. 

Fustic. 

Greenheart or Sipiri.... 

Gum, blue. 

“ water.. 

Hackmatack. 

Hawthorn... 

Hazel. 

Hemlock. 

Hickory, pig-nut. ...... 

“ shell-bark. 

Holly. 

Iron-wood. 

Jasmine. 

Juniper. 

Khair, India... 

Lancewood, mean. 

Larch. j 

Lemon... 


Lignum-vitse.j 

Lime... 

Linden.. 

Locust.. 

Logwood. 

Mahogany. j 

“ Honduras.... 

“ Spanish. 

Maple. 

“ bird’s-eye. 

Mastic. 

Mulberry.| 


Oak, African... 
“ Canadian. 
“ Dantzic... 


Specific 

Gravity. 

Weight 
of a Cube 
Foot. 

9 T 3 

Lbs. 

57.062 

561 

35.062 

1315 

82.157 

441 

27.562 

380 

2 3-75 

1573 

98.312 

280 

* 7-5 

1380 

86.25 

7 i 5 

44.687 

610 

38.125 

726 

45-375 

IO4O 

65 

240 

*5 

644 

40.25 

756 

47-25 

I 33 i 

8 3- i8 7 

1209 

75-562 

695 

43-437 

570 

35 625 

671 

4 i 937 

800 

5 o 

1014 

63-375 

600 

37-5 

5i2 

32 

582 

36.375 

970 

60.625 

1055 

65-95 

843 

52.687 

IOOO 

62.5 

592 

37 

910 

56.875 

860 

53-75 

368 

23 

792 

49-5 

690 

43- 12 5 

760 

47-5 

990 

61.875 

770 

48.125 

566 

35-375 

II7I 

73-187 

720 

45 

544 

34 

560 

35 

703 

43-937 

650 

40.625 

I 333 

83-312 

804 

50.25 

604 

37-75 

728 

45-5 

9 X 3 

57 062 

720 

45 

1063 

66.437 

560 

35 

852 

53-25 

75 o 

46-875 

576 

36 

849 

53.062 

561 

35.062 

897 

56.062 

823 

51-437 

872 

54-5 

759 

47-437 

* U. 

S. Ordnanc 


Substances. 


Woods (Dry). 

Oak, English. j 

“ green. 

“ heart, 60 years.... 

“ live, green. 

u “ seasoned...,. 

“ white. 

Olive. 

Orange..... 

Pear. 

Persimmon... 

Plum. 

Pine, pitch. 

“ red. 

“ white.. 

“ yellow. 

“ Norway. 

Pomegranate. 

Poon... 

Poplar. 

“ white. 

Quince_'..... 

Rosewood. 

Sassafras. 

Satinwood. 

Spruce . 

Sycamore. 

Tamarack. 

Teak (African oak).... j 

Walnut. 

black. 

Willow.| 

Yew, Dutch. 

“ Spanish. 

(Well Seasoned.*) 

Ash. 

Beech... 

Cherry. 

Cypress. 

Hickory, red. 

Mahogany, St.Domingo. 

Pine, white. 

‘ ‘ yellow. 

Poplar.. 

White Oak, upland. 

“ “ James River 


Stones, Earths, 
etc. 

Alabaster, white----- 

“ yellow....... 

Alum..... 

Amber. 

Ambergris. 

Asbestos, starry........ 

Asphalte.. 

Barytes, sulphate .... j 
Beton, N. Y. St.Con’g Co. 


Specific 

Gravity. 


858 

932 

1146 

1170 

1260 

1068 

860 

680 

705 

661 

710 

785 

660 

59 ° 

554 

461 

740 

I 354 

580 

383 

529 

7°5 

728 

482 

885 

500 

62^ 

383 

657 

980 

671 

500 

486 

585 

788 

807 


722 

624 

606 

441 

838 

720 

473 

54 i 

587 

687 

759 


2730 

2699 

1714 

1078 

866 

3073 

2250 

4000 

4865 

2305 


Weight 
of a Cube 
Foot. 


Lbs. 

53-625 

58-25 

71.625 
73-125 

78-75 

66.75 

53-75 

42-5 

44.062 

41.312 

44- 375 
49.062 

41.25 

36-875 

34-625 

28.812 

46.25 

84.625 

3.6.25 

23-937 

33.062 

44.062 

45- 5 

30- 125 

55-312 

31- 25 
38-937 
23-937 
41.062 

61.25 
4i 937 
3 1 - 2 5 
30-375 
3.6- 562 

49- 2 5 

50- 437 


45-125 

39 

37-875 

27.562 
52-375 
45 

29.562 
5^.812 
36.687 
42-937 
42-437 


170.625 
168.687 
X07.125 

67-375 

192.062 

140.625 
250 

304.062 

144.06 




















































































































212 


SPECIFIC GRAVITY AND WEIGHT, 


Substances. 


Stones, ICartlis, 
etc. 


Basalt.| 

Bitumen, red. 

“ brown. 

Borax. 

Brick.| 

“ pressed . 

“ fire. 

“ work in cement... 

“ “ “ mortar. | 

Carbon. 

Cement, Portland. 

“ Roman. 

Chalk. j 

Clay. 

“ with gravel. 

Coal, Anthracite. j 

“ Borneo. 

“ Cannel. j 

“ Caking. 

“ Cherry. 

“ Chili. 

“ Derbyshire. 

“ Lancaster. 

11 Maryland. 

“ Newcastle. 

“ Rive de Gier. 

“ Scotch.| 

“ Splint. 

“ Wales, mean. 

Coke... 


“ Nat’l, Ya. 

Concrete, in cement.... 

“ mean. 

Earth,* common soil,dry 

“ loose. 

“ moist sand. 

“ mold, fresh. 

“ rammed. 

“ rough sand..... 

“ with gravel.... 

“ Potters’. 

“ light vegetable.. 

Emery. 

Feldspar. 

Flint, black.. 

“ white. 

Fluorine.». 

Fuel, Warlich’s. 

“ Lignite. 

Glass, bottle. 

“ Crown. 

“ flint.I 


Specific 

Gravity. 

Weight 
of a Cube 
Foot. 

Substances. 

Specific 

Gravity 

Weight 
of a Cube 
Foot. 

2740 

Lbs. 

171.25 

Stoives, Earths, 
etc. 

Glass, green. 

2642 

Lbs. 

165.125 

2864 

179 

“ optical. 

3450 

215.625 

1160 

72-3 

“ white. 

2892 

180.75 

830 

5 i -7 

“ window. 

2642 

165.125 

1714 

107.125 

■ “ soluble. 

1250 

78.125 

1367 

85-437 

Gniess, common. 

270 

I 5-875 

1900 

118.75 

Granite, Egyptian red.. 

2654 

165.875 

2400 

150 

“* Patapsco. 

2640 

165 

2201 

137-562 

“ Quincy. 

2652 

165.75 

1800 

112.5 

“ Scotch.... 

2625 

164.062 

1600 

IOO 

“ Susquehanna.. 

2704 

169 

2000 

125 

“ “ gray 

2800 

W 5 

3500 

2x8.75 

Graphite. 

2200 

137-5 

1300 

81.25 

Gravel, common. . 

1749 

IOQ. 312 

1560 

97-25 

Grindstone. 

2143 

133-937 

1520 

95 

Gypsum, opaque. 

2168 

135-5 

2784 

J 74 

Hone, white, razor. 

2876 

179-75 

1930 

120.625 

Hornblende. 

354 ° 

221.25 

2480 

155 

Iodine. 

4940 

— 

1350 

84-375 

Lava, Vesuvius... j 

1710 

106.875 

1436 

89-75 

2810 

175-625 

1640 

102.5 

Lias. 

i 35 o 

146.875 

I29O 

80.625 

Lime, quick. 

804 

5 o- 25 

1238 

77-375 

“ hydraulic. 

2745 

171.562 

1318 

82.375 

Limestone, white. 

3156 

197-25 

1277 

79.812 

“ green . 

3180 

198-75 

1276 

79-75 

Magnesia, carbonate.... 

2400 

150 

1290 

80.625 

Magnetic ore. 

5094 

317-6 

1292 

80.75 

Marble, Adelaide. 

2715 

169.687 

1273 

79.562 

“ African. 

2708 

169.25 

1 355 

84.687 

“ Biscayan, black. 

2695 

168.437 

1270 

79-375 

“ Carrara. 

2716 

169.75 

1300 

81.25 

“ common. 

2686 

167.875 

1259 

78.687 

“ Egyptian. 

2668 

166.75 

1300 

81.25 

“ French. 

2649 

165.562 

1302 

8i- 375 

“ Italian, white... 

2708 

169.25 

I 3 I 5 

82.187 

“ Parian. 

2838 

177-375 

IOOO 

62.5 

“ Vermont, white. 

2650 

i 65-57 

746 

46.64 

“ Silesian. 

2730 

170.625 

2200 

137-5 

Marl, mean. 

1750 

109-375 

2000 

125 

“ tough. 

2340 

146.25 

1216 

76 

Masonry, rubble. 

2050 

128.125 

1500 

93-75 

“ Granite. 

2640 

165 

2050 

128.125 

“ Limestone.... 

2640 

165 

2050 

128.125 

“ Sandstone.... 

2160 

135 

1600 

IOO 

“ Brick. 

2240 

140 

I92O 

120 

“ “ rough work 

1600 

IOO 

2020 

126.25 

Mica. 

2800 

175 

I9OO 

118.75 

Millstone. 

2484 

155-25 

I4OO 

87-5 

“ Quartz.. 

1260 

78-75 

4000 

250 

Mortar. j 

1384 

86.5 

2600 

162.5 

1750 

109.375 

2582 

i 6 i -375 

Mud.. 

1630 

101.875 

2594 

162.125 

“ wet and fluid. 

1782 

112 

1320 

82.5 

“ “ “ pressed... 

1920 

120 

1150 

71-875 

Nitre...'. 

1900 

118.75 

1300 

81.25 

Oyster-shell. 

2092 

130-75 

2732 

170.75 

Paving-stone. 

2416 

151 

2487 

155-437 

Peat, Irish, light. 

278 

17-375 

2933 

183.312 

“ “ dense. 

562 

35-125 

3200 

196 

“ very “ . 

675 

42. x87 


* Specific gravity of earth is estimated at from 1320 to 2200. 



















































































































SPECIFIC GRAVITY AND WEIGHT, 


213 


Substances. 


Stones, Earths, 
etc. 

Peat, black. j 

Phosphorus.. 

Plaster of Paris.| 

“ “ “ dry. 

Plumbago. 

Porcelain, China. 

Porphyry, red. 

Pumice-stone. 

Quartz. 

Red lead.. 

Resin.. 

Rock, crystal. 

Rotten-stone. 

Salt, common. 

“ rock. 

Saltpetre. 

Sand, coarse. 

“ common. 

“ damp and loose... 
dried “ <k ... 

“ dry...... 

“ mortar,Ft.Richm’d 
“ “ Brooklyn.. 

“ silicious.v_ 

Sandstone, mean.. 

“ Sydney. 

Schorl. 

Scoria, volcanic . 

Sewer pipe, mean.__ 

Shale. 


Specific 

Gravity. 


1058 

1329 

1770 

1176 

3400 

1400 

2100 

2300 

2765 


8940 

1089 

2735 

1981 

2130 

2200 

2090 

1800 

1670 

1392 

1560 

1420 

1659 

1716 

1701 

2200 

2237 

3170 

830 

2250 

2600 


Slate.j 

“ purple. 

Smalt. 

Soapstone. 

Spar, calcareous.... 

“ Feld, blue. 

“ “ green. 

“ Fluor. 

Specular ore. 

Stalactite. 

Stone, Bath, Engl. 

“ Blue Hill.| 

“ Bluestone (basalt) 

“ Breakneck, N.Y.. 

“ Bristol, Engl. 

“ Caen, Normandy. 

“ common_.... 

“ Craigleith, Scotl. . 

“ Kentish rag, “... 

“ Kip’s Bay, N.Y.. 

“ Norfolk (Parlia¬ 

ment House)... 
“ Portland, Engl... 

“ Staten Isl’d, N.Y. 

“ Sullivan Co., “ 

Sulphur, native ... 

Terra Cotta. 

Tile.. 

Trap....1 


2672 

2900 

2784 

2440 

2730 

2735 

2603 

2704 

3400 

5251 

24*5 

1961 

2640 

2625 

2704 

2510 

2076 

2520' 

2316 

2651 

2759 

2304 

2368 

2976 

2688 

2033 

1952 

1815 

2720 


Weight 
of a Cube 
Foot. 

Substances. 

Specific 

Gravity. 

Weight 
of a Cube 
Foot. 

Lbs. 

Granite. 


Lbs. 

66.123 

(Gen'l Gillmore , U. S. A.) 
Duluth, Minn., dark.... 

2780 

* 73-7 

83.062 

Fall River, Mass., gray.. 

2635 

164.7 

110.625 

Garrison’s, N. Y. “ .. 
Jersey City, N. J., soap.. 

2580 

161.2 

73-5 

3 ° 3 ° 

189.3 

212.5 

Keene, N. H., bluish gray 

2656 

166 

87-5 

Maine. 

2635 

164.7 

131-25 

Millstone Pt., Conn. 

2706 

169.1 

* 43-75 

New London, “ . 

2660 

166.25 

172.812 

Quincy, Mass., light.... 

2695 

168.5 

57 -i 8 7 

Richmond, Va. 

2727 

170-5 

166.25 

“ “ gray. 

2630 

164.4 

558.75 

Staten Island, N. Y. 

2861 

178.8 

68.062 

Westchester Co., N. Y.. 

2655 

165.9 

170.937 

Westerly, R. I., gray_ 

2670 

166.9 

123.812 

133-125 

137-5 

130.625 

Limestone. 
(Gen'l Gillmore , U. S. A.) 
Bardstown, Ky., dark .. 

2670 

166.9 


Caen, France. 

1900 

118.8 

io 4 - 375 

87 

97-5 

88.75 

103.66 

107.25 

106.33 

137-5 

I 39 - Sl 

iq8. 125 

51-875 

140.625 
162.5 

167 

Canajoharie, N. Y. 

2685 

167.8 

Cooper Co., Mo., d'k drab 

2320 

* 4*-3 

Erie Co., N. Y , blue.... 

2640 

165 

Garrison’s, N. Y. 

2635 

164.7 

Gleqs’ Falls, “ . 

2700 

168.7 

Joliet, Ill., white. 

Kingston, N. Y. 

2540 

158-7 

2690 

168.1 

Lake Champlain, N. Y.. 

2750 

* 7*-9 

Lime Island, Mich., drab 

2500 

*56-3 

Marblehead, Ohio, white 

2400 

*50 

Marquette, Mich., drab . 

2340 

146.25 

* Sturgeon Bay, Wis., blu¬ 
ish drab. 

2780 

* 73-7 

181.25 

*74 

152-5 

170.625 

IVIarble. 

(Gen'l Gillmore , U. S. A. 
Dorset, Vt. 

2635 

164.7 

170.937 

168.312 

East Chester, N. Y. 

Italian, common. 

2875 

2690 

179.7 
168.1 

169 

Mill Creek, Ill., drab- 

2570 

* 7*-9 

212.5 

North Bay, Wis., “ - 

2800 

*75 

328.187 
150-937 
122.562 
165 

164.062 

Sandstone. 
(Gen'l Gillmore , U. S. A.) 
Albion, N.Y., brown- 

2420 

*51-25 

169 

Belleville, N. J., gray... 

2259 

I4I.2 

*56.875 

Berea. Ohio, drab. 

2110 

* 3*-9 

129.75 

Cleveland, “ olive green 

224O 

140 

* 57-5 

Edinb’h,Sc’tl., Craigleith 

2260 

141.25 

* 44-75 

Fond du Lac, Wis., purple 

2220 

*38-7 

16^.687 

Fontenac,Minn.,l’g’tbuff 

2325 

* 45 - 3 * 

172 

Haverstraw, N. Y., red.. 

2130 

* 33 - 1 

Kasota., Minn., pink- 

2630 

i 64-375 

*44 

Little Falls, N.Y., brown 
Marquette, Mich., purple 

2250 

140.6 

148 

2285 

142.5 

186 

Masillon, 0 ., yellow drab 

2110 

131.87 

168 

Medina, N. Y., pink. 

241O 

150.6 

127.062 

Middletown, Ct., brown. 

2360 

* 47-5 

122 

Seneca, Ohio, red “ 

2390 

149.4 

i* 3-437 

Vermillion, Ohio, drab.. 

2160 

*35 

170 

j Warrensburgh, Mo. 

214O 

* 33-75 
























































































214 


SPECIFIC GRAVITY AND WEIGHT, 


Spec. Grav. 


Agate. 2590 

Amethyst. 3920 

Carnelian. 2613 

Chrysolite. 2782 

Diamond, Oriental... 3521 
“ Brazilian.. 3444 

“ pure. 3520 

Emerald. 3950 


Precious Stones. 

Spec. Gray. 

Emerald, aqua ma¬ 


rine.’. 2730 

Garnet.'4189 

“ black... 3750 

Jasper.. 2600 

Jet. 1300 

Lapis lazuli. 2960 

Malachite ..... 4020 


Spec. Grav. 


Onyx. 2700 

Opal. 2090 

Pearl, Oriental. 2650 

Ruby.. 3980 

Sapphire. 3994 

Topaz. 3500 

Tourmaline.. 3070 

Turquoise. 2750 


Substances. 

Specific 

Gravity. 

IVEiscellaneoxxs. 
Amber. 

IO9O 

Atmospheric Air. 

Beeswax. 

9 6 5 " 
1900 
942 . 
988 
930 
950 
1650 
.1090 

9 2 3 

936 

9 2 3 

1790 

1222 

1261 

59 ° 

750 

500 

1452 

900 

1000 

1550 

1800 

980 

Bone. 

Butter. 

Camphor. 

Caoutchouc. 

Cotton. 

Dynamite. 

Egg. 

Fat of Beef. 

“ Hogs. 

“ Mutton. 

Flax. 

Gamboge... 

Glycerine, 6o°. 

Grain, Barley. 

“ Wheat. 

“ Oats. 

Gum Arabic. 

Gunpowder, loose. 

‘ ‘ shaken .... 

“ solid.| 

Gutta-percha. 

Hay, old compact...... 

128.8 

Horn. 

1689 

IO7O 

922 

IOO9 

Human body. 

Ice, at 32 0 . 

Indigo. 

Isinglass. 

I I I I 

Ivory. 

1825 

Q 47 

Lard. 

Leather . 

960 

1074 

1360 

1600 

Mastic. 

Myrrh. 

Nitro-Glycerine. 

Opium. 

1336 

2100 

Potash. 

Resin. 

1089 

.0822 

Snow. 

Soap, Castile. 

IO7I 

943 

950 

1606 

Spermaceti. 

Starch. 

Sugar. 

“ .66.{ 

Tallow. 

972 

1326 

941 

964 

970 

Wax.| 


Weight 
of a Cube 
Foot. 

Substances. 

Lbs. 

Hiiqxxicls. 

68.125 

Acid, Acetic. 

.080 728 

■ Benzoic. 

60.312 

“ Citric.. 

118.75 

- “ Concentrated. 

58-875 

“ Fluoric. 

6 i .75 

“ Muriatic. 

58.125 

“ Nitric. 

59-375 

“ Nitrous. 

103.125 

“ Phosphoric. 

— 

“ “ solid.. 

57.687 

“ Sulphuric. 

58.5 

Alcohol, pure, 6o°. 

57.687 

“ 95 per cent.... 

111.875 

“ 80 “ 

— 

“ 5 ° “ •••• 

78.752 

“ 40 “ .... 

3 6 .875 

“ 25 “ - 

46.875 

“ 10 “ 

31.25 

“ 5 “ •••• 

9°-75 

“ proof spirit,* 50 

56-25 

per cent., 6o° 

62.5 

“ proof spirit, 50 

96.875 

per cent., 8o° 

112.5 

Ammonia, 27.9 percent. 

61.25 

Aquafortis, double. 

8.05 

“ single. 

IO 1 ^. ^62 

Beer. 

66.935 

Benzine. 

57-5 

Bitumen, liquid. 

63.062 

Blood (human). 

69 A 37 

Brandy, .83 or .5 of spirit 

114.062 

Bromine . 

SQ. 187 

Cider. 

60 

Ether, Acetic. 

67.125 

“ Muriatic. 

85 

“ Nitric. 

IOO 

Sulphuric. 

83.5 

Honey. . . 

131.25 

Milk.'.. 

68.062 

Oil, Anise-seed. 

5-2 

“ Codfish . 

56.937 

“ Whale. 

58.937 

“ Linseed. 

59-375 

“ Naphtha. 

100.375 

“ Olive . 

60.25 

“ Palm. 

82.875 

“ Petroleum. 

58.812 

“ Rape . 

60.25 

“ Sunflower . 

60.625 

“ Turpentine . 


Specific 

Gravity. 

Weight 
of a Cube 
Foot. 


Lbs. 

1062 

66.375 

667 

41.687 

1034 

64.625 

1521 

95.062 

1500 

93-75 

1200 

75 

1217 

76.062 

1550 

96.875 

1558 

97-375 

2800 

175 

1849 

115.562 

794 

49.622 

816 

5 i 

863 

53-937 

934 

58.375 

9 Si 

59-437 

970 

60.625 

986 

61.625 

992 

62 

} 934 

58.375 

j 875 

54.687 

891 

55.687 

1300 

81.25 

1200 

75 

1034 

64.625 

850 

53-125 

848 

53 

1054 

65-875 

924 

57-75 

2966 

185.375 

1018 

63.625 

866 

54-125 

845 

52.812 

iiio 

69-375 

715 

44.687 

1450 

90.625 

1032 

64-5 

986 

61.625 

923 

57.687 

923 

57.687 

940 

58.75 

850 

53.125 

9 J 5 

57-187 

969 

60.562 

880 

55 

914 

57-125 

926 

57.875 

■ 870 

54-375 


* Specific gravity of proof spirit according to Ure’s Table for Sykes’s Hydrometer, 920. 



























































































































SPECIFIC GRAVITY AND WEIGHT, 


215 


Substances. 


3 Aiqu.id.s. 
Spirit, rectified.... 
Steam, at 212 0 .... 

Tar. 

Vinegar . 

Water, at 32 0 . 


“ 39 -^ 
6 2 °f. 
“ 212 °. 


distilled, at 39 0 . 
.03818. 


\ 

Specific 

Gravity. 

Weight 
of a Cube 
Foot. 

Substances. 

Specific 

Gravity. 

Weight 
of a Cube 
Foot. 


Lbs. 

IMqLciids. 


Lbs. 

824 

51-5 

Water, Dead Sea. 

1240 

77-5 

.00061 

.038* 

“ Mediterranean... 

1029 

64.312 

1015 

63-437 

‘ ‘ sea. 

1029 

64.312 

1080 

67-5 

“ Black Sea. 

1016 

63-5 

998.7 

62.418 

“ rain. 

IOOO 

62.5 

998.8 

62.425 

Wine, Burgundy. 

992 

62 

997-7 

62.355 

“ Champagne. 

997 

64-375 

956-4 

59-64 

“ Madeira. 

1038 

62.312 

998 

62.379 

“ Port. 

997 

62.312 


+ 1 cube inch at standard temperature = 232.5954 grains. 


Compression of following fluids under a pressure of 15 lbs. per square inch: 
Alcohol.. .0000216 | Mercury.. .00000265 I Water.. .00004663 | Ether.. .00006158 


Elastic ETuids. 

1 Cube Foot of Atmospheric Air at 32 0 weighs 080728 lbs. Avoirdupois. 

grains , and at 62° 532.679 grains. 

Its assumed Gravity of 1 is Unit for Elastic Fluids. 


565.096 


Spec. Grav. 

Spec. Grav. 

Acetic Ether. 

3-°4 

Nitric acid. 

I.217 

Ammonia. 

•589 

“ oxide. 

1.094 

Atmos, air, at 32 0 .. 

I 

Nitrogen... 

•974 

Azote. 

.976 

Nitrous acid. 

2.638 

Carbonic acid. 

i -53 

Nitrous oxide.... 

1-527 

“ oxide.... 

.972 

Olefiant gas. 

.9672 

Carburet’d Hydrog. 
Chlorine. 

•559 

2.421 

Oxygen. 

Phosphurett’d Hy- 

1 106 

Chloro-carbonic... 
Chloroform ..._ 

3-389 

5-3 

drogen. 

Sulphuretted Hy- 

1.77 

Cyanogen. 

Gas, coal.j 

Hydrochloric acid. 

1.815 

•438 

drogen. 

Sulphurous acid.. 

I.I7 

2.21 

•752 

1.278 

Steam, J at 212 0 ... 
Smoke. 

•47295 

Hydrocyanic “ . 

.942 

Bitum. Coal_ 

. 102 

Hydrogen. 

.0692 

Coke. 

TO 

O 

H 

Muriatic acid. 

1.247 

Wood. 

09 

f Weight of a cube foot 267.26 grains, and compared with water at 


Spec. 

Vapor. 

Alcohol. 

Bisulphuret of 

Carbon . 

Bromine___ 

Chloric Ether.... 

Chloroform. 

Ether. 

Hydrochlor. Ether 

Iodine. 

Nitric acid. 

Spirits of Turpen¬ 
tine. 

Sulphuric acid ... 

“ Ether.. 

Sulphur. 

Water. 

62° specific gravity = .000 


Grav. 

1.613 

2.64 

5-4 

3-44 

4.2 

2.586 

2- 255 
8.716 

3 - 75 

5-oi3 

2-7 

2.586 

2.2x4 

.623 

6123. 


Weight of a Cube Foot of Gases at 32 0 F., and under Pressure of one Atmos¬ 
phere, or 2116.4 lbs. per Square Foot. 

Lbs. 

Hydrogen.005 594 

Nitrogen.. .078596 

Olefiant gas.079 5 

Oxygen.089 256 

Steam.050 22 


Lbs. 

Air, at 32 0 .080728 

<< u g 2 o.076097 

Alcohol.130 2 

Carbonic acid.123 44 

Carburet. Hydrog. .04462 


Lbs. 

Chlorine.197 

Chloroform.428 

Coal gas.03536 

Ether, Sulphuric.. .2093 
Gaseous steam.05022 


Sulphurous acid.. .1814 lbs. 


To Compute Weight of a Body or Substance wlierx 
Specific Gravity is given. 

Rule. —Multiply specific gravity by unit or standard of body or sub¬ 
stance, and product is the weight. 

Or, Divide specific gravity of body or substance by 16, and quotient will 
give weight of a cube foot of it in lbs. 

Example. —Specific gravity is 2250; what is weight of a cube foot of it? 

2250 X 62.5 = 140.625 lbs. 











































































2l6 


WEIGHTS OF VARIOUS SUBSTANCES, 


Weights and. Volumes of various Substances in 

Ordinary TTse. 


Substances. 


HVIetals. 

B-,.. { Sr %} 

“ gun metal. 

“ sheets. 

“ wire...... 

Copper, cast. 

“ plates..._ 

Iron, cast. 

“ gun metal. 

“ heavy forging.. 

“ plates. 

“ wrought bars... 

Lead, cast. 

“ rolled. 

Mercury, 6o°. 

Steel, plates. 

“ soft. 

Tin. 

Zinc, cast. 

“ rolled. 

W oods. 

Ash. 

Bay. 

Blue Gum. 

Cork. 

Cedar. 

Chestnut. 

Hickory, pig nut. 

“ shell-bark.. 

Lignum-vitae. 

Logwood. 

Mahoga’y,Hondur ! s | 

Oak, Canadian. 

“ English. 

“ live, seasoned... 

“ white, dry. 

“ “ upland... 

Pine, pitch. 

y- red. 

“ white. 

“ well-seasoned.. 
Pine, yellow. 


Cube Foot. 

Cube Inch. 

Lbs. 

Lbs. 

488.75 

.2829 

543-75 

• 3 M 7 

5 I 3-6 

.297 

524.16 

•3033 

547-25 

•3179 

543-625 

•3167 

450.437 

.2607 

466.5 

■27 

479-5 

■2775 

481.5 

.2787 

486^75 

.2816 

7 ° 9 - 5 

.4106 

711-75 

.4119 

848.7487 

.491174 

487-75 

.2823 

489.562 

•2833 

455-687 

•2637 

428.812 

.2482 

449-437 

, 2601 

Cube Feet 
in a Ton. 

52.812 

42.414 

51-375 

43.601 

64-3 

34-837 

15 

149-333 

35.062 

63.886 

38.125 

58.754 

49-5 

45-252 

43-125 

51.942 

83- 3 12 

26.886 

57.062 

39- 2 55 

35 

64 

66-437 

33 - 7 I 4 

54-5 

41. IOI 

58-25 

38-455 

66.75 

33-558 

53-75 

41.674 

42-937 

52.169 

41.25 

54-303 

3 6 - 875 

60.745 

34-625 

64.693 

29.562 

75-773 

33 - 8 i 2 

66.248 


Substances. 

Cube Foot. 

Woods. 

Lbs. 

Spruce. 

3 r -25 

Walnut, black, dry... 

3 I -25 

Willow. 

36.562 

“ dry. 

30-375 

IVIiscellaneous. 


Air. 

•075291 


Basalt, mean. 

1 75 

Brick, fire. 

137.562 

“ mean. 

102 

Cbal, anthracite.... I 

89-75 

1 

102.5 

“ bitumin., mean. 

80 

“ Cannel. 

Q 4 . 87S 

“ Cumberland.... 

84.687 

“ Welsh, mean... 

81.25 

Coke. 

62. S 

Cotton, bale, mean... 

14-5 

“ “ pressed j 

20 

25 

Earth, clay.. 

126.625 

u common soil.. 

I 37- 12 5 

“ “ gravel 

ioq.312 

“ dry, sand. 

120 

“ loose. 

93-75 

“ moist, sand... 

128.125 

“ mold. 

128.125 

“ mud. 

101.875 

“ with gravel... 

126.25 

Granite, Quincy. 

165-75 

“ Susquehanna 

169 

Gypsum. 

135-5 

Hay, bale. 

12 

u hard pressed- 

25 

Ice, at 32 0 . 

57-5 

India rubber. 

56-437 

“ vulcanized 


Limestone. 

197-25 

Marble, mean. 

167.875 

Mortar, dry, mean... 

97.98 

Plaster of Paris. 

73-5 

Water, rain. 

62.5 

“ salt. 

64.312 

“ at 62°. 

62.355 


Cube Feet 
in a Ton. 


71.68 

71.68 

61.265 

73-744 


12.8 

16.284 

21.961 

24.958 

21.854 

28 

23.609 

26.451 

27.569 
35-84 

154.48 

114 

89.6 

18.569 
16.335 
20.49 
18.667 

23- 8 93 

17.482 

17.482 

21.987 

17.742 

13-514 

13-254 

16.531 

186.66 

89.6 

38.95 

39- 6 9 

n-355 

13-343 

22.862 

30.476 

35-84 

34 - 83 

35 - 955 


To Compute Proportions of Two Ingredients in a Com¬ 
pound, or to Discover Adulteration in IVEetals. 


Rule. —Take differences of each specific gravity of ingredients and spe¬ 
cific gravity of compound, then multiply gravity of one by difference of 
other; and, as sum of products is to respective products, so is specific 
gravity of body to proportions of the ingredients. 

Example. — A compound of gold (spec. grav. — 18.888) and silver (spec. grav. = 
10.535) has a specific gravity of 14; what is proportion of each metal? 

18.888—14=4.888X 10.535 = 51.495. 14—10.535= : 3-465X 18.888=65.447. 

65- 447+51-495:65.447; 114:7.835 gold , 65.447+51.495: 51.495;; 14:6.165 silver. 









































































WEIGHTS OF VARIOUS SUBSTANCES IN BULK. 21 J 


"Weiglits of Various Svfbstances per Cube Foot in Bulk. 


Lbs. 


Lead, in pigs.567 

Iron, “ 360 

Marble, in blocks) 
Limestone, “ ) ' * 172 

Trap.170 

Granite, in blocks .... 164 

Sandstone.141 


Lbs. 

Potters’ clay.130 

Loam.126 

Gravel.109 

Sand. 95 

Bricks, common.... 93 

Ice, at 32 0 . 57-5 

Oak, seasoned. 52 


Lbs. 


Coal, caking.50 

Wheat.48 

Barley.38 

Fruit and vegetables.. 22 

Cotton seeds.12 

Cotton. . 

Hay, old. 8 


Ash, dry, 100 feet BM.175 ton. 

“ white, “ “.141 “ 

Cement, struck bushel and 

packed*.100 lbs. 

Cement, Portland, bushel.no lbs. 

Cherry, dry, 100 BM. .156 ton. 

Chestnut, dry, 100 BM ... tco “ 
Coal, anthracite, 1 cub. yd. 
broken and loose .. 

“ “ “ x ton. 

Coke, ton =. 

Earth, common soil 


Earth,loose. 93.75 lbs. 

Elm, dry, 100 feet BM.13 ton. 

Gypsum, ground, str. bush. 70 lbs. 

“ “ well shaken 80 “ 

Hemlock, dry, 100 feet BM. .093 ton. 

Hickory, “ “ “ . .197 “ 

Masonry, Granite,dressed.. 165 lbs. 

“ “ rough... 126 “ 

“ Limestone, dres’d 165 “ 

“ Sandstone.135 “ 

“ Brick, pressed ... 140 “ 

“ “ com’n, rough. 100 “ 


1.75 yds. 

41.5 cub. feet. 

80 to 97 cub. feet. 

.137 - 12 5 lbs - 

* One.,paeked bushel = 1.43 loose. 


Comparative Weight of Green and Seasoned Timber. 


Timber. 

Weight of a 
Green. 

Cube Foot. 
Seasoned. 

Timber. 

Weight of a 
Green. 

Cube Foot. 
Seasoned. 

American Pine. 

Ash. 

Beech . 

Lbs. 

44-75 

58.18 

60 

Lbs. 

3°-7 

5 ° 

53-37 

Cedar . 

English Oak. 

Riga Fir. 

Lbs. 

3 2 

71.6 

4 8 -75 

Lbs. 

28.25 

43-5 

35-5 


-Application. of tlie Ta"bles. 


When Weight of a Solid or Liquid Substance is required. Rule. —Ascer¬ 
tain volume of substance in cube feet; multiply it by unit in second column 
of tables (its specific gravity), and divide product by 16; quotient will give 
weight in lbs. 

When Volume is given or ascertained in Inches. Rule. —Multiply it by 
unit in third column of tables (weight of a cube inch), and product will give 
weight in lbs. 

Example.— What is weight of a Cube of Italian marble, sides being 3 feet? 

33 x 2708 = 73 116 oz ., which - 4 -16 = 4569.75 lbs. 

Or of a sphere of cast iron 2 inches in diameter? 

23 x -5236 X .2607 weight of a cube inch— 1.092 lbs. 


When Weight of an Elastic Fluid is required. Rule. —Multiply specific 
gravity of fluid by 532.679 (weight of a cube foot of air at 62° in grains), 
divide product by 7000 (grains in a lb. Avoirdupois), and quotient will give 
weight of a cube foot in lbs. 

Example. —What is weight of a cube foot of hydrogen ? 

Specific gravity of hydrogen .0692. 

532.679 X .0692 -i- 7000 = .005 265 9 lbs. 

To Compute Weiglit of Cast iVtetal Toy "Weiglit of Pattern. 

When Pattern is of White Pine. Rule. —Multiply weight of pattern in 
lbs. by following multipliers, and product will give weight of casting: 

Iron, 14; Brass, 15; Lead, 22 ; Tin, 14; Zinc, 13.5. 

When there are Circular Cores or Prints. Multiply square of diameter of 
core or print by its length in inches, the product by .0175, and result is 
weight of pattern of core or print to be deducted from weight of pattern. 














































2 l8 balloons, shrinkage of castings, etc. 


To Compute "Weights of Ingredients, that of Compound 

heing given. 

Rule.—A s specific gravity of compound is to weight of compound, so are 
each of the proportions to weight of its material. 

Example. —Weight, as above, being 28 lbs., what.are we'ghts of the ingredients? 


ia • 28 •• f7- 8 35 : 15-67 9 ? ld , 

* ‘ " (6.165 : I2 -33 silver. 

Note. —Specific gravity of alloys does not usually follow ratio of their compo¬ 
nents, it being sometimes greater and sometimes less than their mean. 


To Compute Capacity of a Balloon. 

Rule. —From specific gravity of air in grains per cube foot, subtract that 
of the gas with which it is inflated ; multiply remainder by volume of bal¬ 
loon in cube feet; divide product by 7000, and from quotient subtract weight 
of balloon and its attachments. 


Example. —Diameter of a balloon is 26.6 feet, its weight is 100 lbs., and specific 
gravity of the gas with which it is inflated is .07 (air being assumed at 1); what is 
its capacity, specific gravity of air assumed at 527.04 grains. 


527.04 — (527.04 X -07) 36-89 X 26.63 x .5236 
7000 


100 = 590.04 lbs. 


To Compute Diameter of a Balloon. 
Weight to be raised being given .—By inversion of preceding rule. 


3 /W X 7000 -}- s — s' 


</ 


•5236 


(I. s and s' representing iveight of air and gas 


in grains per cube foot , W weight to be raised in lbs., and d diameter of bal¬ 
loon in feet. 

Illustration. —Given elements in preceding case. 


Then 3/ ^°'°^ + 100 X 700 0 -i- 527-04 36.89 = 3 / q8s 4 .69 = 

V -5236 V -5236 

Proof of Spirituous Liquors. 

A cube inch of Proof Spirits weighs 234 grains; then, if an immersed 
cube inch of any heavy body weighs 234 grains less in spirits than air, it 
shows that the spirit in which it was weighed is Proof 

If it lose less of its weight, the spirit is above proof; and if it lose more, 
it is below proof. 

Illustration. —A cube inch of glass weighing 700 grains weighs 500 grains when 
weighed in a certain spirit; what is the proof of it? 

700 — 500 = 200 = grains = weight lost in spirit. 

Then 200 : 234 1 : 1.17 = ratio of proof of spirits compared to proof spirits, or 

1 —.17 above proof. 

Note. —For Hydrometers and Rules for ascertaining Proof of Spirits, see page 
67; and for a very full treatise on Specific Gravities and on Floatation, see Jamie¬ 
son’s Mechanics of Fluids. Lond., 1837. 


Shrinkage of Castings. 

It is customary, in making of patterns for castings, to allow for shrinkage 
per lineal foot of pattern as follows: 

Iron, small cylinders ... = Xo in. per ft. 

“ Pipes. — Y “ 

“ Girders, beams, etc. = % in 15 ins. 

“ Large cylinders,'! 

the contraction > —X* P er foot. 


of diam.at top.J 
Ditto at bottom 


_ 1 

— T8 


Ditto in length. ... = % in 16 ins. 

Brass, thin. — Y in 9 ins. 

thick. — Y in 10 ins. 


Zinc. 
Lead . .. 
Copper.. 
Bismuth 


in a foot. 

u 


_ 5 

— 32 

















GEOMETRY. 


219 


GEOMETRY. 

Definitions. 

Point has position, but not magnitude. 

Line is length without breadth, and is either Right, Curved, or Mixed. 
Right Line is shortest distance between two points. 

Curved Line is one that continually changes its direction. 

Mixed Line is composed of a right and a curved line. 

Superficies has length and breadth only, and is plane or curved. 

Solid has length, breadth, and thickness, or depth. 

Angle is opening of two lines having different directions, and is either 
Right, A cute, or Obtuse. 

Right Angle is made by a line perpendicular to another falling upon it. 
Acute Angle is less than a right angle. 

Obtuse Angle is greater than a right angle. 

• Triangle is a figure of three sides. 

Equilateral Triangle has all its sides equal. 

Isosceles Triangle has two of its sides equal. 

Scalene Triangle has all its sides unequal. 

Right-angled Triangle has one right angle. 

Obtuse-angled Triangle has one obtuse angle. 

A cute-angled Triangle has all its angles acute. 

Oblique-angled Triangle has no right angle. 

Quadrangle or Quadrilateral is a figure of four sides, and has following 
particular designations—viz,, 

Parallelogram, having its opposite sides parallel. 

Square, having length and breadth equal. 

Rectangle, a parallelogram having a right angle. 

Rhombus or Lozenge , having equal sides, but its angles not right angles. 
Rhomboid , a parallelogram, its angles not being right angles. 

Trapezium , having unequal sides. 

Trapezoid, having only one pair of opposite sides parallel. 

Note. — Triangle is sometimes termed a Trigon, and a Square a Tetragon. 
Gnomon' is space included between the lines forming two similar parallelo¬ 
grams, of which smaller is inscribed within larger, so as to have one angle 
in each common to both. 


Polygons are plane figures having more than four sides, and are either 
Regular or Irregular, according as their sides and angles are equal or un¬ 
equal, and they are named from number of their sides or angles. Thus: 


Pentagon has five sides. 
Hexagon six 
Heptagon “ seven 
Octagon u eight 


Nonagon has nine sides. 
Decagon “ ten “ 
Undecagon “ eleven “ 
Dodecagon “ twelve u 


Circle is a plane figure bounded by a'curved line, termed Circumference 
or Periphery. 

Diameter is a right line passing through centre of a circle or sphere, and 
terminated at each end by periphery or surface. 

Arc is any part of circumference of a circle. 

Chord is a right line joining extremities of an arc. 

Segment of a circle is any part bounded by an arc and its chord. 

Radius of a circle is a line drawn from centre to circumference. 

Sector is any part of a circle bounded by an arc and its two radii. 

. Semicircle is half a circle. 

• Quadrant is a quarter of a circle. 

Zone is a part of a circle included between two parallel cords. 

Lune is space between the intersecting arcs of two eccentric circles. 



220 


GEOMETRY. 


Secant is line running from centre of circle to extremity of tangent of arc. 

Cosecant is secant of complement of an arc, or line running from centre of 
circle to extremity of cotangent of arc. 

Sine of an arc is a line running from one extremity of an arc perpendicu¬ 
lar to a diameter passing through other extremity, and sine of an angle is 
sine of arc that measures that angle. 

Versed Sine of an arc or angle is part of diameter intercepted between sine 
and arc. 

Cosine of an arc or angle is part of diameter intercepted between sine and 
centre. 

Coversed Sine of an arc or angle is part of secondary radius intercepted 
between cosine and circumference. 

Tangent is a right line that touches a circle without cutting it. 

Cotangent is tangent of complement of arc.’ 

Circumference of every circle is supposed to be divided into 360 equal 
parts,termed Degrees', each degree into 60 Minutes, and each minute into 60 
Seconds , and so on. * - 

Complement of an angle is what remains after subtracting angle from 90 
degrees. 

Supplement of an angle is what remains after subtracting angle from 180 
degrees. 

To exemplify these definitions , let Ac b, in following Figure , be an assumed 
arc of a circle described with radius B A: 

A c b, an Arc of circle A C E D. 

A 6, Chord of that arc. 

B A, an Initial radius. 

B C, a Secondary radius. 
cDd,a Segment of the circle. 

A B b, a Sector. 

A I) E, a Semicircle. 

C B E, a Quadrant. 

A e d E, a Zone. 
n oh, a Lune. 

B g , Secant of arc Act; written Sec. 
b k, Sine of arc A c b ; written Sin. 

A k, Versed Sine of arc A cb; written Versin. 

B k or m b , Cosine of arc A cb. 

A <7, Tangent of arc A cb. 

CB6, Complement, and b B E, Supplement of 
arc A cb. 

B s, Cosecant of arc; written Cosec. 

to C, Coversed sine of arc, or, by convention, of angle A B b ; written Coversin. 

Vertex of a figure is its top or upper point. In Conic Sections it is point 
through which generating line of the conical surface always passes. 

Altitude , or height of a figure, is a perpendicular let fall from its vertex 
to opposite side, termed base. 

Measure of an angle is an arc of a circle contained between the two lines 
that form the angle, and is estimated by number of degrees in arc. 

Segment is a part cut off by a plane, parallel to base. 

Frustum is the part remaining after segment is cut off. 

Perimeter of a figure is the sum of all its sides. 

Problem is something proposed to be done. 

Postulate is something required. 

Theorem is something proposed to be demonstrated. 

Lemma is something premised, to render what follows more easy. 

Corollary is a truth consequent upon a preceding demonstration. 

Scholium is a remark upon something going before it. 

For other definitions see Mensuration of Surfaces and Solids, and Conic Sections. 



C s , Cotangent of arc, written Cot. 










GEOMETRY, 


221 


Lengths of* following Elements, Radins = 1 . 



Angle 45 0 . 

Angle 6o°. 


Angle 45 0 . 

Angle 6o°. 

Sine. 

.707 107 

.866 025 

Cosecant. 

1.414 214 

I - I 54 7 

Cosine. 

.707 107 

•5 

Tangent. 

1 

1.732 05 

Versed Sine.. 

.292 893 

•5 

Cotangent . . . 

1 

•577 349 

Coversed “ .. 

.292 893 

•133975 

Chord. 

.765 366 

1 

Secant. 

1.414214 

2 

Arc. 

•785 39 s 

1.047 2 



Scales. 

To Divide a Line, as A B, with any- reqnired USTnmher 
of Equal Parts.—Fig. 1 . 

From A and B draw two parallel lines, 
A o, B r, to an indefinite length, and upon 
them point off required number of equal 
parts, as A i, 2, 3, 4, and B 1, 2, 3, 4; join 
0 B, 4 1, etc. 

Or, point off on A 0, join 0 B, and draw 
the other lines parallel thereto. 

Diagonal Scale, as A B.—Fig. 2 . 

Divide a line into as many di¬ 
visions as there are hundreds of 
feet, spaces of ten feet, feet, or 
inches required. 

Draw perpendiculars from each 
division to a parallel line, C D. 
Divide them and one of divisions, 
A E, C F, into spaces of ten if for 
feet and hundredths, and twelve 
if for feet and inches; draw the 


To 


2. A at 


Construct 
E G 


123 


F 



lines A 1, a 2, 6 3, etc., and they will complete scale. 

Thus: Line A B representing ten feet; A to E, E to G, etc., will measure one 
foot; A to a, C to 1, 1 to 2, etc., will measure i-ioth of a foot. The several lines 
A 1, a 2, etc., will measure upon lines k, Z, etc., i-iooth of a foot; and op will 
measure upon k, ?, etc., divisions of i-ioth of a foot. 


To D 

3. 


Lines. 

raw a Perpendicular to a Right Line, 

P as 0 r, Fig. 3 , c A, Fig. d, or 

from a Point external to 
it, as A, Fig. £ 5 , and. from 
any two Points, as c d , 

Fig. G. 

With any radius as r A, r B, cut line 
at A and B; then with a longer radius, 
as A 0, B 0, describe arcs cutting each 
other at 0, and connect 0 r. (Fig. 3.) 

Or, from A, set off A B equal to 3 B 
parts by scale; from A B, with radii 
of 4 and 5 parts, describe arcs cut¬ 
ting at c, and connect c A. (Fig. 4.) 

Note. — This method is useful 
where straight edges are inappli¬ 
cable. 'Any multiples of numbers 
3, 4, 5 may be taken with same ef¬ 
fect, as 6, 8,10, or 9,12,15. ^- 

From A, with a sufficient radius, c 
cut line at 0 c, and from them de¬ 
scribe arcs cutting at r, and connect 
A r. (Fig. 5.) 

From any two points, as c d, at a proper 
distance apart, describe arcs cutting at A B, 
and connect them. (Fig. 6. j 




y- 

'3 


































































222 


GEOMETRY, 


7. 





To Bisect a Big lit Line or an Arc of a 
Circle, and. to Draw, a Perpendicu¬ 
lar to a Circnlar or Biglit Line, or a 
Badial Arc.—>Big. 7 . 

From A B as centres describe arcs cutting each other 
at c and d, connect c d, and line and arc are bisected 
at e and o. 

Line c d is also perpendicular to a right line as A B, 
and radial to a circular arc as A o B. 


8 . 


To Draw a Line 
Biglit I 


Barallel to 
c d, Big. 


a Given 

8 . 




ane, as v, 

From A B describe arcs Ac, Ad, and draw a line par¬ 
allel thereto, touching arcs c and d. 


To Describe Angles 


-Angles. 

of 30 ° and 
Big. lO. 


60 °, Big. 9 , and 45 °, 



From A, with any radius, A o, de¬ 
scribe o r, and from o with a like ra¬ 
dius cut it at r, let fall perpendicular 
r s ; then o A r = 6 o°, and A rs — 30 0 . 

(Fig. 9-) 

Set off any distance, as A B, erect 
perpendicular Ao=A B, and connect 
0 B. (Fig. 10.,' 




To Bisect Inclination of Two Lines, 
when Boint of Intersection is Inac¬ 
cessible.—Big. 11. 

Upon given lines, A B, C D, at any points draw perpen¬ 
diculars eo, sr, of equal lengths, and from 0 and 5 draw 
parallels to their respective lines, cutting at n\ bisect 
angle ons , connect nm, and line will bisect lines as re¬ 
quired. 


Rectilineal ZEUigunes. 


To Describe an Octagon upon a Line, as A B.—Big. IS. 



From points A B erect indefinite perpendiculars A/, Be; 
produce A B to m and n, and bisect angles m A 0 and n Bp 
with A u and B r. 

Make A u and B r equal to A B, and draw u z, r v parallel 
to A/, and equal to A B. 

From z and v , as centres, with a radius equal to A B, de¬ 
scribe arcs cutting A/, Be, in / and e. Connect zffe, 
and e v. 



To Inscribe any Begular Polygon in a 
Circle, or to Divide Circumference into 
a given 3 STnnaber of Equal Barts.—Big. 13 . 

If Circle is to contain a Heptagon. — Draw angle A 0 B at 
centre o for 360° -4-7 = 51° 42 ' 51"+) or 51-f, then sct off upon 
circumference distance A B or remaining angles A 0 B, 

















GEOMETRY. 


223 


To Inscribe a, Hexagon in 
a Circle.—Pig. 14 . 



'»■—h 



Draw a diam¬ 
eter, AoB. From 
A and B as cen¬ 
tres, with Ao and 
B 0, cut circle at 
c m and en, and 
connect. 


To Inscribe a Pentagon in 
a Circle.—Pig. 16 . 


16 . A 

Draw diameters 


A c and m n , at 

/yf jv\\ 

right angles to 

~Bt / i \ * ^ 

\ each other; bisect 

A\ / 1 \i 

(\ on in r, and with 

MV si); jr / 

~l n r A describe As; 

\\ ! V 

/ from A with A s 

\\ 1 l / 

\\ 1 // 

describe s B. 


To Describe a Hexagon 
about a Circle.—Pig. 15 . 

Draw a diam¬ 
eter as aob ; and 
with ao cut circle 
ate; join ac,and 


dius or, through 
r draw e r paral¬ 
lel to c a, cutting 
diameter at m ; 
then with radius 


m 0 



/ 

r VV / 

/ / 
/ / 

/ / 

Mm 

/7 ✓ 
P/✓ 


e - 


0 m describe circle, within which describe 
a hexagon as above. 


one side of a pentagon. 

To Describe 
upon a Line 


Connect AB, and 
distance is equal to 

a Pentagon 
as A B — Pig. 



Draw B m per¬ 
pendicular to A B, 
and equal to one 
half of it; extend 
A m until in n is 
equal to B in. 

Fi'om A and B, 
with radius Bn, de¬ 
scribe arcs cutting 
each other in 0: 


then from 0, with radius 0 B, describe 
circle AC B, and line A B is equal to one 
side of a pentagon upon circle described. 

To Describe a Regular Polygon of any recpuirecl Number 

of Sides.—Pig. 18 . 

From point 0 , with distance o B, describe semicircle 
B b A, which divide into as many equal parts, A a, ab, b c, 
etc., as the polygon is to have sides. 

Thus, let a Hexagon be required: 

From 0 to second point b of six divisions draw 0 b, and 
through other points, c, d, and e , draw 0 C, 0 D, etc. 

Apply distance b B, from B to E, from E to D, from D to 
C, etc. Join these points, as Z> C, C D, etc. 

To Construct a Hexa 



E 


To Construct a Square or 
a Rectangle on a given 

Line.—Pig. 19 . 


19 . m;y,' 

r 


0 >'" 


X.n 



A B 

B 11, and join 0 r. 


On A B as cen¬ 
tres, with AB as 
radius, describe 
arcs cutting at 
c; on c describe 
arcs cutting at 
o r; and on 0 r 
describe others, 
cutting at mn\ 
draw A in and 


upon 

SO. 



a nexagon 
a given Line.—Pig. 


From ends of line, 
A B, describe arcs 
cutting each other 
at 0 , and from 0 as 
a centre, with radius 
0 A, describe a cir¬ 
cle, and with same 
radius set off A c, 
c d, B ffe, and con¬ 
nect them. 



m a 


Inscribe an Octagon 

Draw diameters, A C, B D, at right 
angles, bisect arcs, A B, B C, etc., at s, r, 
0, e, and join Ao, 0 B, etc. (Fig. 21.) 

To Describe an Octagon 
about a Circle.—Pig. SS. 
Describe a square about circle A B, 
draw diagonals cf, e d, draw 0 i , etc., 
perpendicular to ’diagonals and touch¬ 
ing circle. (Fig. 22.) 


Circle.—Pig 

C,_ 1 


SI. 






























224 


GEOMETRY, 


To Inscribe a Square in a Circle.—Fig. 33. 

Draw line A B through centre of circle; 
take any radius, as A e, and describe the 
arcs Aee, Bee; connect ee, continuing- 
line to C and D; join AC, AD,etc. (Fig.23.) 

To Describe a Square about 
a Circle.—Fig. 3T. 

Draw line A B through centre of circle. 

Take any radius, as A e; describe arcs 
Aee, Bee; connect ee, continuing line 
to CD. 

Describe B r and D r\ draw and extend B r and D ?*, and sides A and C parallel to 
them. (Fig. 24.) 

To Describe an Octaj 




25. 


;on in a Square.—Fig. 25. 



-rB 





y—G 


Let A B C D be given square. 

Describe A orr, B 0 r r, etc.; join in¬ 
tersections rrrr, etc., and figure formed 
is octagon required. (Fig. 25.) 

To Inscribe an Ecqnilateral 
Triangle in a Circle. — 
Fig. 36. 

From point A, with A 0 equal to radius 



of circle, describe 00; from 0 and 0 describe or, or; join A r, rr, and r A. (Fig. 26.) 

Note. —All figures of 10 or 20 sides are readily determined from side of a pentagon, 
being halved or quartered; and in like manner, all figures of 6, 12, or 24 sides are 
readily determined from radius of a circle, being equal to the side of a hexagon. 

Circles. 

To Describe an Arc of a Circle, 
through Two given Points, 

-witb a given Radius.—Fig. 

27. 

On A B as centres, with given radius, de¬ 
scribe arcs cutting at 0, and from 0 with 
same radius describe arc A B. (Fig. 27.) 


To Ascertain Centre of a Circle 
or of an Arc of a Circle.—Fig. 

28. 

Draw chord A B, bisect it with perpendicular c d, then bisect c d for centre 0. 
(Fig. 28.) 

To Describe a Circular Segment that 
-will both fill the angle between two 
diverging lines and touch them.— 
Fig. 39. 

Bisect inclined lines, A B, D E, by line e f and connect 
perpendicular thereto, B D, to define boundary of seg¬ 
ment to be described. Bisect angles at B and I) by lines 
cutting at 0, and from 0, with radius 0 e, describe arc 
men. *> 





To Draw a Series 
Dines, touching 

30 . 



_, - . — - -] 


j y h 



t Y * • 

1 

r uK x 

A B J c 


of Circles between Two Inclined 
them and each other.—F ig. 30. 

Bisect given lines AB, CD, by line oc. 
From a point r in this line erect r s perpen¬ 
dicular to A B, and on r describe circle s m, 
cutting centre line at u ; from u erect u n 
perpendicular to centre line, cutting AB at 
n, and from n describe-an arc n u v, cutting 
A B at v, erect x v parallel to r s, making x 
centre of next circle to be described, with 


Note.—L argest circle may be described first. 


radius x u, and so on. 
















GEOMETRY. 


225 


Eo Tlcscnib© 9 Circle that sliall pass tlmoT-igli any three 
given Points, as A B C.— Pigs. 31 and. 32. 



Upon points A and B, 
with any opening of a 
dividers, describe arcs 
cutting each other at ee. 

On points B C describe 
two more cutting each 
other in points c c. 

Draw lines ee, and cc, 
and intersection of these 
lines, 0, is centre of circle 
ABC. (Fig. 31.) 



When Centre is not attainable. — From A B as centres, describe arcs A g, B h; 
through C draw A e. B c. Divide A e and B c into any number of equal parts, also 
eg and B h into a like number. Draw A 1, 2, 3, etc., and B 1, 2, etc., and intersec¬ 
tion of these lines as at 0 are points in the circle required. (Fig. 32.) 



Or, let A B C be given points, connect 
A B, A C, C B, and draw e c parallel to A B. 
Divide C A into a number of equal parts, 
as at 1, 2, and 3, and from C describe arcs 
through these points to meet right lines 
from C to points 1, 2, and 3, or A e, and 
these are points in a circle, to be drawn as 



Through point A draw radial line A 0, 
and erect perpendicular ef (Fig. 34.) 


To Draw Tangents to a 
Circle from a Point witb- 
out it.-—Pig. 35. 


To Draw a Tangent to a 
Circle from a given Point 
in Circumference. — Pig. 


From A draw A 0, and bisect it at s ; 
describe arc through 0, cutting circle at 
in n ; join A m or A n. 


To Draw from or to Circumference of a Circle, Lines 
leading to an Inaccessible Centre.—Pig. 3G. 



Divide whole or any given portion of 
circumference into desired number of 
parts; then, with any radius less than 
distance of two divisions, describe arcs 
cutting each other, as A ?*, b r, c r, d r, 
etc.; draw lines b r, c etc., and they 
will lead to centre. 


To draw md lines , as A r, F r. From b describe arc 0, and with radius b 1, from 
A or F as centres, cut arcs A r, etc., and lines A r, F r, will lead to centre. 

To Describe an Arc, or Segment of a Circle, of a large 

Radius.— Pig. • > 7 '. 

Draw chord A c B ; also line h D i 
parallel with chord, and at a distance 
equal to height of segment; bisect 
chord in c, and erect perpendicular 
c D; join A D, D B; draw A h and B i 
perpendicular to A D, BD; erect also perpendiculars A n, B n; divide A B and h i 
into any number of equal parts; draw lines 1 1, 2 2, etc., and divide lines A n, B n, 
each into half number of equal parts in A B; draw lines to D from each division in 
lines A n, B n, and at points of intersection with former lines describe arc or segment. 


















226 


GEOMETKY. 


Ellipse. 

To Describe an Ellipse to any Length and. Breadth. 

given.—Eig. 38. 

Let longest diameter be C D, and shortest E F. Take 
distance G o or o D, and with it, from points E and F, 
describe arcs h and/upon diameter C D. 

Insert pins at h and at f and loop a string around 
them of such a length that when a pencil is introduced 
within it it will just reach to E or F. Bear upon 
string, sweep it around centre o, and it will describe 
ellipse. 

Note. —It is a property of Ellipse that sum of two lines drawn from foci to meet in any point in 
curve is equal to transverse diameter. 

39. 



n 



Bisect transverse axis A B at o, and on centre o 
erect perpendicular C D, making o D and o C each 
equal to half conjugate axis. From C or D, with 
radius A o, cut transverse axis at ss for foci. Divide 
A o into any number of equal parts, as i, 2, 3, etc. 
With radii A 1, B 1, on s and s as centres, describe 
arcs, and repeat this operation for all other divis¬ 
ions 1, 2, 3, etc., and these points of intersection will 
give line of curve. 


To Ascertain Centre and. Two Diameters of an Ellipse. 

—Eig. 4.-0. 

Let A B c u be diameters of an Ellipse. 

Draw at pleasure two lines, q q, 0 m, parallel to 
each other, and equidistant from A and B; bisect 
them in points h n, and draw line ur; bisect it 
B in s, and upon s, as a centre, describe a circle at 
pleasure, as fl v , cutting figure in points/i>. 

Draw right line fv; bisect it in i, and through 
points i s draw greatest diameter A B, and through 
centre, s, draw least diameter c u, parallel to/ v. 

To Describe an Ellipse approximately by Circnlar Arcs. 

q —Eig. 4 rl. 




Set off differences of axes from centre 0 to a and 
c or 0 A and 0 C; draw a c and bisect it, and set off 
its half to r ; draw r s parallel to a c, set off 0 n 
equal to 0 r, connect n s, and draw parallels r m , 
n m ; from m, with radii m s and s to, describe arcs 
through C and D, and from n and r describe arcs 
through A and B. 

Note.— This method is not satisfactory when con¬ 
jugate axis is less than two thirds of transverse axis. 


42. 1 



"Witlr Arcs of Three Daclii.™Eig. 

4rS. 

On transverse axis A B draw rectangle A B c d, 
on height 0 e; to diagonal Ae draw perpendicular 
d h 0; set off 0 r equal to 0 e, describe a semicircle 
on A r, and produce O e to Z; set off o m- equal to 
e Z, and on 0 describe an arc with radius 0 to ; on 
A, with radius 0 Z, cut this arc at a. Thus the five 
centres, 0 , a, a', Z;, Vd, are found, from which arcs 
are described to form ellipse. 

Note.— This process answers for nearly all pro¬ 
portions of ellipses. It is used in striking vaults, 
stone bridges, etc. 












GEOMETRY. 


227 



To Construct an Ellipse from Two 
Circles.—Eig. 43. 

Describe two semicircles, as A B, C D, diameters of 
which are respectively lengths of major and minor 
axes. The intersection of the horizontal and vertical 
lines drawn from any radial line will give a point in 
the curve C D. 



To Construct air Ellipse, when Two 
Diameters are Griven.—Eig. 44. 

Make c o and A v equal to each other, but less 
than half breadth. Draw v 0, and from its centre i 
draw and extend perpendicular at i to d, draw dv to, 
make B u — A v, draw d u r, from u and v describe 
B r and A to, from d describe to c r, extend c z to s, 
and it will be centre for other half of figure. 


To Construct an 



Eighths. 


I — 

.48412 

5 — 

.927 03 

2- 

.66144 

6 — 

.968 24 

3 — 

. 780 63 

7 — 

.99216 

4 — 

.866 03 

8 — 

1 


Ellipse Toy Ordinates.—Fig. 45. 

Divide semi-transverse axis, as A &, into 8 or 10 
divisions, as may be convenient, and erect ordi¬ 
nates, the lengths of which are equal to semi-con¬ 
jugate, multiplied by the units for each division as 
follows: 


Divisions. 


1 —.45389 

2 — .6 

3 — • 7 I 4 4 
4-.8 


Tenths. 

5 — .866 02 

6 — .916 51 

7 — -993 94 

8 —.999 79 


9 — -994 99 
10 — 1 


To Construct an Ellipse when. Diameters do not Inter¬ 
sect at Eight Angles.—Eig. 4G. 



Let A B and C D be given diameters. 

Draw bouudary lines parallel to diameters, 
divide longest diameter into any number of 
equal parts, and divide shortest boundary lines 
into same number of equal parts. 

From one end of shortest diameter, D, draw 
radial lines through divisions of longest diame¬ 
ter, and from opposite end, C, draw radial lines 
to divisions on shortest boundary lines ; the 
intersection of these lines will give points in the 
curve. 



B 


Arcs. 

To Describe a Grothic Arc.—Eig. 47'. 

Take line A B. At points A and B draw arcs B a and Ac, 
and it will describe arc required. 


To Describe air Elliptic Arc, Chord and Height being 

given.—Eig. 48. 



Bisect A B at c ; erect perpendicular A q, and 
draw line q D equal and parallel to A c. 

Bisect A c and A q in r and n; make c I equal to 
c D, and draw line l r q; draw also line wsD; bisect 
s D with a line at right angles, and cutting line 
c D at o] draw line o q; make cp equal to c Jc. and 
draw line 0 p i. 

Then, from 0 as a centre, with radius 0 D, describe 
arc s Di; and from Jc and p as centres, with radius 
A Jc, describe arcs A s and B i. 























228 


GEOMETRY. 


49. 


To Describe a GrotHic _A.ro.—Figs. 4:9 and SO. 

Divide line A B into three equal parts, e c ; from points 
A and B let fall perpendiculars A o and B r, equal ip length 


to two of divisions of line A B; 


J 


draw lines o h and r g from points 
e, c ; with length of c B, describe arcs 

j.l_\ _ /__ _A g and B h, and from points t> and r / \ 

• e '' /c ' describe arcs g i and i h. (Fig. 49.) jJl. -_yl- Ijj 

Or, divide line A B into three a ' ! ^ 

equal parts at a and b , and on points 
A, a , b, and B, with distance of two 
divisions, make four arcs intersect- 
v ‘ ing at c and 0. 

Through points c, 0, and divisions a , &, draw lines c/and 0 <3, on points a and b 
describe arcs A e and B/, and on points c 0 arcs f s and e s. (Fig. 50.) 


\ / 
A 


t 


Cycloid, and Epicycloid. 


To Describe a Cycloid.—Fig. SI. 



51. When a circle, as a wheel, rolls over a 

straight right line, beginning as at A and 
ending at B, it completes one revolution, 
and measures a straight line, A B, exactly 
equal to circumference of circle c e r, which 
is termed the generating circle , and a point 
or pencil fixed at point r in circumference 
traces out a curvilinear path, ArB, termed a 
cycloid. A B is its base and c r its axis. 

Place generating circle in middle of Cy¬ 
cloid, as in figure; draw a line, m «, paral¬ 
lel to base, cutting circle at e; and tangent 
n i to curve at point n. The following are some of properties of Cycloid: 


Horizontal line e n = arc of circle e r. 
Half-base A c=half-circumference cer. 
Arc of Cycloid r n — twice chord r e. 
Half arc of cycloid Ar= twice diameter 
of circle r c. 


Or, whole arc of Cycloid A r B = four 
times axis c r. 

Area of Cycloid A r B A = three times 
area of generating circle r c. 

Tangent n i is parallel to chord e r. 


To Describe Curve of a Cycloid.—Fig. 52. 



On an indefinite line, AB, set off co= 
circumference of generating circle, di¬ 
vide this line into any number of equal 
parts (8 in figure), and at points of divis¬ 
ion erect perpendiculars thereto. Upon 
■p each of these lines describe a circle = 
generating circle. On c 1 take 1 x — 
.25 c 1, and with a; as a centre, with radius xc — .75 c 1, describe an arc cutting circle 
at P; from 2 on next circle, with two distances of 1 x', measured as chords, cut 
circle at 2'; from 3 on next circle, with three distances of 1 1', cut circle at 3', and 
proceed in like manner from each side until figure is complete. 


To Describe air Interior Epicycloid or Hypocycloid.™ 

Fig. S3. 



If generating circle is rolled on inside of fundamental 
circle,*as in Fig. 53, it forms an interior epicycloid , or 
hypocycloid , Ac B, which becomes in this case nearly a 
straight line. Other points of reference in figure cor¬ 
respond to those in Fig .51. When diameter of generat¬ 
ing circle is equal to half that of fundamental circle, 
epicycloid becomes a straight line, being diameter of 
the larger circle. 


* See explanation, Fig. 54. 













GEOMETRY. 


229 



To Describe an Exterior Epicycloid.— 
Eig. 54 . 

An Epicycloid differs from a Cycloid in this, that it is 
generated by a point, o'", in one circle, 0 r, rolling upon 
circumference of another, A r s, instead of upon a right 
line or horizontal surface, former being generating circle 
and latter fundamental circle. 

Generating circle is shown in four positions, in which 
its generating point is indicated by 0 o' 0" o'". A o'" s 
is an Epicycloid. 

Involute. 

To Describe an Involute.—Eig. 55. 

Assume A as centre of a circle, b c 0; a cord laid partly 
upon its circumference, as fee; then the curve eimn, 
described by a tracer at end of cord, when unwound from 
a circle, is an involute. 

This curve can also be defined by a batten, *, rolling on 
a circle, as s u. 



Parabola. 

To Construct a Parabola "by Ordinates or 
Abscissa.—Eigs. 56 and 57 ^. 

By Ordinates. 

Divide ordinate a b into 10 equal parts, and erect perpendicu¬ 
lars, length of which will be determined by multiplying abscissa 
a c by respective units for each perpendicular, as follows: 57 

Divisions. 


■ .19 
•• 3 6 


3 —- 5 i 
4—64 


•75 

.84 


7 — - 9 1 

8 — .96 


9 - 
10 - 


By Abscissa. 

Divide abcissa a c into 8 or 10 equal parts, as may be convenient, 
and draw ordinates thereto, the lengths of which will be deter¬ 
mined by multiplying half ordinate a 6 by respective units for 
each ordinate, as follows: „ . . 

Divisions. Tenths. 




Eighths. 


1 —.31623 

6— .7746 

1 —-3535 

5— .79057 


2 — .447 21 

7— .83666 

2 —-5 

6— .86602 


3 —-547 7 2 

8— .89443 

3 — .612 27 

7 — -935 41 


4 —.63245 

9— .94868 

4 —.707 n 

8 — 1 


5 —.707 11 

10 — 1 

58 . 

"With, a 

Square and Cord. 

—Eig. 58 . 



Place a straight edge to directrix A B, and apply to it a 
square, c 0. 

Attach to end 0 end of a cord equal to 0 A, and attach other 
end to focus e; slide square along straight edge, maintaining 
cord taut against edge of square, by a point or pencil, and curve 
will be traced. (Fig. 58.) 


When Height and. m r __ r 
Base are given.— j !_ 

Eig. 59. a rt 

cKt 


Assume AB axis and cd a double ordinate or base. 

Through A draw m n parallel to c d. and through c e \''} 
and cl draw c m, d n, parallel to axis A B. Divide cm, 0 ?, ' 
d n into any number of equal parts, as at a c e o, also ! 

6 B, B d, into a like number of parts. Through points c" 

1, 2, 3, and 4 draw lines parallel to axis, and through 
aceo draw lines to vertex A, cutting these perpendiculars, and through these points 
curve may be traced. (Fig. 59.) 

tf 







































230 


GEOMETRY. 





To Describe Curve of* a Parabola, Base and. 
Height being given.—Big. 60 . 

Draw an isosceles triangle, as a b d, base of which shall be equal 
to, and its height, c b, twice that of proposed parabola. Divide 
each side, ab,db, into any number of equal parts; then draw lines, 
i i, 2 2, 3 3, etc., and their intersection will define curve. (Fig. 6o.) 

To Describe a Parabola, any- Ordinate to .Ajxis 
and its Abscissa being given.—Fig. 61 . 

Bisect ordinate, as A o in r; join 
B r, and draw r s perpendicular to it, 
meeting axis continued to s.' Setoff 
B c, B e, each equal to o s; draw m 
c u perpendicular to B s , then m u is directrix and 
B e focus; through e and any number of points, i, i, 
i, etc., in axis, draw double ordinates rqfi v, and on 
centre e, with radii e c, i c, etc., cut respective or¬ 
dinates at v v, etc., and trace curve through these 
points. 

Note.— Line v ev passing through focus is parameter. 

Spii*al. 

To Draw a Spiral about a given Point.— 
Fig. 62 . 

Assume c the centre. Draw A h, divide it into twice number 
of parts that there are to be revolutions of line. Upon c de¬ 
scribe r e, o s, A h, and upon e describe r s, o s, etc. 

Hyperbola. 

To Describe a Hyperbola, Transverse and Conjugate 
Diameters being given.—Fig. 63 . 

Let A B represent transverse diameter, and C D 
conjugate. 

Draw C e parallel to A B, and e r parallel to C D; 
draw o e, and with radius o e , with o as a centre, 
describe circle F e r, cutting transverse axis pro¬ 
duced in F and /; then will F and / be foci of fig¬ 
ure. 

In o B produced take any number of points, n, n , 
etc., and from F and/as centres, with A n and B n 
as radii, describe arcs cutting each other in s, s, 
etc. Through s, s, etc., draw curve ssssBssss. 

Note. —If straight lines, as o e y and o r y. are drawn from centre o through ex¬ 
tremities e v , they will be asymptotes of hyperbola, property of which is to ap¬ 
proach continually to curve, and yet never to touch it. 

When Foci and Conjugate Axis are given. —Let F and /be foci, and C D conjugate 
axis, as in preceding figure. 

Through C draw g C e parallel to F and /; then, with o as a centre and oFasa 
radius, describe an arc cutting g C e at g and e; from these points let fall perpen¬ 
diculars upon line connecting F and/, and part intercepted between them as A B 
will be transverse axis. 

Catenary. 

To Delineate a Catenary, Span and "Versed Sine being 
(*4 # given. — Fig. 64 . (W. Hildenbrand.) 

Divide half span, as A B, into any required 
number of equal parts, as i, 2 , 3 , and let fall B C 
and Ao, each equal to versed sine of curve; divide 
Ao into like number of parts, i', 2 ', 3', as A B. 
Connect C i', C 2 7 , and C 3', and points of intersec¬ 
tion of perpendiculars let fall from A B will give 
points through which curve is to be drawn. 

Or, suspend a finely linked chain against a ver- 
tical plane, trace curve from it on the plane in accordance with conditions of given 
length and height, or of given width or length of arc. 

Note.— For other methods see D. R. Clark’s Manual, pp. 18,19. 




















AREAS OF CIRCLES. 


231 


Areas of Circles, from to 150. 


Diam. 

Area. 

Diam. 

Area. | 

Diam. | 

Area. 

Diam. 

Area. 

it 

.OOO 192 

3 

7.0686 

7 

38.4846 

14 

153-938 


.000 767 

1/ 

/16 

7.3662 

Z 

39-8713 

Z 

156.7 

K 

7.6699 

■ Z 

41.2826 

Z 

159-485 

Xg 

.003 068 

3/ 

/ir, 

7.9798 

% 

42.7184 

z 

162.296 

H 

.012 272 

Z 

8.2958 

X 

44.1787 

z 

165-13 

.027612 


8.618 

z 

45.6636 

z 

167.99 


% 

8.9462 

z 

47-1731 

z 

170.874 

X 

.049 087 

1/ 

'MR 

9.2807 

z 

48.7071 

z , 

173.782 

6/ 

Ag 

.076 699 

X 

9/ 

As 

9.6211 

9.968 

8 

z 

=50.2656 

51.8487 

15 

z 

176.715 

179.673 

Z 

.IIO447 

% 

10.3206 

z 

53-4563 

X 

182.655 

X, 

Z 

•15033 

•19635 

% 

10.679 

1 z 

55.0884 

% 

185.661 

% 

n.0447 

% 

56.7451 

X 

188.692 

13/ 

Ag 

n.416 

z 

58.4264 

z 

191.748 

9/ 

Ag 

.248 505 

% 

1 1-7933 

z 

60.1322 

z 

194.828 

% 

.306 796 

% 

12.177 

% 

61.8625 | 

z 

197-933 

4 

12.5664 

9 

63.6174 

16 

201.062 

1 Xg 

.371224 

a; 

12.962 

z 

65.3968 

z 

204.216 

% 

.441787 

Z 

i 3-364 i 

H 

67.2008 

z 

207.395 

13 / 

3/ 

Ag 

I3-772 

% 

69.0293 

z 

210.598 

AG 

•5 i 8487 

X 

14.1863 

X 

70.8823 

z 

213.825 

Z 

.601322 

Xg 

14 606 

z 

72.7599 

z 

217.O77 

% 

.690 292 

% 

15-033 

z 

74.6621 

z 

220.354 

X 

15-465 

z 

76.5888 

z 

223.655 

I 

•7854 


I5-9043 

10 

78-54 

17 

226.981 

a; 

.8866 

9/ 

/16 

16.349 

z 

80.5158 

z 

230.331 

z 

.99402 

Z 

16.8002 

z 

82.5161 

z 

233.706 

x 

1.1075 

% 

Z 

I7-257 

z 

84.5409 

z 

237.105 

x 

1.2272 

17.7206 

z 

86.5903 

z 

240.529 

X 

1-353 

13/ 

/16 

18.19 

z 

88.6643 

z 

243-977 

% 

1.4849 

% 

18.6655 

z 

90.7628 

z 

247-45 

X 

1.6229 

15/ 

/1G 

19.147 

z 

92.8858 

z 

250.948 

X 

1.767 1 

5 

I9-635 

II 

95-0334 

18 

254-47 

9/ 

/16 

I -9 I 7 5 

a; 

20.129 

z 

97-2055 

z 

258.016 

% 

2.0739 

z 

20.629 

z 

99.4022 

% 

261.587 

% 

2.2365 

3 / 

/1G 

2i-i35 

z 

101.6234 

% 

265.183 

Z 

2-405 3 

X 

21.6476 

z 

103.8691 

X 

268.803 

% 

2.58 

X 

22.166 

z 

106.1394 

z 

272.448 

% 

2.761 2 

% 

22.6907 

z 

108.4343 

z 

276.117 

% 

2.9483 

X 

23.221 

z 

no.7537 

z 

279.811 

2 

3.1416 

X 

23-7583 

12 

113.098 

19 

283.529 

% 

3-338 

9/ 

/16 

24.301 

z 

115.466 

Vs 

287.272 


3-5466 

Z 

24.8505 

z 

117.859 

X 

291.04 

3/ 

/i6 

37584 

% 

25.406 

z 

120.277 

z 

294.832 

x 

3.9761 

z 

25-9673 

z 

122.719 

X 

298.648 

X 

4.2 

13/ 

/16 

26-535 

z 

125.185 

z 

302.489 

% 

4.4301 

% 

27-1086 

z 

127.677 

z 

3o6-355 

X 

4.7066 

% 

27.688 

z 

130.192 

z 

310.245 

X 

4.908 7 

6 

28.2744 

13 

132.733 

20 

3 i 4 *i 6 

97 

As 

5- T 57 3 

z 

29.4648 

z 

135-297 

% 

318.099 

% 

5-4ii9 

X 

30.6797 

z 

137.887 

/i 

322.063 

% 

5-6723 

X 

3i-9i9i 

z 

140.501 

z 

326.051 

* z 

5-9396 

X 

33-1831 

z 

I43-I39 

K 

330.064 

13/ 

/16 

6.2126 

X 

34-47I7 

z 

145.802 

z 

z 

334.102 

% 

6.491 8 


35-7848 

z 

148.49 

338.164 

16/ 

/10 

6.777 2 

1 z 

37.1224 

z 

151.202 

z 

342.25 





































232 


AREAS OF CIRCLES. 


Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

21 

346.361 

28 

615.754 

35 

962.115 

42 

1385.45 

X 

350.497 

X 

621.264 

X 

969 

yi 

1393.7 

X 

354-657 

X 

626.798 

yi 

975.909 

yi 

1401.99 

X 

358.842 

X 

632.357 

X 

982.842 

X 

1410.3 

X 

363-05 1 

X 

637.941 

X 

989.8 

xi 

1418.63 

X 

367.285 

X 

643.549 

X 

996.783 

X 

1426.99 

X 

371-543 

X 

649.182 

X 

1003.79 

X 

1435.37 

X 

375.826 

X 

654.84 

X 

1010.822 

X 

1443.77 

22 

3,80.134 

29 

660.521 

36 

1017.878 

43 

1452.2 

X 

384.466 

X 

666.228 

X 

1024.96 

X 

1460.66 

X 

388.822 

X 

671.959 

yi 

1032.065 

yi 

1469.14 

X 

393203 

X 

677.714 

X 

1039.195 

X 

1477.64 

X 

397.609 

X 

683.494 

yi 

1046.349 

yi 

1486.17 

X 

402.038 

X 

689.299 

X 

1053.528 


1494.73 

H 

406.494 

X 

695.128 

X 

1060.732 

X 

1503.3 

X 

410.973 

X 

700.982 

X 

1067.96 

X 

1511.91 

2 3 

415477 

30 

706.86 

37 

1075.213 

44 

1520.53 

X 

42O.OO4 

X 

712.763 

X 

1082.49 

yi 

1529.19 

X 

424-558 

X 

718.69 

X 

1089.792 

% 

1537.86 

X 

429.135 

X 

724.642 

X 

1097.118 

X 

1546.56 

X 

433.737 

X 

730.618 

X 

1104.469 

x? 

1555.29 

X 

438.364 

X 

736.619 

X 

1111.844 

X 

1564.04 

X 

443.015 

X 

742.645 

X 

1119.244 

X 

1572.81 

X 

447.69 

X 

748.695 

X 

II26.669 

X 

1581.61 

24 

452.39 

31 

754.769 

38 

1134.118 

45 

1590.43 

X 

457-115 

X 

760.869 

X 

1141.591 

x^ 

1599.28 

X 

461.864 

X 

766.992 

X 

1149.089 

x£ 

1608.16 

X 

466.638 

X 

773.14 

X 

1156.612 

X 

1617.05 

X 

471.436 

X 

779.313 

X 

1164.159 

X 

1625.97 

X 

476.259 

X 

785.51 

X 

II 7 I -73 I 

X 

1634.92 

X 

481.107 

X 

791.732 

X 

1179.327 

X 

1643.89 

X 

485-979 

X 

797.979 

X 

1186.948 

X 

1652.89 

25 

490.875 

32 

804.25 

39 

H94-593 

46 

1661.91 

X 

495-796 

X 

810.545 

X 

1202.263 

X 

1670.95 

X 

500.742 

X 

816.865 

X 

1209.958 

X 

1680.02 

X 

505-712 

X 

823.21 

X 

1217.677 

X 

1689.11 

X 

510.706 

X 

829.579 

X 

1225.42 

X 

1698.23 

X 

515-726 

X 

835.972 

X 

1233.188 

X 

1707.37 

X 

520.769 

X 

842.391 

X 

I24O.98I 

X 

1716.54 

X 

525-838 

X 

848.833 

X 

1248.798 

X 

1725.73 

26 

530.93 

33 

855.301 

40 

1256.64 

47 

1734.95 

X 

536.048 

X 

861.792 

X 

1264.506 

X 

1744.19 

Tl 

541-19 

X 

868.309 

yi 

1272.397 

X 

1753.45 

X 

546.356 

X 

874.85 

X 

1280.312 

X 

1762.74 

yi 

551-547 

X 

881.415 

X 

1288.252 

X 

1772.06 

X 

556.763 

X 

888.005 

X 

1296.217 

X 

1781.4 

X 

562.003 

X 

894.62 

X 

1304.206 

X 

1790.76 

X 

567.267 

X 

901.259 

X 

1312.219 

X 

1800.15 

27 

572.557 

34 

907.922 

41 

1320.257 

48 

1809.56 

X 

577-87 

X 

914.611 

X 

1328.32 

X 

1819 


583.209 

VC 

921.323 

X 

. 1336-407 

X 

1828.46 

X 

588.571 

X 

928.061 

X 

I344.5I9 

X 

1837.95 

X 

593.959 

X 

934.822 

yi 

1352.655 

X 

1847.46 

X 

599.371 

X 

941.609 

X 

1360.816 

X 

1856.99 

X 

604.807 

X 

948.42 

X 

1369.001 

X 

1866.55 

X 

610.268 

X 

955-255 

X 

1377.211 

X 

1876.14 





























AREAS OF CIRCLES. 


233 


Diam. I 

Area. 

Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

49 

1885.75 I 

56 

2463.01 

63 

3117-25 

70 

3848.46 

X 

1895-38 1 

X 

2474.02 

X 

3129.64 

X 

3862.22 

X 

1905.04 

X 

2485.05 

X 

3142.04 


3876 

X 

1914.72 

X 

2496.11 

X 

3154-47 

X 

3889.8 

H 

1924.43 

% 

2507.19 

X 

3166.93 

X 

3903-63 

X 

1934.16 

X 

2518.3 

X 

3179-41 

X 

3917-49 

X 

1943.91 

X 

2529-43 

X 

3191.91 

X 

3931-37 

X 

1953-69 

X 

2540.58 

X 

3204.44 

X 

3945-27 

50 

1963-5 

57 

2551-76 

64 

3217 

71 

3959-2 

X 

1973-33 

% 

2562.97 

X 

3229.58 

X 

3973-15 

X 

19S3.18 

X 

2574.2 

X 

3242.18 

X 

3987.13 

X 

1993.06 

X 

2585-45 

X 

3254.81 

X 

4001.13 

X 

2OO2.97 

X 

2596-73 

X 

3267.46 

X 

4015.16 

X 

2012.89 

X 

2608.03 

X 

3280.14 

X 

4029.21 

X 

2022.85 

X 

2619.36 

X 

3292.84 

X 

4043.29 

X 

2032.82 

X 

2630.71 

X 

3305-56 

X 

4057-39 

51 

2042.83 

58 

2642.09 

65 

3318.31 

72 

4071.51 

X 

2052.85 

X 

2653-49 

X 

3331-09 

X 

4085.66 

X 

2062.9 

3^ 

2664.91 

X 

3343-89 

/£ 

4099.84 

X 

2072.98 

X 

2676.36 

X 

3356.71 

X 

4II4.O4 

X 

2083.08 

X 

2687.84 

X 

3369-56 

X 

4128.26 

X 

2093.2 

X 

2699.33 

X 

3382.44 

X 

4 I 42-5 I 

X 

2103.35 

X 

27IO.86 

X 

3395-33 

X 

4 i 56-78 

X 

2113.52 

X 

2722.4I 

X 

3408.26 

X 

4171.08 

52 

2123.72 

59 

2733-98 

66 

3421.2 

73 

4185.4 

X 

2133-94 

% 

2745-57 


3434-17 


4199.74 

X 

2144.19 

/i 

2757.2 


3447-17 


42i4.II 

X 

2154.46 

X 

2768.84 

X 

3460.19 

X 

4228.51 

X 

2164.76 

}4 

2780.51 

yi 

3473-24 

X 

4242.93 

X 

2175.08 

X 

2792.2I 

X 

3486.3 

X 

4257-37 

X 

2185.42 

X 

2803.93 

X 

3499-4 

X 

4271.84 

X 

2195.79 

X 

2815.67 

X 

3512.52 

X 

4286.33 

53 

2206.19 

60 

2827.44 

67 

3525-66 

74 

4300.85 

X 

2216.61 

/i 

2839.23 

X 

3538.83 

X 

4315-39 

X 

2227.05 

X 

2851.05 

yi 

3552.02 

X 

4329.96 

X 

2237.52 

X 

2862.89 

X 

3565-24 

X 

4344-55 

X 

2248.OI 

X 

2874.76 

X 

3578.48 

X 

4359- 1 7 

X 

2258.53 

X 

2886.65 

X 

3591-74 

X 

4373-Si 

X 

2269.07 

X 

2898.57 

X 

3605.04 

X 

4388.47 

X 

2279.64 

X 

2910.51 

X 

3618.35 

X 

4403.16 

54 

2290.23 

61 

2922.47 

68 

3631.69 

75 

4417.87 

X 

2300.84 

X 

2934.46 


3645-05 

K 

4432.6 i 

X 

231 x.48 

X 

2946.48 


3658.44 

X 

4447-38 

X 

2322.15 

X 

2958.52 

X 

3671.86 

X 

4462.16 

X 

2332-83 

X 

2970.58 

K 

3685.29 

X 

4476.98 

X 

2343-55 

X 

2982.67 

X 

3698.76 

X 

4491.81 

X 

2354-29 

X 

2994.78 

X 

37x2.24 • 

X 

4506.67 

X 

2365-05 

X 

3006.92 

X 

3725-75 

X 

4521.56 

55 

2375-83 

62 

3019.08 

69 

3739-29 

76 

4536.47 

X 

2386.65 

X 

3031.26 

X 

3752.85 

X 

4551-41 

/i 

2397.48 

X 

3043-47 

K 

3766.43 

X 

4566.36 

X 

2408.34 

X 

3055-71 

X 

3780.04 

X 

458i.35 

' X 

2419.23 

X 

3067.97 

X 

3793-68 

X 

4596-36 

X 

2430.14 

X 

3080.25 

X 

3807.34 

X 

4611.39 

X 

2441.07 

X 

3092.56 

X 

3821.02 

X 

4626.45 

X 

2452.03 

X 

3104.89 

X 

3834-73 

X 

464I-53 


u* 









































234 


AEEAS OF CIRCLES. 


Diam. 

Area. 

Diam. 

Area. 

| Diam. 

Area. 

Diam. 

Area. 

77 

4656.64 

84 

5541-78 

9 1 

6503-9 

98 

7542.98 

A 

467 1 *77 

A 

5558.29 

X 

6521.78 

X 

7562.24 

A 

4686.92 

A 

5574-82 

X 

6539.68 

yi 

7581.52 

As 

4702.1 

% 

5591-37 

% 

. 6557.61 

% 

7600.82 

A 

47I7-3I 

A 

5607.95 

A 

6575-56 

A 

7620.15 

% 

4732-54 

% 

5624.56 

% 

6593-54 

% 

7639-5 

A 

4747-79 

% 

5641.18 

A 

6611.55 

A 

7658.88 

% 

4763.07 

A 

5657-84 

A 

6629.57 

A 

7678.28 

78 

4778.37 

85 

5674-51 

92 

6647.63 

99 

7697.71 

X 

4793-7 

X 

5691.22 

X 

6665.7 

A 

7717.16 

X 

4809.05 

X 

5707-94 

X 

6683.8 

A 

7736.63 

% 

4824.43 

% 

5724.69 

X 

6701.93 

As 

7756.13 

A 

4839-83 

A . 

5741-47 

A 

6720.08 

A 

7775.66 

% 

4855.26 

% 

5758.27 

% 

6738.25 

As 

7795.21 

% 

4870.71 

A 

5775-1 

X 

6756.45 

A 

7814.78 

% 

4886.18 

A 

5791-94 

A 

6774.68 

A 

7834-38 

79 

4901.68 

86 

5808.82 

93 

6792.92 

IOO 

7854 

X 

4917.21 

% 

5825.72 

X 

6811.2 

X 

7893-32 

X 

4932.75 

X 

5842.64 

X 

6829.49 

X 

7932.74 

% 

4948.33 

% 

5859-59 

% 

6847.82 

A 

7972.25 

A 

4963.92 

A 

5876.56 

A 

6866.16 

IOI 

8011.87 

% 

4979-55 

% 

5893-55 

% 

6884.53 

X 

8051.58 

A 

4995-19 

% 

59 io -58 

A 

6902.93 

A 

8091.39 

A 

5010.86 

A 

5927.62 

A 

6921.35 

A 

8131-3 

80 

5026.56 

87 

5944.69 

94 

6939.79 

102 

8171.3 

X 

5042.28 

X 

5961.79 

X 

6958.26 

X 

8211.41 

X 

5058.03 

X 

5978.91 

X 

6976.76 

% 

8251.61 

% 

5073-79 

% 

5996.05 

% 

6995.28 

A 

8291.91 

% 

5089.59 

A 

6013.22 

A 

7013.82 

103 

8332-31 

% 

5105.41 

% 

6030.41 

% 

7032.39 

X 

8372.81 

% 

5121.25 

A 

6047.63 

A 

7050.98 

X 

8413.4 

A 

5 r 37 - 12 

A 

6064.87 

A 

7069.59 

X 

8454.09 

81 

5 i 53 .oi 

88 

6082.14 

95 

7088.23 

104 

8494.89 

A 

5168.93 

A 

6099.43 

X 

7106.9 

X 

8535-78 

/i 

5184.87 

A 

6116.74 

X 

7125-59 

X 

8576.76 

% 

5200.83 

% 

6134.08 

% 

7144-31 

A 

8617.85 


5216.82 

A 

615145 

X 

7 i6 3-°4 

105 

8659.03 

% 

5232.84 

% 

6168.84 

% 

7l8l.8l 

X 

8700.^2 

% 

5248.88 

A 

6186.25 

A 

7200.6 

X 

8741.7 

% 

5264.94 

A 

6203.69 

A 

7219.41 

X 

8783.18 

82 

5281.03 

89 

6221.15 

96 

7238.25 

106 

8824.75 


5297.14 1 

A 

6238.64 

A 

7257. 11 

X 

8866.43 

% 

53 i 3-28 ; 

A 

6256.15 

A 

7275-99 

X 

8908.2 

A 

5329-44 

% 

6273.69 

As 

7294.91 

X 

8950.07 

% 

5345.63 

% 

6291.25 

A 

73 I 3-84 

107 

8QQ2.04 

X 

5361.84 

% 

6308.84 

% 

7332.8 

X 

9034.11 

% 

5378.08 

' A 

6326.45 

A 

7351-79 

X 

9076.28 

A 

5394-34 

A 

6344.08 

A 

7370-79 

X 

9H8.54 

83 

5410.62 

90 

6361.74 

97 

7389-83 

108 

9160.91 

A 

5426.93 

% 

6379.42 

A 

7408.89 

X 

9203.37 

/i 

5443-26 

% 

6397 .I 3 

A 

7427.97 

X 

9245-93 

% 

5459-62 

% 

64I4.86 

% 

7447.08 

X 

9288.58 

% 

5476 -ox 

A 

6432.62 

A 

7466.21 

109 

9331-34 

% 

5492.41 

% 

6450.4 

% 

7485-37 

X 

9374-19 

% 

5508.84 

A 

6468.21 

X 

7504-55 

X 

9417.14 

A 

5525-3 

A 

6486.04 

A 

7523-75 

X 

9460.19 


































AREAS OF CIRCLES. 


235 


Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

iio 

9503-34 

120 

11309.76 

I.30 

13273.26 

140 

15393-84 

X 

9546.59 

X 

11356.93 

X 

13324-36 

X 

15448.87 

'A 

9589-93 

X 

11404.2 

X' 

13375-56 

X 

15 503-99 

X 

9633-37 

X 

11451-57 

X 

13426.85 

X 

15559-22 

III 

9676.91 

121 

11499.04 

131 

13478.25 

141 

15614-54 

X 

9 720.55 

X 

II 546.61 

X 

13529-74 

X 

15 669.96 

X 

9 764-29 

X 

11594.27 

X 

13581-33 

X 

15725-48 

X 

9808.12 

X 

II 642.03 

X 

13633.02 

X 

15781.09 

112 

9852.06 

122 

II 689.89 

132 

13684.81 

142 

15836.81 

X 

9 896.09 

X 

II737-85 

X 

13 73669 

X 

15 802.62 

X 

9940.22 

X 

II 785.91 

X 

13 788.68 

X 

15948.53 

X 

9984.45 

X 

11 834 06 

X 

13840.76 

X 

16004.54 


10028.77 

123 

II 882.32 

133 

13892.94 

143 

16060.64 

x 

10073.2 

X 

11930.67 

X 

13945.22 

X 

16 116.85 

A 

10117.72 

X 

II 979.12 

X 

13997.6 

X 

16173.15 

x 

10162.34 

X 

12027.66 

X 

14050.07 

X 

16229.55 

114 

10207.06 

124 

12076.31 

134 

14 102.64 

144 

16 286.05 

x 

10251.88 

X 

12 125.05 

X 

14155-31 

X 

16342.65 

x 

10296.79 

X 

12 173.9 

X 

14208.08 

X 

1639935 

x 

10341.8 

X 

12 222.84 

X 

14253.09 

X 

16456.14 

ns 

10386.91 

125 

12 271.87 

135 

14313-91 

145 

16513.03 

x 

10432.12 

X 

12 321.01 

X 

14366.98 

X 

16570.02 

A 

10 477-43 

X 

12370.25 

X 

14420.14 

X 

16627. II 

X 

10522.84 

X 

12419.58 

X 

14473-4 

X 

16684.3 

116 

10568.34 

126 

12 469.OI 

136 

14526.76 

146 

16741.59 

X 

10613.94 

X 

12518.54 

X 

14580.21 

X 

16 798-97 

X 

10659.65 

X 

12 568.17 

X 

14633-77 

X 

16856.45 

X 

10705.44 

X 

12 618.09 

X 

14687.42 

X 

16914.03 

117 

!o 751-34 

127 

12 667.72 

137 

14 741.17 

147 

16971.71 

X 

10 797-34 

X 

12 717.64 

X 

14795.02 

X 

17029.48 

X 

10843.43 

X 

12 767.66 

X 

14848.97 

X 

17087.36 

X 

10889.62 

X 

12 817.78 

X 

14903.01 

X 

17145-33 

118 

10935.91 

128 

12 867.99 

138 

14957.16 

148 

17203.4 

X 

10982.3 

X 

12918.31 

X 

15011.4 

X 

17261.57 

X 

11028.78 

X 

12968.72 

X 

15065.74 

X 

17319.84 

X 

11075.37 

X 

13019.23 

X 

15 120.18 

X 

17378.2 

119 

11122.05 

129 

13069.84 

139 

15174-71 

149 

17436.67 

X 

11168.83 

X 

13120.55 

X 

15229.35 

X 

17495-23 

X 

11 215.71 

X 

I3I7I-35 

X 

15 284.08 

X 

17553-89 

X 

11 262.69 

X 

13 222.26 

X 

15338-91 

150 

17671.5 


To Compute -Area, of a Circle greater than any in. Table. 

Rule. —Divide dimension by two, three, four, etc., if practicable to do so, 
until it is reduced to a diameter to be found in table. 

Take tabular area for this diameter, multiply it by square of divisor, and 
product will give area required. 

Example.— What is area for a diameter of 1050? 

1050-^7 = 150; tab. area, 150 = 17 671.5, which X 7 2 = 865903.5, area. 

To Compute Area of a Circle in Feet and. Inches, etc., 
"by- preceding Table. 

Rule. —Reduce dimension to inches or eighths, as the case may be, and 
take area in that term from table for that number. 

































236 


AREAS OF CIRCLES, 


Divide this number by 64 (square of 8) if it is in eighths, and quotient will 
give area in inches, and divide again by 144 (square of 12) if it is in inches, 
and quotient will give area in feet. 

Example.—W liat is area of 1 foot 6.375 ins.? 

1 foot 6.375 ins. = 18.375 ins. =147 eighths. Area of 147 = 16071.71, which = 64 
= 265.18125 ins.; and by 144 = 1.84 125 feet. 

To Compute Area of a Circle Composed of an Integer 

and. a Fraction. 

Rule. —Double, treble, or quadruple dimension given, until fraction is in¬ 
creased to a whole number, or to one of those in the table, as etc., 

provided it is practicable to do so. 

Take area for this diameter; and if it is double of that for which area is 
required, take one fourth of it; if treble, take one sixteenth of it, etc. 

Example. — Required area for a circle of 2.1875 ins. 

2.1875 X 2 = 4.375, area for which = 15.0331, which 4-4 = 3.758 ins. 


When Diameter is composed of Integers and Fractions contained in Table. 

Rule. —Point off a decimal to a diameter from table, and add twice as 
many figures or ciphers to the right of the area as there are figures cut off" 
from the diameter. 

Example i, —What is area of 9675 feet diameter? 

Area of 96.75 = 7351-79; hence, area = 73 517 900 feet. 

2.—What is area of 24375 feet diameter? 

Area of 2.4375 = 4-6664; hence, area = 466 640 000 feet. 

To Ascertain Area of a Circle as 300, 3000, etc., not 
contained in Talale. 

Rule. Take area of 3 or 30, and add twice the excess of ciphers to the 
result. 

Example.— What is area of a circle 3000 feet in diameter? 

Area of 30 = 706.86, hence area of 3000 = 7 068 600 feet. 


is lo 


To Compute Area of a Circle Toy Logarithms. 

Rule.— To twice log. of diameter add r.895091 (log. of .7854), and sum 
1 ''" of area, for which take number. 


Example.— What is area of a circle 1200 feet in diameter? 

Log. 1200 X 2 + 1.895091 = 6.158362 + 1.895091 = 6.053453 and number for 
which = 1130976 feet. 


Areas of Birmingham Wire Grans?. 


Diam. 

A rea. 

Diam. 

Area. 

No. 

Sq. Inch. 

No. 

Sq.Inch. 

I 

.070 686 

IO 

.014 103 

2 

• 0 6 3347 

II 

.011 qoq 

3 

.052 685 

12 

.009 331 

4 

.044 488 

13 

.007 088 

5 

.038 013 

14 

.005411 

6 

•032 365 

15 

.004071 

7 

.025 447 

l6 

.003318 

8 

.021 382 

17 

.002 642 

9 

.017 203 

18 

.001 886 


Diam. 

Area. 

Diam. 

Area. 

No. 

Sq. Inch. 

No. 

Sq. Inch. 

19 

.OOI385 

28 

.OOO 154 

20 

.OOO 962 

29 

.OOO 133 

21 

-OOO 804 

30 

.OOO 11 5 

22 

.OO0616 

31 

.OOOO78 

23 

^.000 491 

32 

.OOO 064 

24 

.OOO 38 

33 

.000 05 

25 

.OOO314 

34 

.OOOO38 

26 

•OOO 254 

35 

-OOO 02 

27 

•OOO 201 1 

36 

.OOOOI3 















CIRCUMFERENCES OF CIRCLES. 


237 


Circumferences of Circles, from ^ to 150. 


Diam, 

ClRCUM. 

Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

1 

64 

.O49O9 

3 

9.4248 

8 

25.1328 

15 

47-124 

34 

i / 

.098 18 

/16 

9.6211 

X 

25-5255 

X 

47.5167 


98175 

X 

25.9182 

X 

47.9094 

34 

•19635 

3/ 

/lfi 

10.014 

% 

26.3109 

X 

48.3021 

X 

•39 2 7 

3^ 

10.2102 

X 

26.7036 

X 

48.6948 

3 / 


/16 

10.406 

A 

27.0963 

X 

49.0875 

Ag 

•5 6 9 

% 

10.6029 

A 

27.489 

X 

49.4802 

x 

•7854 

Xg 

IO -799 

A 

27.8817 

X 

49.8729 

6/ 

.081 7s; 


10.9956 

9 

28.2744 

16 

50.2656 

/\§ 

•y uA /o 


II.I9I 

A 

28.6671 

X 

50.6583 

A 

1.17S 1 

% 

11.3883 

A 

29.0598 

X 

51.051 

X 

1-374 45 

% 

11.584 

A 

29-4525 

X 

51-4437 

1/ 


X 

11.781 

X 

29.8452 

X 

51.8364 

A 

I.57 0 O 

13 / 

/ 1 G 

11.977 

X 

30.2379 

X 

52.2291 

% 

1.76715 

X 

12.1737 

X 

30.6306 

X 

52.6218 

X 

I.Q67 K 

15 / 

/ 1 G 

12.369 

X 

3 1 0233 

X 

53-0145 

7 J j 

4 

12.5664 

10 

31.416 

17 

53-4072 

Xl6 

2.159 85 

X 

12.762 

X 

31.8087 

X 

53-7999 

X 

2.3562 

X 

12.9591 

X 

32.2014 

X 

54.1926 

13 / 

3 / 

/lG 

13-155 

X 

32-5941 

X 

54-5853 

/Vo 

2-552 55 

A 

I3-35I8 

X 

32.9868 

X 

54-978 

% 

2.748 9 

X 

13-547 . 

X 

33-3795 

X 

55-3707 

15 / 

2.945 25 

% 

13-7445 

X 

33.7722 

X 

55-7634 

Ag 

X 

13 94 

X 

34.1649 

X 

56.1561 

I 

3.1416 

A 

14.1372 | 

II 

34-5576 

18 

56.5488 

/VG 

3-337 9 

Xg 

14333 

X 

349503 

X 

56.9415 

% 

3-534 3 

% 

14.5299 

X 

35-343 

X 

57-3342 

3 / 

/lG 

3-73o6 

% 

14-725 

X 

35-7357 

X 

57.7269 

% 

3-927 

% 

14.9226 

X 

36.1284 

X 

58.1196 

6/ 

/\G 

4-1233 

13 / 

/Vo 

15.119 

X 

36.5211 

X 

58.5123 

Vs 

4-3I9 7 

% 

I5-3I53 

X 

36.9138 

X 

58.905 

Xe 

4.516 

. 15 / 

Ag 

i5-5ii 

X 

37-3065 

X 

59-2977 

A 

4.7124 

5 

15.708 

12 

37.6992 

19 

59.6904 

9 / 

A 16 

4.908 7 

% 

16.1007 

X 

38.0919 

X 

60.0831 

A 

5-105 1 

A 

16.4934 

X 

38.4846 

X 

60.4758 

% 

5-301 4 

% 

l6.886l 

X 

38-8773 

X 

60.8685 

X 

5-497 8 

A 

17.2788 

X 

39-27 

X 

61.2612 

% 

5.694 1 

% 

17.6715 

X 

39.6627 

X 

61.6539 

% 

5-8905 

% 

18.0642 

X 

40-0554 

X 

62.0466 

% 

6.086 8 

A 

18.4569 

X 

40.4481 

X 

62.4393 

2 

6.283 2 

6 

18.8496 

13 

40.8408 

20 

62.832 

X 

6-479 5 

As 

19.2423 

X 

41-2335 I 

X 

63.2247 

A 

6.6759 

X 

I9-635 

X 

41.6262 

X 

63.6174 

X 

6.872 2 

% 

20.0277 

X 

42.0189 

X 

64.0101 

A 

7.068 6 

A 

20.4204 

X 

42.4x16 

X 

64.4028 

6/ 

/ 1 G 

7.2649 

A 

20.8131 

X 

42.8043 

X 

64-7955 

% 

7-46 i 3 

' X 

21.2058 

X 

43-197 

X 

65.1882 

7 '/ 

/16 

7.6576 

A 

21.5985 

X 

43-5897 

X 

65.5809 

X 

7-854 

7 

21.9912 

14 

43.9824 

21 

65.9736 

97 

/16 

8.050 3 

A 

22.3839 

X 

44-3751 

/i 

66.3663 

% 

8.246 7 

A 

22.7766 

X 

44.7678 

X 

66.759 

% 

8-443 

X 

23.1693 

X 

45.1605 

X 

67.1517 

X 

8.6394 

X 

23.562 

X 

45-5532 

A 

67.5444 

13 / 

Ag 

8.835 7 

X 

23-9547 

X 

45-9459 

A 

67.9371 

% 

9.032 1 

X 

24-3474 

X 

46.3386 

X 

68.3298 

% 

9.2284 

X 

24.7401 

X 

46.7313 

X 

68.7225 




































238 


CIRCUMFERENCES OF CIRCLES. 


Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

•Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

22 

69.1152 

29 

91.1064 

36 

113.098 

43 

135.089 

X 

69.5079 

X 

9 I - 499 I 

X 

113-49 

A 

135-481 

X 

69.9006 

X 

91.8918 

X 

113.883 

A 

135-874 

X 

70-2933 

X 

92.2845 

X* 

114.276 

% 

136.267 

X 

70.686 

X 

92.6772 

X' 

114.668 

A 

136.66 

% 

71.0787 

X 

93.0699 

X 

115.061 

X 

137.052 

X 

71.4714 

X 

93.4626 

X 

115-454 

X 

137-445 

X 

71.8641 

X 

93-8553 

X 

115.846 

X 

137-838 

23 

72.2568 

30 

94.248 

37 

116.239 

44 

138.23 

X 

72.6495 

X 

94.6407 

■ X 

116.632 

X 

138.623 

X 

73.0422 

X 

95-0334 

A 

117.025 

A 

139.016 

X 

73-4349 

X 

95.4261 

.X 

117.417 

X 

139.408 

X 

73.8276 

X 

95.8188 

A 

117.81 


139.801 

X 

74.2203 

X 

96.2115 

X 

118.203 

X 

140.194 

% 

74-613 

X 

96.6042 

X 

118.595 

X 

140.587 

X 

75-0057 

X 

96.9969 

X 

118.988 

X 

140.979 

24 

75-3984 

31 

97.3896 

38 

119.381 

45 

141-372 

X 

75 - 79 11 

X 

97.7823 

X 

119-773 

3 ^ 

141.765 

A 

76.1838 

X 

98.175 

A 

120.166 

3^ 

142.157 

X 

76-5765 

X 

98.5677 

X 

120.559 

X 

142.55 

X 

76.9692 

X 

98.9604 

A 

120.952 

A 

142.943 

X 

77.3619 

X 

99-3531 

X 

121.344 

X 

143-335 

X 

77-7546 

X 

99-7458 

X 

121.737 

X 

143.728 

X 

78-1473 

X 

100.1385 

X 

122.13 

X 

144.121 

25 

78.54 

32 

100.5312 

39 

122.522 

46 

144-514 

X 

78.9327 

X 

100.9239 

X’ 

122.915 

X 

144.906 

X 

79-3254 

X 

101.3166 

A 

123.308 

X 

145.299 

X 

79.7181 

X 

XOI.7093 

X 

123.7 

X 

145.692 

X 

80.1108 

X 

102.102 

X 

124.093 

X 

146.084 

X 

80.5035 

X 

102.4947 

X 

124.486 

X 

146.477 

X 

80.8962 

X 

102.8874 

X 

124.879 

X 

146.87 

X 

81.2889 

X 

103.2801 

X 

125.271 

X 

147.262 

26 

81.6816 

33 

103.673 

40 

125.664 

47 

147-655 

X 

82.0743 

X 

104.065 

X 

126.057 

X 

147.048 

X 

82.467 

X 

104.458 

A 

126.449 

X 

148.441 

X 

82.8597 

X 

104.851 

X 

126.842 

X 

148.833 

X 

83.2524 

X 

105.244 

X 

127.235 

X 

149.226 

X 

83.6451 

X 

105.636 

X 

127.627 

X 

I 4 Q. 6 IQ 

X 

84.0378 

X 

106.029 

X 

128.02 

X 

150.011 

X 

84-4305 

X 

106.422 

X 

128.413 

X 

150.404 

27 

84.8232 

34 

106.814 

41 

128.806 

48 

150.797 

X 

85.2159 

X 

107.207 

X 

129.198 

X 

151.189 

X 

85.6086 

A 

107.6 

A 

129-591 

X 

151.582 

X 

86.0013 

X 

107.992 

X 

129.984 

X 

151-975 

X 

86.394 

X 

108.385 

A 

130.376 

X 

152.368 

X 

86.7867 

X 

108.778 

X 

130.769 

X 

152.76 

X 

87.1794 

X 

109.171 

X 

131.162 

X 

153-153 

X 

87.5721 

X 

109.563 

X 

131-554 

X 

153-546 

28 

87.9648 

35 

109.956 

42 

131-947 

49 

153-938 

% 

88.3575 

A 

110.349 


132.34 

Vs 

154-331 

A 

88.7502 

A 

110.741 

X 

132.733 

% 

154.724 

X 

89.1429 

X 

111.134 

X" 

133-125 

X 

155-116 

X 

89-5356 

X 

111-527 

X 

133-518 

X 

155-509 

X 

89.9283 

X 

111.919 

X 

133-911 

X 

155.902 

X 

90.321 

X 

112.312 

X 

134-303 

X 

156.295 

X 

90.7137 

X 

I12.705 

X 

1 34.696 

X 

156.687 




























CIRCUMFERENCES OF CIRCLES. 


239 


Diam. 

ClRCUM. | 

Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

50 

157.08 

57 

179.071 

64 

201.062 

71 

223.054 

% 

157-473 

% 

179.464 

X 

201.455 

X 

223.446 

X 

157-865 

X 

I 79-857 

X 

201.848 


223.839 

X 

158.258 

X 

180.249 

X 

202.24 

X 

224.232 

A 

158.651 

A 

180.642 

X 

202.633 

X 

224.624 

% 

159-043 

% 

181.035 

X 

203.026 

X 

225.017 

X 

I 59-436 

% 

181.427 

X 

203.419 

X 

225.41 

% 

159.829 

A 

181.82 

X 

203.811 

X 

225.802 

51 

160.222 

58 

182.213 

65 

204.204 

72 

226.195 

A 

160.614 

A 

182.605 

X 

204.597 

X 

226.588 

X 

161.007 

X 

182.998 

X 

204.989 

X 

226.981 

% 

161.4 

% 

183-391 

X 

205.382 

X 

227-373 

A 

161.792 

A 

183.784 

X 

205.775 

X 

227.766 

% 

162.185 

% 

184.176 

X 

206.167 

X 

228.159 

% 

162.578 

% 

184.569 

X 

206.56 

X 

228.551 

A 

162.97 

A 

184.962 

X 

206.953 

X 

228.944 

52 

1 63-363 

59 

185.354 

66 

207.346 

73 

229.337 

X 

163.756 

X 

185.747 

X 

207.738 

X 

229.729 

% 

164.149 

X 

186.14 

X 

208.131 

X 

230.122 

% 

164.541 

% 

186.532 

X 

208.524 

X 

230-515 

A 

164.934 

A 

186.925 

X 

208.916 

X 

230.908 

% 

165.327 

% 

187.318 

X 

209.309 

X 

231-3 

% 

165.719 

% 

187.711 

X 

209.702 

X 

231.693 

% 

166.112 

A 

188.103 

X 

210.094 

X 

232.086 

53 

166.505 

60 

188.496 

67 

210.487 

74 

232.478 

A 

166.897 i 

A 

188.889 

X 

210.88 

X 

232.871 

X 

167.29 

X 

189.281 

X 

211.273 

X 

233.264 

% 

167.683 

Vs 

189.674 

X 

211.665 

X 

233-656 

A 

168.076 

A 

190.067 

X 

212.058 

X 

234.049 

% 

168.468 

A 

190.459 

X 

212.451 

X 

234.442 

% 

l68.86l 

% 

190.852 

X 

212.843 

X 

234-835 

% 

169.254 

A 

I 9 I - 2 45 

X 

213.236 

X 

235.227 

54 

169.646 

6l 

191.638 

68 

213.629 

75 

235.62 

As 

170.039 

A 

192.03 

X 

214.021 

X 

236.013 

X 

170.432 

X 

192.423 

X 

214.414 

X 

236.405 

% 

170.824 

S A 

192.816 

X 

214.807 

X 

236.798 

A 

I7I.2I7 

A 

193.208 

X 

215.2 

X 

237.191 

% 

171.61 

As 

193.601 

X 

215.592 

X 

237-583 

% 

172.003 

X 

193-994 

X 

215.985 

X 

237.976 

A 

I 72-395 

A 

194.386 

X 

216.378 

X 

238.369 

55 

172.788 

62 

194.779 

69 

216.77 

76 

238.762 

X 

173.181 

% 

I 95 -I 72 


217.163 

X 

239- 1 54 

X 

173-573 

X 

I 95-565 

/i 

217-556 

X 

239-547 

% 

173.966 

% 

195-957 

X 

217.948 

X 

239-94 

A 

174-359 

A 

196.35 

% 

218.341 

X 

240.332 

% 

I 74 - 75 I 

% 

196.743 

X 

218.734 

X 

240.725 

% 

I 75 -I 44 

% 

I 97 -I 35 

X 

219.127 

X 

241. xi8 

% 

175-537 

X 

197.528 

X 

219-519 

X 

241.51 

56 

175-93 

63 

197.921 

70 

219.912 

77 

241.903 

Vs 

176.322 

X 

198-313 

X 

220.305 

X 

242.296 

X 

176.715 

X 

198.706 

X 

220.697 

X 

242.689 

% 

177.108 

X 

199.099 

X 

221.09 

X 

243.081 

' A 

177-5 

X 

199.492 

X 

221.483 


243-474 

% 

177.893 

X 

199.884 

X 

221.875 

X 

243.867 

% 

178.286 

X 

200.277 

X 

222.268 

X 

244.259 

Vs 

178.678 

A 

200.67 

X 

222.661 

X 

244.652 





























240 


CIRCUMFERENCES OF CIRCLES. 


Diam. 

ClBCUM. 

| Diam. 

ClBCUM. 

I Diam. 

I ClBCUM. 

Diam. 

ClBCUM. 

78 

245-045 

85 

267.036 

92 

289.027 

99 

311.018 

X 

245-437 

X 

267.429 

X 

289.42 

X 

3*1.411 

% 

245-83 

X 

267.821 

X 

289.813 

% 

311.804 

% 

246.223 

% 

268.214 


290.205 

% 

312.196 

X 

246.616 

X 

268.607 

X 

290.598 

y 

312.589 

% 

247.008 

X 

268.999 

X 

290.991 

% 

312.982 

% 

247401 

% 

269.392 

M 

291.383 

% 

3 * 3-375 

X 

247.794 

X 

269.785 

% 

291.776 

X 

3*3-767 

79 

248.186 

86 

270.178 

93 

292.169 

100 

3*4.16 

% 

248.579 


270.57 

K 

292.562 

/i 

3 * 4-945 

X 

248.Q72 

X 

270.963 

X 

292.954 

y 

3 * 5 - 73 * 

% 

249.364 

% 

27 *-356 

# 

293-347 

% 

316.516 

X 

249-757 

X 

271.748 

K 

293-74 

IOI 

317.302 

% 

250.15 

% 

272.I4I 

K 

294.132 

% 

318.087 

% 

250.543 

% 

272.534 

M 

294-525 


318.872 

X 

250.935 

X 

272.926 

% 

294.918 

% 

319.658 

80 

251.328 

87 

273 - 3 I 9 

94 

295 - 3 1 

102 

320.443 

% 

251-721 

% 

273.712 

X 

295-703 

X 

321.229 

X 

252.113 

X 

274.105 

X 

296.096 


322.014 

% 

252.506 

% 

274.497 

% 

296.488 

% 

322.799 

X 

252.899 

X 

274.89 

X 

296.881 

103 

323-585 

% 

253.291 

% 

275.283 

% 

297.274 

/i 

324-37 

% 

253.684 

% 

275-675 

% 

297.667 


325 -I 5 6 

% 

254.077 

% 

276.068 

% 

298.059 

H 

325 - 94 I 

Si 

254-47 

88 

276.461 

95 

298.452 

IO4 

326.726 

% 

254.862 

X 

276.853 

X 

298.845 


327.512 

/i 

255-255 

X 

277.246 

X 

299.237 


328.297 

% 

255.648 

% 

277.629 

% 

299.63 

X 

329-083 

jf 

256.04 

X 

278.032 

X 

300.023 

105 

329.868 

% 

25 6 -433 

% 

278.424 

% 

300.415 

y 

■330.653 

% 

256.826 

% 

278.817 

% 

300.808 

X 

33*-439 

x 

257.218 

% 

279.21 

X 

301.201 

H 

332.224 

82 

257.611 

89 

279.602 

96 

3 OI -594 

106 

333 -oi 

H 

258.004 


279-995 

X 

301.986 

% 

333-795 

X 

258.397 

X 

280.388 

X 

302.379 


334-58 

% 

258.789 

% 

280.78 

% 

302.772 

M 

335-366 

jf 

259.182 

X 

281.173 

X 

303.164 

IO7 

336.151 

% 

259-575 

% 

281.566 

X 

303-557 

/i 

336.937 

% 

259.967 

% 

281.959 

K 

303-95 

X 

337-722 

X 

260.36 

% 

282.351 

X 

304-342 

% 

338.507 

83 

260.753 

90 

282.744 

97 

304.735 

108 

339-293 

y 

261.145 

% 

283-137 

X 

305.128 

% 

340.078 

y 

261.538 

X, 

283.529 

X 

305-521 

X 

340.864 

% 

26l.93I 

% 

283.922 

% 

305 - 9*3 

% 

341.649 

K 

262.324 

X 

284.315 

X 

306.306 

109 

342-434 

%. 

262.716 

% 

284.707 

% 

306.699 

* 

343-22 

X 

263.109 

% 

285.1 

% 

307-09* 


344.005 

X 

263.502 

X 

285.493 

X 

307.484 

9* 

344 - 79 * 

84 

263.894 

91 

285.886 

98 

307.877 

no 

345.576 


264.287 


286.278 

X 

308.27 

X 

346.361 

X 

264.68 

X 

286.671 

X 

308.662 


347-*47 

% 

265.072 

% 

287.064 

% 

309-055 

% 

347-932 

s4 

265.465 

/4 

287.456 

X 

309.448 

III 

348.718 

% 

265.858 

% 

287.849 

% 

309.84 

X 

349-503 

% 

266.251 

% 

288.242 

% 

3*0.233 

X 

350.288 

x 

266.643 

% 

288.634 

X 

310.626 

% 

351-074 





































CIRCUMFERENCES OF CIRCLES. 


241 


Diam, 

Circum. 

| Diam. 

Circum. 

Diam. 

Circum. 

Diam. 

Circum. 

112 

35 I -859 

121 

380.134 

130 

408.408 

L 39 

436.682 

A 

352.645 

A 

380.919 

X 

40 Q. 1 Q 2 

3^ 

437-467 

A 

353-43 

A 

381.704 

A 

409.979 

A 

438-253 

A 

354-215 

% 

382.49 

A 

410.763 

A 

439 037 

113 

355 -ooi 

122 

383-275 

131 

411-55 

140 

439.824 

A 

355-786 

H 

384.061 

X 

412.334 

X 

440.608 

A 

356.572 

A 

384.846 

X 

4 I 3 - 12 

A 

441-395 

A 

357-357 

A 

385 - 63 I 

% 

4^905 

% 

442.179 

114 

358-142 

123 

386.417 

152 

414.691 

141 

442.966 

A 

358.928 

A 

387.202 

X 

415.476 

A 

443-75 

A 

359 - 7 L 3 

A 

387.988 

X 

416.262 

A 

444-536 

A 

360.499 

% 

388.773 

A 

4x7.046 

A 

445-321 

115 

361.284 

124 

389-558 

133 

4 I 7-833 

142 

446.107 

A 

362.069 

' A 

390-344 


418.617 

X 

446.891 

A 

362.855 

A 

391.129 

A 

419.404 

A 

447.678 

A 

363-64 

% 

391-9*5 

A 

420.188 

A 

448.462 

116 

364.426 

12 5 

392-7 

134 

420.974 

143 

449.249 

A 

365.211 

A 

393-484 

A 

421.759 

3€ 

450.033 

A 

365-996 

A 

394.271 

A 

422.545 

3¥ 

450.82 

% 

366.782 

A 

395-055 

A 

423-33 

% 

451.604 

n 7 

367-567 

126 

395-842 

135 

424.116 

144 

452.39 

3^ 

368.353 

A 

396.626 


424.9 

X 

453 i 75 

A 

369.138 

A. 

397.412 

A 

425.687 

A 

453-96i 

A 

369.923 

%: 

398.197 

A 

426.471 

A 

454-745 

n8 

370.709 

127 

398.983 

136 

427.258 

145 

455-532 

3^ 

371-494 


399.768 

A 

428.042 

3€ 

456.316 

A 

372.28 

A 

400.554 

A 

428.828 


457-103 

% 

373 -o 65 

A 

401.338 

A 

429.613 

146 

458.674 

119 

373-85 

128 

402.125 

137 

430.399 

3^ 

460.244 

A 

374-636 

A 

402.909 

3^ 

431-183 

147 

461.815 

A 

375-421 

A 

403.696 

A 

431-97 

A 

463-386 

% 

376.207 

• A 

404.48 

A 

432.754 

148 

464-957 

120 

376.992 

129 

405.266 

X38 

433-541 

A 

466.528 

A 

377-777 


406.051 

A 

434-325 

149 

468.098 

A 

378-563 

X 

406.837 

A 

435 -112 

A 

469.669 

A 

37^-348 

■% 

407.622 

A 

435-896 

150 

471.24 


To Compute Circumference of* a Diameter greater tlraxi 
any in preceding Table. 

Rule.— Divide dimension by two, three, four, etc., if practicable to do so, 
until it is reduced to a diameter in table. 

Take tabular circumference for this dimension, multiply it by divisor, 
according as it was divided, and product will give circumference required. 

Example.— What is circumference for a diameter of 1050? 

I05 o -7- 7 — 1 5 o; tab. circum., 150 = 471.24, which X 7 — 329 s - 68 , circumference. 

To Compute Circumference of a Diameter in Feet and 
Inches, etc., by preceding Table. 

Rule —Reduce dimension to inches or eighths, as the case may be, and 
take circumference in that term from table for that number. 

Divide this number by 8 if it is in eighths, and by 12 if in inches, and 
nuoti'ent will give circumference in feet. 

X 

























242 


CIRCUMFERENCES OF CIRCLES. 


Example.— Required circumference of a circle of x foot 6.375 ins. 

1 foot 6.375 ins. = 18.375 ins. = 147 eighths. Circum. of 147 = 461.815. which -f- 8 
= 57.727 ins.; and by 12 = 4.810 6 feet. 

To Compirte Circumference for a f).iameter composed of 
air Integer and. a Fraction. 

Rule. —Double, treble, or quadruple dimension given, until fraction is in¬ 
creased to a whole number or to one of those in the table, as 3^, etc., pro¬ 
vided it is practicable to do so. 

Take circumference for this diameter ; and if it is double of that for which 
circumference is required, take one half of it; if treble, take one third of it ; 
and if quadruple, one fourth of it. 

Example. —Required circumference of 2.21875 ins. 

2.21875 X 2 = 4.4375, which X 2 = 8.875; circum. for which = 27.8817, which-f-4 
= 6.9704 ins. 

When Diameter consists of Integers and Fractions contained in Table. 

Rule. —Point a decimal to a diameter in table, take circumference from 
table, and add as many figures to the right as there are figures cut off. 

Example. —What is circumference of a circle 9675 feet in diameter? 
Circumference of 96.75 = 303.95; hence, circumference of 9675 = 30395/^. 

To Ascertain Circumference for a Diameter, as 500, 
5000, etc., not contained in Table. 

Rule.—Take circumference of 5 or 50 from table, and add the excess of 
ciphers to the result. 

Example.—W hat is circumference of a circle 8000 feet in diameter? 

Circumference of 80 = 251.38; hence, circumference of 8000 = 25 138 feet. 

To Compute Circnmference of* a Circle by Logarithms. 

Rule. —To log. of diameter add .49701 (log. of 3.1416), and sum is log. 
of circumference, from which take number. 

Example.—W hat is circumference of a circle 1200 feet in diameter? 

Log. 1200 = 3.07918 -}-.497 ox = 3.576 19, and number for which = 3769.91 feet. 


Circumferences of* Birmingham Wire Grange. 


Diam. 

Circum. 

Diam. 

Circum. 

Diam. 

Circum. 

Diam. 

Circum. 

No. 

Ins. 

No. 

Ins. 

No. 

Ins. 

No. 

Ins. 

I 

.942 48 

10 

.42097 

19 

•I 3 I 95 

28 

.04398 

2 

.892 21 

11 

.37699 

20 

.10995 

29 

.040 84 

3 

.81367 

12 

•342 43 

21 

•IOO53 

30 

•037 7 

4 

•747 7 

13 

.298 45 

22 

.087 96 

31 

.031 41 

5 

.691 15 

14 

.260 75 

23 

.078 54 

32 

.028 27 

6 

•637 74 

15 

.226 19 

24 

“.069 11 

33 

•025 13 

7 

•56s 49 

16 

.204 2 

25 

.062 83 

34 

.021 99 

8 

•51836 

17 

.182 21 

26 

•056 55 

35 

.OI57I 

9 

.46495 

18 

•15394 

27 

.050 26 

36 

.OI257 




















cr> t-'X) O' h n nt 10 t>oo O' h n n't iovo r^co O' h m n^t n'O t^»oo O' h w ro tj- iovq t>oo O' m cj to ^t- ^ 


AREAS AND CIRCUMFERENCES OF CIRCLES. 


Areas and. Circumferences, 

Diam. Area. Circum. II Diam. 


.007 854 
.031416 
.070 686 
.125664 

•19635 

.282 744 
.384846 
.502656 
.636174 
•7854 
•950 3 
1.131 

I -3273 

1 - 5394 
1.767 1 
2.0106 
2.2698 

2 - 544 7 

2.8353 

3- i4i6 

3-4636 
3-8013 
4.1548 
45239 
4.908 7 

5-3093 

5- 7256 

6 - 1575 
6.605 2 
7.0686 

7 - 547 7 
8.042,5 

8 - 553 
9.0792 
9.621 x 

10.1788 
10.7521 
11.341 2 

n -9459 

12.5664 

13.2026 

I 3-8545 

14.522 

15-2053 

I 5-9043 

16.619 1 

17-3495 
18.095 6 

18.8575 

I9-635 
20.4283 
21.237 2 
22.061 9 
22.9023 
23-7583 


.31416 
.628 32 
.942 48 
1.2566 
1.5708 

1.885 

2.199 1 

2 - 5 I 33 
2.8274 
3.1416 

3 - 455 8 
3.7699 
4.0841 

4 398 2 

4 - 7 I2 4 

5 0266 

5 - 340 7 
5-6549 

5- 969 

6.2832 

6 - 5974 
6.9115 

7 - 225 7 
7-5398 
7-854 
8.1682 
8.4823 

8.7965 

9.1106 
9 4248 

9-739 
10.053 1 
10.3673 
10.681 4 
10.995 6 
11.3098 
11.6239 
11.9381 
12.252 2 
12.5664 
12.8806 
I3-I94 7 

13-5089 

1 3 - 823 
14.1372 
14.4514 

1 4 - 765 5 
15.0797 

1 5 - 3938 
15.708 
16.022 2 
16.3363 
16.6505 
16.9646 
17.2788 


(Advancing by Tenths.) 


Area. 

Circum. 

24.63OI 

17-593 

2 5-5 i 76 

17.9071 

26.4209 

18.2213 

27-3398 

18.5354 

28.2744 

18.8496 

29.2247 

19.1638 

30.1908 

19.4779 

3 I - I 7 2 5 

19.7921 

32.17 

20.1062 

33-1831 

20.4204 

34.212 

20.7346 

35-2566 

21.0487 

36.3x69 

21.3629 

37-39 2 9 

21.677 

38.4846 

21.9912 

39-592 

22.3054 

40.7151 

22.6195 

41.854 

22-9337 

43.0085 

23.2478 

44.1787 

23.562 

45-3647 

23.8762 

46.5664 

24- i 9°3 

47-7837 

24-5045 

49.0168 

24.8186 

50.2656 

25.1328 

51-5301 

25-447 

52.8103 

25.7611 

54.1062 

26.0753 

55-4I78 

26.3894 

56.7451 

26.7036 

58.0882 

27.0178 

59.4469 

. 27.3319 

60.8214 

27.6461 

62.2115 

27.9602 

63.6174 

28.2744 

65-039 

28.5886 

66.4763 

28.9027 

67.9292 

29.2169 

69.3979 

29-53I 

70.8823 

29.8452 

72.3825 

30.1594 

73-8983 

30.4735 

75.4298 

30.7877 

76.9771 

31.1018 

78-54 

31.416 

80.1187 

31.7302 

81.713 

32.0443 

83-3231 

32-3585 

84.9489 

32.6726 

86.5903 

32.9868 

88.2475 

33-301 

89.9204 

33-6i5i 

91.6091 

33-9293 

93-3I34 

34-2434 















244 AREAS and circumferences of circles. 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 

ClRCUM. 

II 

95-0334 

34-5576 

•5 

213.8251 

5I.8364 

.1 

96.7691 

34.8718 

.6 

216.4248 

52 -T 505 

.2 

98.5206 

35 -I 859 

•7 

219.0402 

52.4647 

•3 

100.2877 

35 - 5 ooi 

.8 . 

221.6713 

52.7789 

•4 

102.0706 

35.8142 

•9 

224.3181 

53-093 

•5 

103.8691 

36.1284 

17 

226.9806 

53-4072 

.6 

105.6834 

36.4426 

.1 

229.6588 

53 - 72 I 4 

•7 

107-5134 

36-7567 

.2 

232-3527 

54-0355 

.8 

109.3591 

37.0709 

•3 

235.0624 

54-3497 

•9 

111.2205 

37-385 

•4 

237.7877 

54.6638 

12 

113.0976 

37.6992 

•5 

240.5287 

54-978 

.1 

114.9904 

38.0134 

- .6 

243 2855 

55.2922 

.2 

116.8989 

38-3275 

•7 

246.058 

55.6063 

•3 

118.8232 

38.6417 

.8 

248.8461 

55-9205 

•4 

120.7631 

38-9558 

•9 

25 I -65 

56.2346 

•5 

122.7187 

39-27 

18 

254.4696 

56.5488 

.6 

124.6901 

39-5842 

.1 

257-3049 

56.863 

•7 

126.6772 

39-8983 

.2 

260.1559 

57 -I 77 I 

.8 

128.6799 

40.2I25 

-3 

263.0226 

57 - 49 L 3 

•9 

130.6984 

40.5266 

•4 

265.905 

57-8054 

13 

132.7326 

40.8408 

•5 

268.8031 

58.1196 

.i 

134-7825 

4 I-I 55 

.6 

271.717 

58.4338 

.2 

136.8481 

41.4691 

•7 

274.6465 

58-7479 

•3 

138.9294 

4 I -7833 

.8 

277.5918 

59.0621 

•4 

141.0264 

42.0974 

•9 

280.5527 

59-3762 

•5 

I 43 -I 39 I 

42.4116 

!9 

283.5294 

59.6904 

.6 

145.2676 

42.7258 

.1 

286.5218 

60.0046 

•7 

147.4H7 

43-0399 

.2 

289.5299 

60.3187 

.8 

149.5716 

43-3541 

•3 

292.5536 

60.6329 

•9 

i 5 i- 747 i 

43.6682 

•4 

295 5931 

60.947 

14 

I 53-9384 

43.9824 

•5 

298.6483 

61.2612 

.i 

156-1454 

44.2966 

.6 

30 I- 7 I 93 

6 i -5754 

.2 

158.3681 

44 6107 

•7 

304.806 

61.8895 

•3 

160.6064 

44.9249 

.8 

307.9082 

62.2037 

•4 

162.8605 

45-239 

•9 

311.0263 

62.5178 

•5 

165.1303 

45-5532 

20 

3 i 4 -i 6 

62.832 

.6 

167.4159 

45.8674 

.1 

317.3094 

63.1462 

•7 

169.7171 

46.1815 

.2 

320 4746 

63.4603 

.8 

172 034 

46-4957 

•3 

323-6555 

63-7745 

•9 

174.3667 

46.8098 

•4 

326.8521 

64.0886 

15 

176.715 

47.124 

•5 

330.0643 

64.4028 

• i 

179.0791 

47 4382 

.6 

333-2923 

64 - 7 i 7 

.2 

181.4588 

47-7523 

•7 

336-536 

65.0311 

•3 

183.8543 

4S.0665 

.8 

339-7955 

65 3453 

•4 

186.2655 

48.3806 

•9 

343 0706 

65-6594 

•5 

188 6924 

48.6948 

21 

346.3614 

65-9736 

.6 

I 9 I - I 349 

49.009 

.1 

349 6679 

66.2878 

•7 

I 93 - 593 2 

49 - 323 I 

.2 

352.9902 

66.6019 

.8 

196.0673 

49-6373 

•3 

356 3281 

66.9161 

•9 

I 9 s -557 

49 - 95 I 4 

•4 

359.6818 

67.2302 

16 

201.0624 

50.2656 

•5 

363 0511 

67-5444 

•i 

203 5835 

50-5797 

.6 - 

366.4362 

67.8586 

.2 

206 1204 

50-8939 

•7 

369-837 

68.1727 

•3 

208.6729 

51.2081 

.8 

373-2535 

68.4869 

•4 

211.2412 

51.5222 

•9 

376.6857 

68.801 




















AREAS AND CIRCUMFERENCES OF CIRCLES. 


245 


Diam. 

Area. 

ClRCUM. 

22 

380.1336 

69.II52 

.1 

383-5972 

69.4294 

.2 

387.0765 

69-7435 

•3 

390.5716 

70.0577 

•4 

394.0823 

70.3718 

•5 

397.6087 

70.686 

.6 

401.I509 

71.0002 

•7 

404.7088 

7 I- 3 I 43 

.8 

408.2823 

71.6285 

•9 

411.8716 

71.9426 

23 

415.4766 

72.2568 

.1 

4 I 9-°973 

72.571 

.2 

422.7337 

72.8851 

•3 

426.3858 

73-1993 

•4 

430.0536 

73-5134 

•5 

433-7371 

73.8276 

.6 

437-4364 

74.1418 

•7 

44 i-i 5 i 3 

74-4559 

.8 

444.882 

74.7701 

•9 

448.6283 

75.0842 

24 

452.3904 

75-3984 

.1 

456.1682 

75 - 7 126 

.2 

459.9617 

76.0267 

•3 

463.7708 

76.3409 

•4 

467-5957 

76-655 

•5 

47 I -4363 

76.9692 

.6 

475.2927 

77.2834 

•7 

479.1647 

77-5975 

.8 

483.0524 

77 - 9 XI 7 

•9 

486.9559 

78.2258 

25 

490.875 

78.54 

.1 

494.8099 

78.8542 

.2 

498.7604 

78.1683 

•3 

502.7267 

79.4825 

•4 

506.7087 

79.7966 

•5 

510.7063 

80.1108 

.6 

514.7196 

80.425 

•7 

518.7488 

80.7391 

.8 

522.7937 

81.0533 

•9 

526.8542 

81.3674 

26 

530-9304 

81.6816 

.1 

535.0223 

81.9958 

.2 

539-13 

82.3099 

•3 

543-2533 

82.6241 

•4 

547-3924 

82.9382 

•5 

55 i- 547 i 

83.2524 

.6 

555 - 7 x 7 6 

83.5666 

•7 

559-9038 

83.8807 

.8 

564-1057 

84.1949 

•9 

568.3233 

84.509 

27 

572.5566 

84.8232 

.1 

576.8056 

85 -I 374 

.2 

581.0703 

85 - 45 I 5 

•3 

585.3508 

85-7657 

•4 

589.6469 

86.0798 


Diam. 

Area. 

ClRCUM. 

•5 

593-9587 

86.394 

.6 

598.2863 

86.7082 

•7 

602.6296 

87.0223 

.8 

606.9885 

87-3365 

•9 

611.3632 

87.6506 

28 

6 I 5-7536 

87.9648 

.1 

620.1597 

88.279 

.2 

624.5815 

88.593I 

•3 

629.019 

88.9073 

•4 

633-4722 

89.2214 

•5 

637.9411 

89-5356 

.6 

642.4258 

89.8498 

•7 

646.9261 

9 °. 1 639 

.8 

651.4422 

9O.4781 

•9 

655-9739 

90.7922 

29 

660.5214 

9I.IO64 

.1 

665.0846 

91.4206 

.2 

669.6635 

91-7347 

•3 

674.258 

92.O489 

•4 

678.8683 

92.363 

•5 

683-4943 

92.6772 

.6 

688.1361 

92.9914 

•7 

692.7935 

93-3055 

.8 

697.4666 

93.6197 

•9 

702.1555 

93-9338 

30 

706.86 

94.248 

.1 

711.5803 

94.5622 

.2 

716.3162 

94.8763 

•3 

721.0679 

95- X 905 

•4 

725-8353 

95-5046 

•5 

730.6183 

95 . 8 l 88 

.6 

735 - 4 I 7 I 

96 -I 33 

•7 

740.2316 

96.447I 

.8 

745.0619 

96.7613 

•9 

749.9078 

97-0754 

3 i 

754.7694 

97.3896 

.1 

759.6467 

97.7038 

.2 

764-5398 

98.0179 

•3 

769.4485 

98.3321 

•4 

774-373 

98.6462 

•5 

779 - 3 x 3 i 

98.9604 

.6 

784.269 

99.2746 

•7 

789.2406 

99.5887 

.8 

794.2279 

99.9029 

•9 

799.2309 

100.217 

32 

804.2496 

100.5312 

.1 

809.284 

100.8454 

.2 

8 i 4 - 334 i 

101.1595 

•3 

819.4 

ioi .4737 

•4 

824.4815 

101.7878 

•5 

829.5787 

102.102 

.6 

834.6917 

102.4162 

•7 

839.8204 

102.7303 

.8 

844 9647 

1 03-0445 

•9 

850.1248 

103.3586 


x* 















246 AREAS AND CIRCUMFERENCES OF CIRCLES. 


Diam. 

Area. 

ClRCUM. 

33 

855 - 3 °° 6 

IO3.6728 

.1 

860.4921 

IO3.987 

.2 

865.6993 

IO4.3011 

•3 

870.9222 

IO4.6153 

•4 

876.1608 

IO4.9294 

•5 

881.4151 

IO5.2436 

.6 

886.6852 

I 05-5578 

•7 

891.9709 

IO5.87I9 

.8 

897.2724 

106. l86l 

•9 

902.5895 

106.5002 

34 

907.9224 

106.8144 

.1 

9 I 3 - 2 7 I 

107.1286 

.2 

9 i8 -6353 

107.4427 

•3 

924.0152 

IO7.7569 

•4 

929.4109 

108.071 

•5 

934.8223 

I08.3852 

.6 

940.2495 

I08.6994 

•7 

945.6923 

IO9.OI35 

.8 

951.1508 

IO9.3277 

•9 

956.6251 

IO9.6418 

35 

962.115 

IO9.956 

.1 

967.6207 

I 10.2702 

.2 

973.142 

IIO.5843 

•3 

978.6791 

IIO.8985 

•4 

984.2319 

III.2I26 

•5 

989.8003 

III.5268 

.6 

995-3845 

III.84I 

•7 

1000.9844 

II 2 .I 55 I 

.8 

1006.6001 

II2.4693 

•9 

1012.2314 

II2.7834 

36 

1017.8784 

II3.O976 

.1 

I02 3 - 54 n 

II3.4II8 

.2 

1029.2196 

II 3-7259 

•3 

I ° 34 - 9 I 37 

I I4.O4OI 

•4 

1040.6236 

II 4-3542 

•5 

1046.3491 

II4.6684 

.6 

1052.0904 

II4.9826 

•7 

1057.8474 

II5.2967 

.8 

1063.6201 

II5.61O9 

•9 

1069.4085 

II 5-925 

37 

1075.2126 

Il6.2392 

.1 

1081.0324 

H6.5534 

.2 

1086.8679 

H6.8675 

•3 

1092.7192 

II7.1817 

•4 

1098.5861 

II7.4958 

•5 

1104.4687 

II 7 . 8 l 

.6 

mo 3671 

118.1242 

•7 

1116.2812 

H8.4383 

.8 

1122.2109 

118.7525 

•9 

1128.1564 

II9.0666 

38 

1134.1176 

II9.3808 

.1 

1140.0945 

II9.695 

.2 

1146.0871 

120.OO9I 

•3 

1152.0954 

I2O.3233 

•4 

1158.1194 

I2O.6374 


Diam. 

Area. 

ClRCUM. 

•5 

1164.1591 

120.9516 

.6 

II7O.2146 

121.2658 

•7 

1176.2857 

121.5799 

.8 ‘ 

1182.3726 

I2I.894I 

•9 

1188.4751 

122.2082 

39 

H 94-5934 

122.5224 

.1 

1200.7274 

I22.8366 

.2 

1206.8771 

I 23 -I 507 

•3 

1213.0424 

I23.4649 

•4 

1219.2235 

I23.779 

•5 

1225.4203 

124.0932 

'.6 

1231.6329 

I24.4O74 

•7 

1237.8611 

124.7215 

.8 

1244.105 

125-0357 

•9 

1250.3647 

I 25-3498 

40 

1256.64 

I25.664 

.1 

1262.9311 

I25.9782 

.2 

1269.2378 

126.2923 

•3 

1275.5603 

I26.6065 

•4 

1281.8985 

126.9206 

•5 

1288.2523 

I27.2348 

.6 

1294.6219 

I27.549 

•7 

1301.0072 

I27.863I 

.8 

1307.4083 

128.1773 

•9 

I 3 I 3 825 

I28.49I4 

4 i 

1320.2574 

I28.8056 

.1 

1326.7055 

I29.II98 

.2 

1333-1694 

129.4339 

•3 

1339-6489 

I29.748I 

•4 

1346.1442 

130.0622 

•5 

1352.6551 

130.3764 

.6 

1359.1818 

I3O.6906 

•7 

1365-7242 

I3I.OO47 

.8 

1372.2823 

I 3 I - 3 I 89 

•9 

1378.8561 

I 3 I - 6 33 

42 

I 385-4456 

131.9472 

.1 

1392.0508 

132.2614 

.2 

1398.6717 

I 32-5755 

•3 

1405.3084 

132.8897 

•4 

1411.9607 

133-2038 

•5 

1418.6287 

i 33 - 5 i 8 

.6 

1425.3125 

133.8322 

•7 

1432.012 

134-146 3 

.8 

1438.7271 

134-4605 

•9 

1445.458 

I 34-7746 

43 

1452.2046 

135.0888 

.1 

1458.9669 

135-403 

.2 

1465.7449 

I 35 - 7 I 7 1 

•3 

1472.5386 

I 3 6 - 03 i 3 

•4 

I 479-348 

I 36-3454 

•5 

, 1486.1731 

136.6596 

.6 

I 493 -°i 4 

136.9738 

•7 

1499.8705 

! 37-2879 

.8 

1506.7428 

137.6021 

•9 

1513-6307 

137.9162 


















AREAS AND CIRCUMFERENCES OF CIRCLES. 247 


PlAM. 

Area. 

ClRCUM. 

44 

1520.5344 

138.2304 

.1 

1527.4538 

I38.5446 

.2 

1534.3889 

I38.8587 

•3 

1 54 I -3396 

I 39 -I 729 

•4 

1548.3061 

I39.487 

•5 

1555.2883 

139.8012 

.6 

1562.2863 

140.1 I54 

•7 

1569.2999 

140.4295 

.8 

1576.3292 

140-7437 

•9 

I 583-3743 

I4I.O578 

45 

1590-435 

I 4 I -372 

.1 

I 597 - 5 II 5 

141.6862 

.2 

1604.6036 

I42.OOO3 

•3 

16x1.7115 

142 3145 

•4 

16x3.8351 

142.6286 

•5 

1625.9743 

I42.9428 

.6 

1633.1293 

I 43-257 

•7 

1640.3 

143 5711 

.8 

1647.4865 

I 43-8853 

•9 

1654.6886 

I 44 -I 994 

46 

1661.9064 

144.5136 

.1 

1669.1399 

I44.8278 

.2 

1676.3892 

145.1419 

•3 

1683.6541 

I 45 - 456 I 

•4 

1690.9348 

145.7702 

•5 

1698.2311 

I46.0844 

.6 

I 705-5432 

I46.3986 

•7 

1712.871 

I46.7I27 

.8 

1720.2145 

I47.O269 

•9 

I 727-5737 

I 47 - 34 I 

47 

I 734-9486 

147.6552 

.1 

1742.3392 

I47.9694 

.2 

1749-7455 

I48.2835 

•3 

1757. i6 7 6 

I48.5977 

•4 

1764.6053 

I48.9H8 

•5 

1772.0587 

149.226 

.6 

1779.5279 

149.5402 

•7 

1787.0128 

I 49-8543 

.8 

I 794 - 5 I 33 

150.1685 

•9 

1S02.0296 

I5O.4826 

48 

1809.5616 

150.7968 

.1 

1817.1093 

I 5 I-III 

.2 

1824.6727 

I 5 I- 425 I 

•3 

1832.2518 

I 5 I -7393 

•4 

1839.8466 

152.0534 

•5 

1847.4571 

I52.3676 

.6 

1855.0834 

I52.68l8 

•7 

1862.7253 

152.9959 

.8 

1870.383 

I 53 . 3 IOI 

•9 

1888.0563 

153.6242 

49 

1885.7454 

1539384 

.1 

1893.4502 

154.2526 

* .2 

1901.1707 

I 54-5667 

•3 

1908.9068 

154.8809 

•4 

1916.6587 

I 55 -I 95 


Diam. 

Area. 

ClRCUM. 

•5 

1924.4263 

155.5092 

.6 

1932.2097 

I 55-8234 

•7 

1940.0087 

156.1375 

.8 

1947.8234 

156.4517 

•9 

I 955-6539 

I56.7658 

50 

I 963-5 

157.08 

.1 

I 97 1 - 3 6 i 9 

157-3942 

.2 

I 979-2394 

I 57-7083 

•3 

1987.1327 

158.0225 

•4 

1995.0417 

I58.3366 

•5 

2002.9663 

I58.65O9 

.6 

2010.9067 

I58.965 

•7 

2018.8628 

159.2791 

.8 

2026.8347 

159-5933 

•9 

2034.8222 

159.9074 

5 i 

2042.8254 

160.2216 

.1 

2050.8443 

160.5358 

.2 

2058.879 

160.8499 

•3 

2066.9293 

161.1641 

•4 

2074-9954 

161.4782 

•5 

2083.0771 

161.7924 

.6 

2091.1746 

162.1066 

•7 

2099.2878 

162.4207 

.8 

2107.4167 

162.7349 

•9 

2115.5613 

163.049 

52 

2123.7216 

163.3632 

.1 

2131.8976 

163.6774 

.2 

2140.0893 

163.9915 

•3 

2148.2968 

164-3057 

•4 

2156.5199 

164.6198 

•5 

2164.7587 

164.934 

.6 

2 i 73 -oi 33 

165.2482 

•7 

2181.2836 

165.5623 

.8 

2189.5695 

165.8765 

•9 

2197.8712 

166.1906 

53 

2206.1886 

166.5048 

.1 

2214.5217 

166.819 

.2 

2222.8705 

167.1331 

•3 

2231.235 

167.4473 

•4 

2239.6152 

167.7614 

•5 

2248.0111 

168.0756 

.6 

2256.4228 

168.3898 

•7 

2264 8501 

168.7039 

.8 

2273.2932 

169.0181 

•9 

2281.7519 

169.3322 

54 

2290.2264 

169.6464 

.1 

2298.7166 

169.9606 

.2 

2307.2225 

170.2747 

•3 

23 I 5-744 

170.5889 

•4 

2324.2813 

170.903 

•5 

2332-8343 

171.2172 

.6 

2341-4031 

I 7 I- 53 I 4 

•7 

2349-9875 

i 7 I -8455 

.8 

2358.5876 

172.1597 

•9 

2367.2035 

172.4738 















248 AREAS AND CIRCUMFERENCES OF CIRCLES. 


Diam. 

Area. 

ClRCUM. 

55 

2375-835 

I72.788 

.1 

2384.4823 

173.1022 

.2 

2393 -I 452 

i 73 - 4 l6 3 

•3 

2401.8239 

I 73-7305 

•4 

2 4 IO - 5 I 83 

174.0446 

•5 

2419.2283 

174.3588 

.6 

2427.9541 

I 74-673 

•7 

2436.6957 

i 74 - 987 i 

.8 

2445-4529 

i 75 - 3 oi 3 

•9 

2454.2258 

i 75 - 6 i 54 

56 

2463.0144 

175.9296 

.1 

2471.8187 

176.2438 

.2 

2480.6388 

I 76-5579 

•3 

2489.4745 

176.8721 

•4 

2498.326 

177.1862 

•5 

2507.1931 

177.5004 

.6 

2516.076 

177.8146 

•7 

2524-9736 

178.1287 

.8 

2533-8889 

178.4429 

•9 

2542.8189 

178.757 

57 

2551.7646 

179.0712 

.1 

2560.726 

I 79-3854 

.2 

2569.7031 

179.6995 

•3 

2578.696 

180.0137 

•4 

2587.7045 

180.3278 

•5 

2596.7287 

180.642 

.6 

2605.7687 

180.9562 

•7 

2614.8244 

181.2703 

.8 

2623.8957 

181.5845 

•9 

2632.9828 

181.8986 

58 

2642.0856 

182.2128 

.1 

2651.2041 

182.527 

.2 

2660.3383 

182.8411 

•3 

2669.4882 

183.1553 

•4 

2678.6538 

183.4694 

•5 

2687.8351 

183.7836 

.6 

2697.0322 

184.0978 

•7 

2706.2449 

184.4119 

.8 

27 I 5-4734 

184.7261 

•9 

2724-7175 

185.0402 

59 

2733-9774 

I 85-3544 

.1 

2743-253 

185.6686 

.2 

2752.5443 

185.9827 

•3 

2761.8512 

186.2969 

•4 

2 77 I - I 739 

186.611 

•5 

2780.5123 

186.9252 

.6 

2789.8665 

i8 7- 2 394 

•7 

2799.2363 

i 87-5535 

.8 

2808.6218 

187.8677 

•9 

2818.0231 

188.1818 

60 

2827.44 

188.496 

.1 

2836.8727 

188.8102 

.2 

2846.321 

189.1243 

•3 

2855-7851 

189.4385 

•4 

2865.2649 

189.7526 


Diam. 

Area. 

ClRCUM. 

•5 

2874.7603 

190.0668 

.6 

2884.2715 

I9O.381 

•7 

2893.7984 

I9O.695I 

.8*. 

2903 - 34 II 

I 9 I .0093 

•9 

2912.8994 

I 9 I - 3 2 34 

61 

2922.4734 

19:1.6376 

.1 

2932.0631 

191.9518 

.2 

2941.6686 

192.2659 

•3 

2951.2897 

192.5801 

•4 

2960.9266 

192.8942 

•5 

2970 - 579 1 

193.2084 

' .6 

2980.2474 

193.5226 

•7 

2989.9314 

193-8367 

.8 

2999.6311 

194.1509 

•9 

3009.3465 

194.465 

62 

3019.0776 

! 94 - 779 2 

.1 

3028.8244 

I 95-0934 

.2 

3038.5869 

I 95-4075 

•3 

3048.3652 

I 95 - 72 I 7 

•4 

3058.1591 

196.0358 

•5 

3067.9687 

I 96-35 

.6 

3077.7941 

196.6642 

•7 

3087.6341 

196.9783 

.8 

3 ° 97 - 49 x 9 

197.2925 

•9 

3 ! 07-3644 

197.6066 

63 

3117.2526 

197.9208 

.1 

3127.1565 

198.235 

.2 

3137.0761 

198.5491 

•3 

3147.0114 

198.8633 

•4 

3156.9624 

199.1774 

•5 

3166.9291 

I 99 - 49 l6 

.6 

3176.9116 

199.8058 

•7 

3186.9097 

200.1199 

.8 

3196.9236 

200.4341 

•9 

3206.9531 

200.7482 

64 

3216.9984 

201.0624 

.1 

3227.0594 

201.3766 

.2 

3237.1361 

201.6907 

•3 

3247.2284 

202.0049 

•4 

3257-3365 

202.319 

•5 

3267.4603 

202.6332 

.6 

3277-5999 

202.9474 

•7 

3287.7551 

203.2615 

.8 

3297.9261 

203-5757 

•9 

3308.1127 

203.8898 

65 

33 ! 8 . 3 i 5 

204.204 

.1 

3328.5331 

204.5182 

.2 

3338.7668 

204.8323 

•3 

3349.0163 

205.1465 

•4 

3359.2815 

205.4606 

• 5 . 

3369.5623 

205.7748 

.6 

3379.8589 

206.089 

•7 

3390.1712 

206.4031 

.8 

3400.4993 

206.7173 

•9 

34x0.843 

207.0314 
















AREAS AND CIRCUMFERENCES OF CIRCLES. 249 


Area. 

ClRCUM. 

Diam. 

Area. 

ClRCUM. 

3421.2024 

207.3456 

•5 

4015.1611 

224.6244 

3431-5775 

2O7.6598 

.6 

4026.4002 

224.9386 

3441.9684 

207.9739 

•7 

4037-655 

225.2527 

3452.3749 

208.2881 

.8 

4048.9255 

225.5669 

3462.7972 

208.6022 

•9 

4060.2117 

225.88l 

34732351 

208.9164 

72 

4071.5136 

226.1952 

3483.6888 

209.2306 

.1 

4082.8312 

226.50Q4 

3494.1582 

209.5447 

.2 

4094.1645 

226.8235 

3504.6433 

2O9.8589 

•3 

4105.5136 

227.1377 

3515.1441 

2IO.I73 

•4 

41x6.8783 

2274518 

3525.6606 

2IO.4872 

•5 

4128.2587 

227.766 

3536.1928 

210.8014 

.6 

4 I 39-655 

228.0802 

3546.7407 

2 II -1155 

•7 

415X.0668 

228.3943 

3557-3044 

211.4297 

.8 

4162.4943 

228.7085 

3567.8837 

2II.7438 

•9 

4 I 73-9376 

229.0226 

3578.4787 

212.058 

73 

4x85.3966 

220.4468 

3589.0895 

212.3722 

.1 

4196.8713 

22 q.6=;i 

3599.716 

212.6863 

.2 

4208.3617 

229 9651 

3610.3581 

213.OOO5 

•3 

4219.8678 

240.27Q4 

3621.016 

2 I 3 - 3 I 46 

•4 

4231.3896 

230 5934 

3631.6896 

213.6288 

•5 

4242.9271 

230.9076 

3642.3789 

213-943 

.6 

4254.4804 

231.2218 

3653-0839 

214-2571 

•7 

4266.0493 

231-5359 

3663.805 

2 I 4 - 57 I 3 

.8 

4277.634 

231.8501 

3674.541 

214.8454 

•9 

4289.2343 

232.1642 

3685.2931 

215.1996 

74 

4300.8504 

232.4784 

3696.061 

215-5138 

.1 

4312.4822 

242.7026 

3706.8445 

215.8279 

.2 

4324.1297 

233.1067 

3717-6438 

216.1421 

•3 

4335 - 79 2 8 

233.4209 

3728.4587 

216.4562 

•4 

4347 - 47 I 7 

233-735 

3739.2894 

216.7704 

•5 

4359.1663 

234.0492 

3750.1358 

217.0846 

.6 

4370.8767 

234-3634 

3760.9979 

217.3987 

•7 

4382.6027 

234-6775 

3771.8756 

217.7129 

.8 

4394-3444 

234 - 99 I 7 

3782.7691 

218.027 

•9 

4406.1019 

235-3058 

3793-6783 

218.3412 

75 

44 I 7-875 

235.62 

3804.6033 

218.6554 

.1 

4429.6639 

235-9342 

3815-5439 

218.9695 

.2 

4441.4684 

236.2483 

3826.5002 

219.2837 

•3 

4453.2887 

236.5625 

3847.4722 

219.5978 

•4 

4465.1247 

236.8766 

3848.46 

219 912 

•5 

4476.9763 

237.1908 

3859.4635 

220.2262 

.6 

44S8.8437 

237 505 

3870.4826 

220.5403 

•7 

4500.7268 

237-8191 

3881.5175 

220.8545 

.8 

4512.6257 

238.1333 

3892.5681 

221.1686 

•9 

4524.5402 

238 4474 

3903 6343 

221.4828 

76 

4536.4704 

238.7616 

39 r 4 - 7 i 63 

221.797 

.1 

4548.4163 

239.0758 

3925.814 

222.1111 

.2 

4560.378 

239.3899 

3936.9275 

222.4253 

•3 

4572.3553 

239.7041 

3948.9566 

222.7394 

•4 

4584.3484 

240.0182 

3959.2014 

223.0536 

•5 

4596 - 357 I 

240-3324 

3970.3619 

223.3678 

.6 

4608.3816 

240.6466 

3981.5382 

223.6819 

•7 

4620.4218 

240.9607 

3992.7301 

223.9961 

.8 

4632.4777 

241.2749 

4003.9378 1 

224.3102 

•9 

4644.5493 

241.589 





















250 AREAS AND CIRCUMFERENCES OF CIRCLES. 


Diam. I Area. I Circum. 


77 

4656.6366 

241.9032 

.1 

4668.7396 

242.2174 

.2 

46S0.8583 

2 4 2 - 53 I 5 

•3 

4692.9928 

242.8457 

•4 

4705.1429 

2 43 - I 59 8 

•5 

47 I 7 - 3°87 

243-474 

.6 

4729.4903 

243.7882 

•7 

4741.6876 _ 

244.1023 

.8 

4753-9005 

244.4165 

•9 

4766.1292 

244.7306 

78 

477 8 - 373 6 

245.0448 

.1 

4790.6337 

245-359 

.2 

4802.9095 

245-6731 

•3 

4815.201 

245 - 9 8 73 

•4 

4827.5082 

246.3014 

•5 

4839.8311 

246.6156 

.6 

4852.1698 

246.9298 

•7 

4864.5241 

247.2439 

.8 

4876.6942 

247 - 558 i 

•9 

4889.2799 

247.8722 

. 79 

4901.6814 

248.1864 

.1 

4914.0986 

248.5006 

.2 

4926.5315 

248.8147 

•3 

4938.98 

249.1289 

•4 

4951.4443 

249.443 

•5 

4963.9243 

249.7572 

.6 

4976.4201 

250.0714 

•7 

4988.9315 

250.3855 

.8 

5001.4586 

250.6997 

•9 

5014.0015 

251.0138 

80 

5026.56 

251.328 

.1 

5039.1343 

251.6422 

.2 

5051.7242 

251-9563 

•3 

5064.3299 

252.2705 

•4 

5076.9513 

252.5846 

-5 

5089.5883 

252.8988 

.6 

5102.2411 

253-213 

•7 

5114.9096 

253-5271 

.8 

5127.5939 

253-8413 

•9 

5140.2938 

254-1554 

81 

5 I 53-°°94 

254.4696 

.1 

5165.7407 

254.7838 

.2 

5178.4878 

255-0979 

•3 

5191.2505 

255.4121 

•4 

5204.0289 

255.7262 

•5 

5216.8231 

256.0404 

.6 

5229.633 

256.3546 

•7 

5242.4586 

256.6687 

.8 

5255.2999 

256.9829 

•9 

5268.1569 

257.297 

82 

5281.0296 

257.6x12 

.1 

5293 - 9 18 

257.9254 

.2 

5306.8221 

258.2395 

•3 

53 i 9'742 

258.5537 

•4 

533 2.6775 

258 8678 


Diam. 

Area. 

Circum. 

-5 

5345.6287 

259.182 

.6 

5358.5957 

259.4962 

-7 

537 I -5784 

259.8103 

.8 

5384.5767 

260.1245 

• 9 " 

5397-5908 

260.4386 

83 

5410.6206 

260.7528 

.1 

5423.6661 

261.067 

.2 

5436.7273 

261.3811 

•3 

5449.8042 

261.6953 

•4 

5462.8968 

262.0094 

•5 

5476.0051 

262.3236 

i .6 

5489.1292 

262.6378 

•7 

5502.2689 

262.9519 

.8 

5515.4244 

263.2661 

•9 

5528.5955 

263.5802 

84 

5541.7824 

263.8944 

.1 

5554-985 

264.2086 

.2 

5568.2033 

264.5227 

-3 

5581.4372 

264.8369 

•4 

55946869 

265.151 

•5 

5607.9523 

265.4652 

.6 

5621.2335 

265.7794 

•7 

5634.5303 

266.0935 

.8 

5647.8428 

266.4077 

•9 

5661.1711 

266.7218 

85 

5674 - 5 I 5 

267.036 

.1 

5687.8747 

267.3502 

.2 

57 oi - 2 5 

267.6643 

•3 

5714.6411 

267.9785 

•4 

5728.0479 

268.2926 

-5 

5741.4703 

268.6068 

.6 

5754.9085 

268.921 

•7 

5768.3624 

269.2351 

.8 

5781.8321 

269-5493 

•9 

5795 - 3 1 74 

269.8634 

86 

5808.8184 

270.1776 

.1 

5822.3351 

270.4918 

.2 

5835.8676 

270.8059 

•3 

5849.4157 

27I.1201 

•4 

5862.9796 

27I.4342 

•5 

5876.5591 

271.7484 

.6 

5S90.1544 

272.0626 

•7 

5903.7654 

272.3767 

.8 

5917.3921 

272.6909 

•9 

5931.0345 

273.OO5 

87 

5944.6926 

273.3192 

.1 

5958.3644 

273-6334 

.2 

5972.0559 

273 9475 

•3 

5985.7612 

274.2617 

•4 

5999.4821 

274-5758 

•5 

6013.2187 

274.89 

.6 ■ 

6026.9711 

275.2042 

•7 

6040.7392 

275-5183 

.8 

6054.5229 

275-8325 

•9 

6068.3224 

275.1466 


















AREAS AND CIRCUMFERENCES OF CIRCLES. 25 I 


Diam. 

Area. 

ClRCUM, 

88 

6082.1376 

276.4608 

.1 

6095.9685 

276.775 

.2 

6109 8151 

277.089I 

•3 

6123.6774 

277-4033 

•4 

6 I 37-5554 

277.7I74 

•5 

6151.4491 

278.O316 

.6 

6165.3586 

278.3458 

•7 

6179.2837 

278.6599 

.8 

6193.2246 

278.974I 

•9 

6207.1811 

279.2882 

89 

6221.1534 

279.6024 

.1 

6235.1414 

279.9166 

.2 

6249.1451 

280.2307 

•3 

6263.1644 

280.5449 

•4 

6277.1995 

280.859 

•5 

6291.2503 

281.1732 

.6 

6305.3169 

281.4874 

•7 

63 I 9 - 399 I 

281.8015 

.8 

6333497 

282.1157 

•9 

6347.6107 

282.4298 

90 

6361.74 

282.744 

.1 

6375 - 885 I 

283 O582 

.2 

6390.0458 

283.3723 

•3 

6404.2223 

283.6865 

•4 

6418.4144 

284.OOO6 

•5 

6432.6223 

284.3148 

.6 

6446.8459 

284.629 

•7 

6461.0852 

284.943I 

.8 

6475-3403 

285-2573 

•9 

6489.61 X 

285 - 57 I 4 

9 1 

6503.8974 

285.8856 

.1 

6518.1995 

286.I998 

.2 

653 2 - 5*74 

286.5139 

•3 

6546.8909 

286.8281 

•4 

6561.2002 

287.1422 

•5 

6575 - 565 I 

287.4564 

.6 

6589-9458 

287.7706 

•7 

6604.3422 

288.0847 

.8 

6618.7543 

288.3989 

•9 

6633.1821 

288.7X3 

92 

6647.6256 

289.O272 

.1 

6662.0848 

289.3414 

.2 

6676.5598 

289.6555 

•3 

6691.0504 

289.9697 

•4 

6705-5567 

29O.2838 

•5 

6720.0787 

29O.598 

.6 

6734.6165 

29O.9121 

•7 

6749.17 

291.2263 

.8 

6763 - 739 I 

29I.5405 

•9 

6778 324 

291.8546 

93 

6792.9246 

292.1688 

.1 

6807.5409 

292.483 

* .2 

6822.1729 

292.797I 

•3 

6836.8206 

293 - 1**3 

•4 

6851484 

293 4254 


Diam. 

Area. 

ClRCUM. 

•5 

6866.1631 

293.7396 

.6 

6880.858 

294.O538 

•7 

6895.5685 

294.3679 

.8 

69IO.2948 

294.6821 

•9 

6925.0367 

294.9962 

94 

6939-7944 

295.3IO4 

.1 

6954-5678 

295.6246 

.2 

6969-3569 

295-9387 

-3 

6984.1616 

296.2529 

•4 

6998.9821 

296.567 

•5 

7013.8183 

296.8812 

.6 

7028.6703 

297.1954 

•7 

7043-5379 

297-5095 

.8 

7058.4212 

297.8237 

•9 

7073.3203 

298.I378 

95 

7088.235 

298.452 

.1 

7103.1655 

298.7662 

.2 

7118. IIl6 

299.0803 

•3 

7 * 33-0735 

299-3945 

•4 

7148.0511 

299.7086 

•5 

7163.0443 

300.0228 

.6 

7*78-0533 

300.337 

•7 

7193.078 

300.6511 

.8 

7208.1185 

3OO.9653 

•9 

7223.1746 

301.2794 

96 

7238.2464 

301.5936 

.1 

7 2 53-3339 

3OI.9O78 

.2 

7268.4372 

302.2219 

• -3 

7283.5561 

302.5361 

•4 

7298.6908 

302.8502 

•5 

7313.8411 

3O3.I644 

.6 

7329.0072 

3O3.4786 

•7 

7344.189 

303.7927 

.8 

7359-3865 

3O4.I069 

•9 

7374-5997 

304.421 

97 

7389.8286 

304.7352 

.1 

7405.0732 

305.0494 

.2 

7420.3335 

305-3635 

•3 

7435.6096 

3O5.6777 

•4 

7450.9013 

305.9918 

•5 

7466.2087 

306.306 

.6 

748 i- 53 i 9 

306.6202 

•7 

7496.8708 

306.9343 

.8 

7512.2253 

3O7.2485 

•9 

7527-5956 

3O7.5626 

98 

7542.9816 

3O7.8768 

.1 

7558.3833 

308.191 

.2 

7573.8007 

308.5051 

•3 

7589-2338 

308.8193 

•4 

7604.6826 

309.1334 

•5 

7620.1471 

3O9.4476 

.6 

7635.6274 

3O9.7618 

•7 

7651-1233 

310.0759 

.8 

7666.635 

310.3901 

•9 

7682.1623 

310.7042 
















252 AREAS AND CIRCUMFERENCES OF CIRCLES. 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 

ClRCUM. 

99 

7697.7054 

3II.OI84 

•5 

7775-6563 

312.5892 

,1 

77 I 3 - 2 ^ 4 2 

3 ti - 33 2 6 

.6 

7791.2937 

312.9034 

.2 

7728.8337 

311.6467 

•7 

7806.9467 

3 I 3- 2I 75 

•3 

7744.4288 

311.9609 

.8 ' 

7822.6154 

3 I 3 - 53 I 7 

•4 

7760.0347 

3 I2<2 75 

•9 

7838.2999 

313-8458 


To Compute Area, or Circumference of a Diameter greater 
tlian any in preceding Table. 

See Rules, pages 235-6 and 241-2. 

Or, If Diameter exceeds 100 and is less than 1001. 

Put a decimal point, and take out area or circumference as for a Whole 
Number by removing decimal point, if for an area, two places to right, and 
if for a circumference, one place. 

Example. —What is area and what circumference of a circle 967 feet in diame¬ 
ter? 

Area of 96.7 is 7344.189; hence, for 967 it is 734 418.9; and circumference of 96.7 
is 303.7927, and for 967 it is 3037.927 

To Compute Area, and. Circumference of a Circle by Log¬ 
arithms. 

See Rules, pages 236, 242. 

Areas and. Circumferences of Circles. 

From i to 50 Feet (advancing by an Inch). 

Or, From i to 50 Inches (advancing by a Twelfth). 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 

ClRCUM. 


Feet. 

Feet. 


Feet. 

Feet. 

1/6 

•7854 

3.1416 

3/6 

7.0686 

9.4248 

1 

.9217 

3-4034 

I 

7.4668 

9.6866 

2 

I.069 

3.6652 

2 

7-8758 

9.9484 

3 

I.2272 

3-927 

3 

8.2958 

10.2102 

4 

I -3963 

4.1888 

4 

8.7267 

IO.472 

5 

!-5763 

4.4506 

5 

9- i6 85 

IO.7338 

6 

1.7671 

4.7124 

6 

9.6211 

IO.9956 

7 

1.969 

4.9742 

7 

10.0848 

1 1.2574 

8 

2.1817 

5-236 

8 

IO -5593 

II. 5 I 92 

9 

2-4053 

5-4978 

9 

11.0447 

II.781 

10 

2.6398 

5-7596 

10 

n.541 

12 0428 

11 

2.8853 

6 0214 

11 

12.0483 

I2.3046 

2 ft. 

3 - T 4 i 6 

6.2832 

4/6 

12.5664 

12.5664 

* 1 

3.4088 

6 545 

1 

1 3 0955 

12.8282 

2 

3.687 

6.8068 

2 

T 3-6354 

r 3 °9 

3 

3 - 976 i 

7.0686 

3 

14.1863 

I 3 - 35 I 8 

4 

4.2761 

7-3304 

4 

14.7481 

13.6136 

5 

4.5869 

7.5922 

5 

15.3208 

138754 

6 

4.9087 

7-854 

6 

1 5 -9043 

I 4 - I 37 2 

7 

5-2415 

8.1158 

7 . 

16.4989 

14-499 

8 

5-5852 

8.3776 

8 

17.1043 

14.6608 

9 

5-9396 

8.6394 

9 

17.7206 

14.9226 

10 

6-305 

8.9012 

10 

18.3478 

15.1844 

11 

6.6814 1 

9 163 

11 

18.9859 

15 4462 































AREAS AND CIRCUMFERENCES OF CIRCLES. 253 


Diam 

Area. 

ClRCUM. 


Feet. 

Feet. 

5 ft - 

I 9-635 

15.708 

1 

20.2949 

15.9698 

2 

20.9658 

16.2316 

3 

21.6476 

16.4934 

4 

22.3403 

16.7552 

5 

23 0439 

I7.OI7 

6 

23-7583 

17.2788 

7 

24.4837 

17.5406 

8 

25.201 

17.8024 

9 

25 - 9 6 73 

18.0642 

10 

26.7254 

18.326 

11 

27.4944 

18.5878 

6 ft . 

28.2744 

18.8496 

1 

29.0653 

I9.III4 

2 

29.867 

193732 

3 

30.6797 

I 9.635 

4 

3*-5033 

19.8968 

5 

32.3378 

20.1586 

6 

33 -I 83 I 

20.4204 

7 

34-0394 

20.6822 

8 

34.9067 

20.944 

9 

35-7848 

21.2058 

10 

36.6738 

21.4676 

11 

37-5738 

21.7294 

7 A 

38.4846 

21.9912 

1 

39.4064 

22.253 

2 

40.339 

22.5148 

3 

41.2826 

22.7766 

4 

42.2371 

23.0384 

5 

43.2025 

23.3002 

6 

44-1787 

23.562 

7 

45-1659 

23.8238 

8 

46.164.1 

24.0856 

9 

47 -I 73 I 

24-3474 

10 

48.193 

24,6092 

11 

49.2238 

24.871 

8 A 

50.2656 

2 5 ' I 3 2 S 

1 

5 i - 3 i 8.3 

25-3946 

2 

52.3818 

25.6564 

3 

53-4563 

25.9182 

4 

54-5417 

26.18 

5 

55-638 

26.4418 

6 

56. 745 1 

26.7036 

7 

57.8632 

26.9654 

8 

58.9923 

27.2272 

9 

60.1322 

27.489 

10 

61.283 

27.7508 

11 

62.4448 

28.0126 

9 A 

63.6174 

28.2744 

1 

64.801 

28.5362 

2 

6 S .9954 

28.798 

3 

67.2008 

29.0598 

4 

68.417 

29.3216 

5 

69.6442 

29-5834 


Diam. 

Area. 

ClRCUM. 

6 

Feet. 

70.8823 

Feet. 

29.8452 

7 

72.1314 

30.107 

8 

73 - 39 T 3 

30.3688 

9 

74.6621 

30.6306 

10 

75-9439 

30.8924 

11 

77-2365 

3 I-I 542 

10 ft . 

78-54 

31.416 

1 

79-8545 

31.6778 

2 

81.1798 

3 I -9396 

3 

82.5161 

32.2014 

4 

83-8633 

32.4632 

5 

85.2214 

32.725 

6 

86.5903 

32.9868 

7 

87.9703 

33.2486 

8 

89.3611 

33 - 5 I 04 

9 

90.7628 

33-7722 

10 

92.1754 

34-034 

11 

93-599 

34-2958 

lift . 

95 0334 

34-5576 

1 

96.4787 

34.8194 

2 

97-935 

35.0812 

3 

99 4022 

35-343 

4 

100.8803 

35.6048 

5 

102.3693 

35-8666 

6 

103 8691 

36.1284 

7 

105.38 

36.3902 

8 

106.9017 

36.652 

9 

108.4343 

36 - 9 i 38 

10 

109.9778 

37 -I 756 

11 

in -5323 

37-4374 

W 

to 

113.0976 

37,6992 

1 

114.6739 

37.961 

2 

116.261 

38.2228 

3 

117.8591 

38.4846 

4 

119.468 

38.7464 

5 

121.088 

39.0082 

6 

122.7187 

39-27 

7 

124,3605 

39 - 53 i 8 

8 

126.0131 

397936 

9 

127.6766 

40.0554 

10 

129-351 

40.3172 

11 

131.0366 

40.579 

13A 

132.7326 

40.8408 

1 

I 34-4398 

41.1026 

2 

136.1578 

41.3644 

3 

137.886S 

41.6262 

4 

139 6267 

41.888 

5 

I 4 I -3774 

42.1498 

6 

i 43 -i 39 i 

42.4116 

7 

144.9117 

42.6734 

8 

146.6953 

42-9352 

9 

148.4897 

43-197 

10 

150.295 

43-4588 

11 

152.1113 

43.7206 


Y 


















254 areas and circumferences of circles. 


Diam. 

Area. 

ClRCUM. 

Diam. 

14 ft - 

Feet. 

I 53-9384 

Feet. 

43.9824 

6 

I 

I 55.7764 

44.2442 

7 

2 

157.6254 

44.506 

8 

3 

I 59-4853 

44.7678 

y 

4 

161.3561 

45.0296 

10 

5 

163.2378 

45.2914 

11 

6 

165.1303 

45-5532 

19 ft . 

I 

7 

167.0338 

45-815 

8 

168.9483 

46.0768 

2 

9 

10 

I7O.8736 

172.8098 

46.3386 

46.6004 

3 

•v A 

11 

174.7569 

46.8622 

X 

5 

15/6 

176.715 

47- I2 4 

6 

1 

178.684 

47-3858 

7 

2 

180.6638 

47.6476 

8 

3 

182.6546 

47.9094 

9 

4 

184.6563 

48.1712 

10 

5 

186.6689 

48.433 

11 

6 

188.6924 

48.6948 

■20 ft . 

I 

7 

190.7267 

48.9566 

8 

192.7721 

49.2184 

2 

9 

10 

194.8283 

196.8954 

49.4802 

49.742 

3 

4 

11 

198.9734 

50.0038 

5 

16/6 

201.0624 

50.2656 

6 

1 

203.1622 

50.5274 

7 

2 

205 273 

50.7892 

8 

3 

207.3947 

51-051 

9 

4 

209 5273 

51.3128 

10 

5 

2II.6707 

5 I -5746 

11 

6 

213.8252 

51.8364 

21 /V. 

7 

215.9904 

52.0982 

j 

8 

218.1667 

52.36 

2 

9 

10 

11 

220.3538 

222.5518 

224.7607 

52.6218 

52.8836 

53-1454 

3 

4 

5 

17/6 

226.9806 

53-4072 

6 

1 

229.2II3 

53-669 

7 

2 

23 I .453 

53 - 93 o 8 

8 

3 

233-7056 

54.1926 

9 

4 

235.9691 

54-4544 

10 

5 

238.2434 

54.7162 

11 

6 

240.5287 

54-978 

22/6 

I 

7 

242.8249 

55-2398 

8 

245.I32I 

55 - 50 i 6 

2 

9 

10 

11 

247.45OI 

249.779 

252.H88 

55-7634 

56.02152 

56.287 

3 

4 

5 

18/6 

254.4696 

56.5488 

6 

1 

256.8312 

56.8106 

7 

2 

259.2038 

57.0724 

8‘ 

3 

261.5873 

57-3342 

9 

4 

263.9817 

57-596 

10 

5 

266.3869 

57-8578 

11 


Area. 

Feet. 

268.8031 

271.2302 

273.6683 

276.1172 

278.577 

281.0477 

283.5294 

286.0219 

288.5255 

291.0398 

293-5651 

296.1012 

298.6483 

301.2064 

303-7753 

306.3551 

308.9458 

311-5475 

3 i 4- i 6 

316.7834 

3 i 9 - 4 i 78 

322.0631 

3 2 4 - 7 1 93 
327.3864 
330.0643 
332.7532 
335 - 453 1 
338.1638 
340.8854 
343.618 

346.3614 

349- II 57 

351.881 

354.6572 

357.4442 

360.2422 

363.0511 

365.8709 

368.7017 

371-5433 

374-3958 

377.2592 

380.1336 
383.0188 
385 915 

388.8221 

391-74 

394.6689 

397.6087 

400.5594 

403.5211 

406.4936 

409.477 

412.4713 


ClRCUM. 


Feet. 

58.1196 

58.3814 

58.6432 

58.905 

59.1668 

59.4286 

59.6904 

59-9522 

60.214 

60.4758 

60.7376 

60.9994 

61.2612 

61-523 

61.7848 

62.0466 

62.3084 

62.5702 

62.832 

63.0938 

63-3556 

63.6174 

63.8792 

64.141 

64.4028 

64.6646 

64.9264 

65.1882 

65-45 

65.7118 

65 9736 

66.2354 

66.4972 

66.759 

67.0208 

67.2826 

67-5444 

67.8062 

68.068 

68.3298 

68.5916 

68.8534 

69.1152 

69.377 

69.6388 

69.9006 

70.1624 

70.4242 

70.686 

70.9478 

71.2096 

71.4714 

7I-7332 

71.995 



















AREAS AND CIRCUMFERENCES OF CIRCLES. 255 


Diam. | 

Area. 

ClRCUM. 


Feet. 

Feet. 

23 A 

415.4766 

72.2568 

1 

418.4927 

72.5186 

2 

421.5198 

72.7804 

3 

424-5578 

73.0422 

4 

427.6067 

73-304 

5 

430.6664 

73-5658 

6 

433-7371 

73.8276 

7 

436.8187 

74.0894 

8 

439 - 9 I 7 

74-3512 

9 

443.OI47 

74613 

10 

446.129 

74.8748 

11 

449.2542 

75.1366 

24 A 

452.3904 

75-3984 

1 

455-5374 

45.6602 

2 

458.6954 

75.922 

3 

461.8643 

76.1838 

4 

465.044 

76.4456 

5 

468.2347 

76.7074 

6 

47 I -4363 

76.9692 

7 

474.6488 

77.231 

8 

477-8723 

77.4928 

9 

481.1066 

77-7546 

10 

484.3518 

78.0164 

11 

487.6076 

78.2782 

25 A 

490.875 

78.54 

1 

494.1529 

78.8018 

2 

497.4418 

79.0636 

3 

500.7416 

79 3254 

4 

504.0523 

79.5872 

5 

507-3738 

79.849 

6 

510.7063 

80.1108 

7 

514.0413 

80.3726 

8 

517.404 

80.6344 

9 

520.7693 

80.8962 

10 

524.1454 

81.158 

11 

527.5324 

81.4198 

26 A 

530.9304 

81.6816 

1 

534 - 33 I 3 

81.9434 

2 

537-759 

82.2052 

3 

541.1897 

82.467 

4 

544 - 63 I 3 

82.7288 

5 

548.0837 

82.9906 

6 

55 I- 547 1 

83.2524 

7 

555-0214 

83.5142 

8 

558.5066 

83.776 

9 

562.0028 

84.0378 

10 

565.5098 

84.2996 

11 

569.0277 

84.5614 

27 A 

572.5566 

84.8232 

1 

576.0963 

85.085 

2 

579.6467 

85.3468 

3 

583.2086 

85.6086 

4 

586.781 

85.8704 

5 

590.3644 

86.1322 


Diam. 

Area. 

ClRCUM. 


Feet. 

Feet. 

6 

593-9587 

86.394 

7 

597-5639 

86.6558 

8 

601.18 

86.9176 

9 

604.8071 

87.1794 

10 

608.445 

87.4412 

11 

612.0938 

87-703 

28 A 

6 I 5-7536 

87.9648 

1 

619.4242 

88.2266 

2 

623.1058 

88.4884 

3 

626.7983 

88.7502 

4 

630.5016 

89.OI2 

5 

634.2159 

89.2738 

6 

637.94II 

89-5356 

7 

641.6772 

89.7974 

8 

645-4243 

90.0592 

9 

649.1822 

90.321 

10 

652.951 

90.5828 

11 

656.7307 

90.8446 

29 A 

660.5214 

91.1064 

1 

664.3229 

91.3682 

2 

668.1354 

9 i - 6 3 

3 

671.9588 

91.8918 

4 

675 - 793 I 

92.1536 

5 

679.6382 

92.4154 

6 

683-4943 

92.6772 

7 

687.3613 

92.939 

8 

69 1 -2393 

93.2008 

9 

695.1281 

93.4626 

10 

699.0278 

93-7244 

11 

702.9384 

93.9862 

30 A 

706.86 

94.248 

1 

710.7924 

94.5098 

2 

7 I 4-7358 

94 - 77 i 6 

3 

718.6901 

95-0334 

4 

722.6553 

95-2952 

5 

726.6313 

95-557 

6 

730.6183 

95.8188 

7 

734.6162 

96.0806 

8 

738.6251 

96.3424 

9 

742.6448 

96.6042 

10 

746.6754 

96.866 

11 

750 7 i6 4 

97.1278 

3 1 A 

754.7694 

97.3896 

1 

758.8327 

97.6514 

2 

762.907 

97 - 9 I 3 2 

3 

766.9922 

98-175 

4 

771.0883 

98.4368 

5 

775 -I 952 

98.6986 

6 

779 - 3 I 3 I 

98.9604 

7 

783.4419 

99.2222 

8 

787.5817 

99.484 

9 

79 I - 73 2 3 

99-7458 

10 

795-8938 

100.0076 

11 

800.0662 

100.2694 





















256 AREAS AND CIRCUMFERENCES OF CIRCLES. 


Diam. 

Area. 

ClRCUM. 


Feet. 

Feet. 

32 ft . 

804.2496 

IOO.5312 

I 

808.4439 

IOO.793 

2 

812.649 

IOI.0548 

3 

816.8651 

IOI.3166 

4 

821.092 

IOI.5784 

5 

825.3299 

IOI.8402 

6 

829.5787 

102.102 

7 

833 8384 

IO2.3638 

S 

838.1091 

IO2.6256 

9 

842.3906 

IO2.8874 

10 

846.683 

I 03 -I 492 

11 

850.9863 

I ° 3 - 4 * 1 

33 A 

855.3006 

103.6728 

1' 

859.6257 

103.9346 

2 

863 9618 

104.1964 

3 

868.3088 

104.4582 

4 

872.6667 

104.72 

5 

877-0354 

104.9818 

6 

881.4151 

105.2436 

7 

885.8057 

105-5054 

8 

890.2073 

1057672 

9 

894.6197 

106.029 

10 

899.043 

106.2908 

11 

903.4772 

106.5526 

34 .A 

907.9224 

106.8144 

1 

9 I2 -3784 

107.0762 

2 

916.8454 

107.338 

3 

9 2I - 3 2 33 

107.5998 

4 

925.812 

107.8616 

5 

930.3117 

108.1234 

6 

934.8223 

108.3852 

7 

939-3439 

108.647 

8 

943-8763 

108.9088 

9 

948.4196 

109.1706 

10 

952.9738 

1094324 

11 

957 - 539 2 

109.6942 

35 A 

962.115 

109.956 

1 

966.7019 

110.2178 

2 

971.2998 

110.4796 

3 

975.9086 

no.7414 

4 

980.5287 

111.0032 

5 

985.15SS 

111.265 

6 

989.8005 

111.5268 

7 

994.4527 

111.7886 

8 

999.116 

II2.0504 

9 

1003.7903 

112.3122 

10 

1008.4754 

112.574 

11 

1013.1714 

112.8358 

36 A 

1017.8784 

113.0976 

1 

1022.5962 

113-3594 

2 

1027.325 

113.6212 

3 

1032.0647 

113.883 

4 

1036.8153 

114.1448 

5 

! 04 i- 57 6 7 

114.4066 


Diam. 

Area. 

Cir.ccM. 


Feet. 

Feet. 

6 

IO46.349I 

II4.6684 

7 

IO51.1324 

114.9302 

8 

IO55.9266 

II5.192 

9 

1060.7318 

1 * 5-4538 

10 

1065.5478 

II 5 - 7 I 56 

11 

IO7O.3747 

** 5-9774 

37 A 

IO75.2126 

116.2392 

1 

1080.0613 

116 501 

2 

1084.921 

116.7628 

3 

1089.7916 

117.0246 

’ 4 

IO94.673I 

117.2864 

5 

IO99.5654 

117.5482 

6 

IIO4.4687 

117.81 

7 

IIO9.3839 

118.0718 

S 

III4.308 

iiS- 333 6 

9 

1119.2441 

**8 5954 

10 

II24.191 

118 8572 

11 

1129.1489 

119.119 

38 A 

II34.II76 

119 3808 

1 

H39.O972 

119.6426 

2 

1144.0878 

1*9 9044 

3 

II49 0893 

120.1662 

4 

1154.IOI7 

120.428 

5 

1159.1249 

120.6898 

6 

1164.1591 

120.9516 

7 

1169.2042 

121.2134 

8 

II74.2603 

121.4758 

9 

II79.3272 

121.737 

10 

1184.405 

121.9988 

11 

1189.4937 

122.2606 

39 A 

U 94-5934 

122.5224 

1 

I199.7039 

122.7848 

2 

1204.8254 

123.046 

3 

1209.9578 

123.3078 

4 

1215.IOI 

123 5696 

5 

1220.2552 

123.8314 

6 

I225.4203 

124.0932 

7 

I23O.5963 

* 24-355 

8 

I 235-7833 

124.6168 

9 

I24O.981I 

124,8786 

10 

I246.1898 

125.1404 

11 

I25I.4094 

125 4022 

40 A 

1256.64 

125.664 

1 

1261.8814 

125.9258 

2 

I267.I338 

126.1876 

3 

I272.397I 

126.4494 

4 

1277.6712 

126.7112 

5 

1282.9563 

126.973 

6 

1288.2523 

127.2348 

7 

12 93-5592 

127.4966 

8 

1298.877 

127.7584 

9 

1304.2058 

128.0202 

10 

* 309-5454 

128.282 

11 

1314.8959 1 

128.5438 




















AREAS AND CIRCUMFERENCES OF CIRCLES. 257 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 


Feet. 

Feet. 


Feet. 

41/6 

1320.2574 

128.8056 

6 

1625.9743 

I 

1325.6297 

129.0674 

7 

1631.9357 

2 

I 33 I - OI 3 

129.3292 

8 

1637.9081 

3 

1336-4072 

129.591 

9 

1643.8913 

4 

1341.8123 

129.8528 

10 

1649.8854 

5 

1347.2282 

13O.II46 

11 

1655.8904 

6 

1352-6551 

130.3764 

46 ft. 

1661.9064 

7 

1358.0929 

130.6382 

1 

1667.9332 

8 

1363-5416 

1309 

2 

1673 971 

9 

1369.0013 

131.1618 

3 

1680.0197 

10 

1374-4718 

I 3 I -4 2 3 6 

4 

16S6.0792 

11 

I379-9532 

131.6854 

5 

1692.1497 

42 ft- 

I385-4456 

I3I-9472 

6 

1698.2311 

1 

1390.9488 

132.209 

7 

I704-3334 

2 

1396463 

132.4708 

8 

I71O.4267 

3 

1401.9881 

132.7326 

9 

I 7 l6 -54°8 

4 

1407.5241 

I32-9944 

10 

1722.6658 

5 

1413.0709 

133.2562 

11 

1728.8017 

6 

1418.6287 

I33-5I8 

47 ft- 

I734-9486 

7 

1424.1974 

I33-7798 

1 

1741.1063 

8 

1429.777 

i 34-°4 i 6 

2 

1747.275 

9 

1435-3676 

I34-3034 

3 

I753-4546 

10 

1440.969 

134.5652 

4 

I759-645I 

11 

1446.5813 

134.827 

5 

1765.8464 

43/6 

1452.2046 

135.0888 

6 

7 

1772.0587 

1778.2819 

1 

1457.8387 

i35-35o6 

8 

1784.516 

1790.7611 

1797-017 

1803.2838 

1809.5616 

1815.8502 

1822.1498 

1828.4603 

2 

1463-4838 

135.6124 

Q 

3 

4 

5 

6 

7 

8 

1469.1398 

1474.8066 

1480.4844 

1486.1731 

1491.8717 

I497-5833 

135- 8742 
136.136 

136- 3978 
1366596 
136.9214 

1 37 - i 83 2 

y 

10 

11 

48/6 

1 

2 

3 

9 

1503-3047 

137-445 

4 

1834.7817 

10 

11 

1509 037 
1514.7802 

137.7068 

137.9786 

5 

6 

1841.1139 

1847.4571 

44 ft- 

1520.5344 

138.2304 

7 

1853.8112 

1 

1526.2994 

138.4922 

8 

1S60.1763 

2 

1532-0754 

I38-754 

9 

1866.5522 

3 

1537-8623 

139 oi 58 

10 

1872.939 

4 

1543.66 

139.2776 

11 

1879-3367 

5 

1549.4687 

139-5394 

49/6 

1885.7454 

6 

1555.2883 

139.8012 

1 

1892.1649 

7 

1561.1188 

140.063 

2 

1898 5954 

8 

1566.9603 

140.3248 

3 

1905 0368 

9 

1572.8126 

140.5866 

4 

1911.4897 

10 

1578-6756 

140.8484 

5 

I 9 I 7-95 22 

11 

I584-5499 

141.1102 

6 

1924.4263 

45/6 

I590-435 

I4I-372 

7 

1930.9113 

1 

1596-3309 

141.6338 

8 

I937-4073 

2 

1602 2378 

141.8956 

9 

I 943-9 I 4 2 

3 

1608.1556 

142.1574 

10 

1950.4318 

4 

1614.0843 

142.4192 

11 

1956.9604 

5 

| 1620.0238 

142.681 

50/6 

1 1963-5 


y* 


ClRCUM. 

Feet. 

142.9428 

143.2046 

143.4664 

143.7282 

143- 99 
144.2518 
144 5136 

1 44- 77S4 
145.0372 
145.299 
145.5608 
145.8226 
146.0844 
146.3462 
146.608 
146.8698 
147.1316 
147-3934 
i47- 6 552 
I47-9I7 
148.1788 
148.4406 
148.7024 
148.9642 
149.226 
149.4878 
149.7496 
150.0114 
150.2732 
I50-535 
150.7968 
151.0586 
151.3204 
151.5822 
151.844 
152.1058 
152.3676 
152.6294 
152.8912 
I53-I53 
I53-4I48 
153.6766 

153 9384 

154 2002 
154 462 
154.7238 

154 9856 

155 2474 
155 5092 

155- 77I 
156.0328 
156.2946 

156- 5564 

156.8182 

157- 08 

























258 SIDES OF SQUARES EQUAL TO AREAS. 


Sides of Squares-equal in. Area to a Circle. 

Diameter from 1 to 100. 


Diam. 

Side of Sq. 

Diam. 

Side of Sq. 

Diam. 

Side of Sq. 

Diam. 

Side of Sq. 

I 

.8862 

14 

12.4072 

27 

23.9281 

40 

35 - 449 1 

X 

1.1078 

X 

12.6287 

X* 

24.1497 

X 

35.6706 

X 

1-3293 

X 

12.8503 

X 

24.3712 

X 

35.8922 

X 

1-5509 

X 

13.0718 

X 

24.5928 

X 

36.1137 

2 

1.7724 

15 

13-2934 

28 

24.8144 

41 

36-3353 

X 

1.994 

X 

13-515 

X 

25-0359 

X 

36-5569 

X 

2.2156 

X 

13-7365 

X 

25-2575 

X 

36.7784 

X 

2 - 437 1 

X 

13-9581 

X 

25-479 

X 

37 

3 

2.6587 

16 

14.1796 

2 9 , 

25.7006 

42 

37.2215 

X 

2.8802 

X 

14.4012 

X 

25.9221 

X 

37 - 443 1 

X 

3.1018 

X 

14.6227 

X 

26.1437 

X 

37.6646 

X 

3 - 3 2 33 

X 

14.8443 

X 

26.3653 

X 

37.8862 

4 

3-5449 

17 

15.0659 

30 

26.5868 

43 

38.1078 

X 

3.7665 

X 

15.2874 

X 

26.8084 

X 

38.3293 

X 

3.988 

X . 

15-509 

X 

27.0299 

X 

38.5509 

X 

4.2096 

X 

15-7305 

X 

27-2515 

X 

38.7724 

5 

4 - 43 11 

18 

15-9521 

31 

27-473 

44 

38.994 

X 

4.6527 

X- 

16.1736 

X 

27.6946 

X 

39-2155 

X 

4.8742 

X 

16.3952 

X 

27.9161 

X 

39-4371 

X 

5-0958 

X 

16.6168 

X 

28.1377 

X 

39-6587 

6 

5 ’ 3 T 74 

19 

16.8383 

32 

28.3593 

45 

39.8802 

X 

5-5389 

X 

17.0599 

X 

28.5808 

X 

40.1018 

X 

5-7605 

X 

17.2814 

X 

28.8024 

X 

40.3233 

X 

5.982 

X 

17-503 

X 

29.0239 

X 

40.5449 

7 

6.2036 

20 

17-7245 

33 

29-2455 

46 

40.7664 

X 

6.4251 

X 

17.9461 

X 

29.467 

X 

40.988 

X 

6.6467 

X 

18.1677 

X 

29.6886 

X 

41.2096 

X 

6.8683 

X 

18.3892 

X 

29.9102 

X 

41.4311 

8 

7.0898 

21 

18.6108 

34 

30.1317 

47 

41.6527 

X 

7 - 3 ii 4 

X 

18.8323 

X 

30-3533 

X 

41.8742 

X 

7-5329 

, X 

19-0539 

X 

30-5748 

X 

42.0958 

X 

7-7545 

X 

19.2754 

X 

30.7964 

X 

42.3173 

9 

7.976 

22 

19.497 

35 

31.0179 

48 

42.539 

X 

8.1976 

X 

19.7185 

X 

3 1 -2395 

X 

42.7604 

X 

8.4192 

X 

19.9401 

-X 

31.4611 

X 

42.982 

X 

8.6407 

X 

20.1617 

X 

31.6826 

X 

43.2036 

10 

8.8623 

23 

20.3832 

36 

31.9042 

49 

434251 

X 

9.0838 

X 

20.6048 

X 

32.1257 

X 

43.6467 

X 

9-3054 

X 

20.8263 

X 

323473 

X 

43.8682 

X 

9.5269 

X 

21.0479 

X 

32.5688 

X 

44.0898 

II 

9-7485 

24 

21.2694 

37 

32.7904 

50 

44-3113 

X 

9-97 

X 

21.491 

X 

33.0112 

X 

44-5329 

X 

10.1916 

X 

21.7126 

X 

33-2335 

X 

44-7545 

X 

10.4132 

X 

21.9341 

X 

33-4551 

X 

44.976 

12 

10.6347 

25 

22.1557 

38 

33.6766 

51 

45.1976 

X 

10.8563 

X 

22.3772 

X 

33.8982 

X 

45.4191 

X 

11.0778 

X 

22.5988 

X 

34-1197 

X 

45.6407 

X 

11.2994 

X 

22.8203 

X 

34 - 34 1 3 

X 

45.8622 

13 

11.5209 

26 

23.0419 

39 

34.5628 

52 

46.0848 

X 

11.7425 

X 

23.2634 

X 

34.7884 

X 

46.3054 

X 

11.9641 

X 

23-485 

X 

35.006 

X 

46.5269 

X 

12.1856 

X 

23.7066 

X 

35-2275 

X 

46.7485 


























SIDES OF SQUARES EQUAL TO AREAS. 259 


Diam. 

Side of Sq. 1 

Diam. 

Side of Sq. 

Diam. 

Side of Sq. 

Diam. 

Side of Sq. 

53 

46.97 

65 

57.6047 

77 

68.2395 

89 

78.8742 

A 

47.1916 

3 ^ 

57.8263 

A 

68.461 

A 

79-0957 

a 

47 - 4 I 3 I 

A 

58.0479 

A 

.68.6826 

A 

79 - 3 I 73 

a 

47-6347 

% 

58.2694 

% 

68.9041 

A 

79-5389 

54 

47.8562 

66 

58.491 

78 

69.1257 

no 


a 

48.0778 

A 

58.7125 

X 

69-3473 

V 

/V* /'- WJ 4 
70 082 

a 

48.2994 

A 

58.9341 

X 

69.5688 

/4 

14 

80 . 207 C: 

a 

48.5209 

% 

59 -I 556 

% 

69.7904 

A 

80.4251 

55 

48.7425 

67 

59-3772 

19 

7O.OII9 

H 

48.964 

X 

59-5988 

A 

70-2335 

9 1 

80.6467 

A 

49.1856 

A 

59.8203 

A 

7°-455 

A 

80.8682 

A 

49.4071 

% 

60.0419 

A 

70.6766 


81.0898 

56 

49.6287 

68 

60.2634 

80 

70.8981 

A 

81.3113 

3 € 

49-8503 

A 

60.485 

A 

71.1197 

92 

Si.5329 

Vi 

50.0718 

A 

60.7065 

A 

7 I- 34 I 3 

A 

81.7544 

A 

50-2934 

% 

60.9281 

% 

71.5628 

A 

81.976 

57 

50 . 5 I 49 

69 

61.1497 

81 

71.7844 

A 

82.1975 

A 

50.7365 

3 ^ 

61.3712 

A 

72.0059 

91 

82.41Q1 

A 

50958 

A 

61.5928 

A 

72.2275 

A 

82.6407 

A 

51.X796 

% 

61.8143 

A 

72.449X 

A 

82.8622 

58 

51.4012 

70 

62.0359 

82 

72.6706 

A 

83.0838 

A 

51.6227 

A 

62.2574 

A 

72.8921 



5 I -8443 

A 

62.479 

A 

73- II 37 

94 

1/ 

83-3053 

A 

59 

52.0658 

52.2874 

\ X 

71 

62.7006 

62.9221 

A 

83 

73-3353 

73-5568 

/i 

A, 

A. 

8q. 5260 
83.7484 

8 q.Q 7 

A 

52.5089 

3 € 

63- i 437 

X 

73-7784 

/4 


X 

52.7305 

A 

63.3652 

X 

73-9999 

95 

84.1916 

A 

52.9521 

% 

63.5868 

A 

74.2215 

3 ^ 

84.4131 

60 

53 .I 736 

72 

63.8083 

84 

74-4431 

K 

84.6347 

A 

53.3952 

; X 

64.0299 

3 ^ 

74.6647 

6 A 

84.8502 

A 

53.6167 

! X 

64.2514 

X 

74.8862 

96 

85.0778 

% 

53-8383 

' % 

644730 

A 

75.1077 

3 £ 

85-2993 

61 

54-0598 

73 

64 6946 

85 

75-3293 

A 

85.5209 

X 

54.2814 

i A 

64.9161 

X 

75-5508 

A 

85-7425 

A 

54-503 

A 

65 .I 377 

A 

75-7724 

91 

8^.0646 

% 

54-7245 

% 

65-3592 

A 

75-9934 

A 

86.185 

62 

54.9461 

74 

65.5808 

86 

76.2155 

x; 

86.4071 

A 

55 -I 676 

3 ^ 

65.8023 

X 

76.4371 

A 

86.6289 

A 

55.3892 

A 

66.02.qq 

A 

76.6586 

rsQ 


A 

63 

55 . 6 I 07 

55-8323 

% 

75 

66.2455 

66.467 

V* 

co\ 

CO 

76.8802 

77.1017 

9 6 

A 

A 

0O.0502 

87.0718 

8- 7 .- 20-2 

X 

X 

56.0538 

56.2754 

A 

A 

66 6886 
66.9104 

X 

• X 

77-3233 

77-5449 

A 

87.5449 

% 

56.497 

X 

67 - i 3 i 7 

A 

77.7664 

99 

87.7364 

64 

56.7185 

76 

67-3532 

88 

77.988 

tC 

1 / 

87.958 

X 

56.9401 

A 

67.5748 

A 

78.2095 


00.1790 

A 

57 - i6 i 6 

V \ 

67.7964 

A 

78.4316 

% 

00.4011 

% 

57-3832 

\ % 

68.0179 

A 

78.6526 

100 

88.6227 


^Application, of Ta/ble. 

To Ascertain a Square tliat Las same Area as a Given 
„ Circle. 

Illus. —If side of a square that has same area as a circle of 73.25 ins. is required. 
By Table of Areas, page 233, opposite to 73.25 is 4214.11; and in this table is 
64.9161, which is side of a square having same area as a circle of that diameter. 






























26 o 


LENGTHS OF CIRCULAR ARCS, 


Lengths of Circular 4Arcs, up to a Semicircle. 
Diameter of a Circle — i, and divided into 1000 equal Parts. 


H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. * 

H’ght. 

Length. 

H’ght. 

Length. 

.1 

1.026 45 

•15 

I.05896 

.2 

I.IO348 

•25 

I-I59I2 

•3 

; 1.22495 

.IOI 

1.026 98 

•151 

1-059 73 

.201 

I. IO4 47 

.251 

I.16033 

.301 

11.22635 

.102 

I.O27 52 

.152 

1.06051 

.202 

I.IO548 

.252 

1.161 57 

.302 

1.227 76 

.103 

I.02806 

•153 

1.0613 

.203 

I.1065 

.253 

1.162 79 

•303 

1.22918 

.104 

1.0286 

•154 

1.062 09 

.204 

I-I07 52 

.254 

1.164 02 

•304 

1.23061 

.105 

I.O29 14 

•155 

1.062 88 

.205 

I.I0855 

•255 

1.165 26 

•305 

1.23205 

.106 

1.029 7 

.156 

1.06368 

.206 

I.IO958 

.256 

1.16649 

•306 

1-23349 

.107 

x.030 26 

•157 

1.06449 

.207 

I.II062 

•257 

1.16774 

•307 

1.23494 

.108 

1.030 82 

.158 

1-0653 

.208 

I.II165 

.258 

1.16899 

.308 

1.23636 

.109 

I-03I39 

•159 

1.06611 

.209 

I.II269 

•259 

1.170 24 

•309 

1.2378 

.11 

I.03I 96 

.16 

1.066 93 

.21 

I-H3 74 

.26 

1-1715 

•3i 

1.23925 

.III 

I.O32 54 

.161 

1.06775 

.211 

1,11479 

.261 

1-172 75 

•311 

1.240 7 

.112 

I.O33 12 

.162 

1.06858 

.212 

1.11584 

.262 

1.17401 

.312 

1.24216 

•113 

1-033 71 

.163 

1.06941 

.213 

1.11692 

•263 

1.17527 

•3i3 

1.2436 

.114 

1-0343 

.164 

1.07025 

.214 

1.11796 

.264 

1-17655 

•3i4 

1.24506 

•US 

1-0349 

.165 

1.07109 

-215 

1.119 04 

.265 

1.17784 

•3i5 

1.24654 

.Il6 

1-035 51 

.166 

1.071 Q4 

.2l6 

1.12011 

.266 

1.179 12 

•3 l6 

1.24801 

.117 

1.036 11 

.167 

1.072 79 

.217 

1.12118 

.267 

1.1804 

•3i7 

1.249 46 

.Il8 

1.036 72 

.168 

1-07365 

.2l8 

1.12225 

.268 

1.1S162 

.318 

1.25095 

.119 

1-037 34 

.169 

1.074 51 

.219 

1.123 34 

.269 

1.18294 

•3i9 

1.25243 

.12 

1-037 97 

•17 

1-07537 

.22 

1.12445 

.27 

1.184 28 

•32 

1-25391 

.121 

1.0386 

.171 

1.076 24 

.221 

1-125 56 

.271 

i-i 85 57 

•321 

1-25539 

.122 

1.03923 

.172 

I.O77 H 

.222 

1.126 63 

.272 

1.186 88 

.322 

1.256 86 

.123 

1.03987 

•173 

I.O7799 

.223 

I-I27 74 

•273 

1.188 19 

•323 

1.25836 

.124 

1.040 51 

.174 

I.O7888 

.224 

1.12885 

.274 

1.18969 

•324 

1.25987 

.125 

1.04116 

•175 

I.O7977 

.225 

1.129 97 

•275 

1.19082 

•325 

1.26137 

.126 

1.04181 

.176 

I.08066 

.226 

1.13108 

.276 

1.192 14 

.326 

1.262 86 

.127 

1.042 47 

.177 

I.o8l 56 

.227 

1-13219 

.277 

1-193 45 

•3 2 7 

1.26437 

.128 

1-04313 

.178 

1.082 46 

.228 

I-I333I 

.278 

I-I94 77 

.328 

1.265 88 

.129 

1.0438 

.179 

I-083 37 

.229 

I-I3444 

•279 

1.1961 

•329 

1.2674 

•13 

1.04447 

.18 

I.08428 

•23 

I-I3557 

.28 

I-I97 43 

•33 

1.268 92 

•131 

1.045 15 

.181 

I.085 19 

.231 

1.13671 

.281 

1.19887 

•33i 

1.270 44 

.132 

1.045 84 

.182 

1.086 II 

.232 

1.13786 

.282 

1.20011 

•332 

1.271 96 

•133 

1.04652 

.183 

I.08704 

•233 

1-13903 

•283 

1.20146 

•333 

1.27349 

.134 

1.04722 

.184 

I.08797 

•234 

1.1402 

.284 

1.202 82 

•334 

1.27502 

•135 

1.04792 

.185 

I .OSS 9 

•235 

1.14136 

.285 

1.204 19 

•335 

1.27656 

.136 

1.048 62 

.186 

1.089 84 

•236 

1.14247 

.286 

1.20558 

•336 

1.278 1 

•137 

1.04932 

.187 

I.090 79 

•237 

1-14363 

.287 

1.20696 

•337 

1.27064 

.138 

1.05003 

.188 

1-091 74 

.238' 

1.1448 

.288 

1.208 28 

•338 

1.281 18 

•139 

1-050 75 

.189 

1.092 69 

•239 

i-145 97 

.289 

1.20967 

•339 

1.282 73 

.14 

1-05147 

.19 

1.09365 

.24 

I-I47I4 

.29 

1.21202 

•34 

1.284 28 

.141 

1.052 2 

.191 

1.09461 

.241 

1.14831 

.291 

1.2123Q 

•34i 

1.285 83 

.142 

1.05293 

.192 

1-095 57 

.242 

1.149 49 

.292 

1.21381 

•342 

1.28739 

•143 

1-05367 

•193 

1.096 54 

•243 

1.15067 

•293 

1.2152 

•343 

1.28895 

.144 

1.054 41 

.194 

1.09752 

•244 

1.151 86 

•294 

1.21658 

•344 

1.200 52 

•145 

1.055 16 

•195 

1.0985 

•245 

1.15308 

•295 

I-2I794 

•345 

I.2Q2 0Q 

.146 

I-0559I 

.196 

1.09949 

.246 

1.154 29 

.296 

1.21926 

•346 

1.293 66 

.147 

1.05667 

.197 

1.10048 

•247 

I-I55 49 

•297 

1.22061 

•347 

I.295 23 

.X48 

1-05743 

.198 

1.10147 

.248 

1.1567 

.298 

1.22203 

•348 

I.29681 

.149 

1.05819 

.199 

1.102 47 

•249 

1-I579I 

•299 

1.22347 

•349 

1-29839 




























LENGTHS OF CIRCULAR ARCS. 


26l 


H’ght. | 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

•35 

I.29997 

.38 

I.348 99 

.41 

I.4OO 77 

•44 

i -455 12 

•47 

1.51185 

• 35 i 

1.301 56 

.381 

I.35068 

• 41 1 

I.402 54 

•441 

1.45697 

•471 

I- 5 I 3 78 

.352 

I- 303 I 5 

.382 

I -352 37 

.412 

I.40432 

442 

145883 

•472 

I. 5 I 5 7 I 

•353 

I.30474 

•383 

I -35406 

•413 

I.406 I 

•443 

1.460 69 

•473 

1.517 64 

•354 

1 - 3 o 6 34 

•384 

i -355 75 

.414 

1.407 88 

•444 

1-46255 

•474 

1.519 58 

•355 

r -30794 

•385 

1-35744 

•415 

1.409 66 

•445 

1.46441 

•475 

I -521 52 

•356 

I -30954 

.386 

I- 359 I 4 

.416 

i- 4 ii 45 

•446 

1.46628 

•476 

I -523 46 

•357 

I - 3 11 15 

•387 

1.36084 

.417 

1-41324 

•447 

1.46815 

•477 

1.525 41 

•358 

1.3L2 76 

.388 

1-362 54 

.418 

1-41503 

•448 

1.470 02 

.478 

1.527 36 

•359 

I- 3 I 437 

•389 

1.36425 

•419 

1.41682 

449 

1.47189 

•479 

I- 5293 I 

•36 

I - 3 T 5 99 

•39 

1.365 96 

42 

1.41861 

•45 

1-473 77 

.48 

.481 

.482 

•483 

.484 

•485 

.486 

.487 

I- 53 I 26 

.361 

1.31761 

• 39 i 

1.36767 

.421 

1.42041 

451 

1475 65 

i -533 22 
i -535 18 

.362 

I - 3 I 9 2 3 

•392 

I -36939 

.422 

1.422 22 

•452 

1-477 53 

•363 

1.320 86 

•393 

r- 37 1 11 i 

•423 

1.42402 

•453 

1.47942 

1-537 14 
i -539 1 
1.541 06 

•364 

1.32249 

•394 

1-37283 

•424 

1.42583 

•454 

1.48131 

•365 

1-32413 

J -395 

1-374 55 

•425 

1.427 64 

•455 

1.4832 

.366 

I -325 77 

• 39 6 

1.37628 

.426 

1.42945 

•456 

1.48509 

I- 543 02 

•367 

1.32741 

•397 

1.37801 

.427 

143127 

•457 

1.486 99 

1-544 99 
1.54696 

i -548 93 
1-5509 

.368 

•369 

1.32905 

1.33069 

• 39 s 

•399 

i -379 74 
1.38148 

.428 

•429 

I -43309 
1.434 91 

•458 

■459 

1.488 89 
1.490 79 

•489 

•49 

•37 

1-332 34 

•4 

1.383 22 

•43 

i- 43 6 73 

.46 

1.492 69 

• 49 i 

1.55288 

• 37 i 

1-33399 

.401 

1.38496 

• 43 i 

1.438 56 

.461 

1.4946 

.492 

1.554 86 

•372 

i -335 64 

.402 

1.38671 

•432 

1.44039 

.462 

1.49651 

•493 

i -556 85 

•373 

1-3373 

•403 

1.388 46 

•433 

1.442 22 

•463 

1.498 42 

•494 

I -558 54 

•374 

1.33896 

.404 

1 39021 

•434 

1.44405 

•464 

1-50033 

•495 

1.56083 

•375 

1.34063 

•405 

I- 39 1 96 

•435 

1.445 89 

•465 

1.502 24 

•496 

1.562 82 

•376 

1.342 29 

.406 

1-393 72 

•436 

1-447 73 

.466 

1.50416 

•497 

1.56481 

•377 

I- 3439 6 

■407 

1-395 48 

•437 

1-449 57 

•467 

1.506 08 

.498 

1.566 8 

•378 

1-345 63 

.408 

1.39724 

•438 

1-45142 

.468 

1.508 

•499 

1.568 79 

•379 

I - 3473 I 

•409 

1 1-399 

•439 

I -453 27 

•469 

1.50992 

•5 

1.570 79 


To ^Ascertain HLeragtli of an ^Ax*c of a Circle "by pre¬ 
ceding Table. 

Rule. —Divide height by base, find quotient in column of heights, take 
length for that height opposite to it in next column on the right hand. 
Multiply length thus obtained by base of arc, and product will give length. 

Example. —What is length of an arc of a circle, base or span of it being 100 feet, 
and height 25 ? 

25 = 100 = . 25; and .25, per table, = 1.15912, length of base, which, multiplied by 
100 = 115.912 feet. 

When , in division of a height by base , the quotient has a remainder after 
third place of decimals , and great accuracy is required. 

Rule. —Take length for first three figures, subtract it from next following 
length; multiply remainder by this fractional remainder, add product to 
first length, and sum will give length for whole quotient. 

Example. —What is length of an arc of a circle, base of which is 35 feet, and 
height or versed sine 8 feet? 

8 -f- 35 = .228 571 4; tabular length for .228 = 1.13331, and for .229 = 1. 13444, 
the difference between which is .00113. Then . 5714 X -ooi 13 = .000 645 682. 

Hence .228 =1.13331, 

and .0005714= .000645682 

1.133955682, the sum by which base of 
arc is to be multiplied; and 1.133955682 X 35 = 39-688 45 feet. 


























262 


LENGTHS OF CIRCULAR ARCS. 


Xjengtlis of Circular .Arcs from 1 ° to 180 °. 
{Radius = i.) 


Degrees. 

Length. 

Degrees. 

Length. 

| j Degrees. 

Length. 

Degrees. 

Length. 

I 

.OI74 

46 

.8028 

9 1 

I.5882 

136 

2.3736 

2 

•0349 

47 

.8203 

92 

• I.6057 

137 

2 - 39 II 

3 

.0524 

48 

•8377 

93 

I.6231 

138 

2.4085 

4 

.0698 

49 

•8552 

94 

I.6406 

139 

2.426 

• 5 

.0873 

50 

.8727 

95 

96 

97 

98 

99 

I.6581 

I -6755 

I.693 

1.7104 
I.7279 

140 

2-4435 

6 

.OI47 

5 i 

.89OI 

I 4 I 

» 2.4609 

t> CO c 

.0222 

.0396 

•0571 

52 

53 

54 

.9076 

•925 

.9424 

142 

143 

144 

2.4784 

2.4958 

2-5133 


55 

•9599 

100 

1-7453 

145 

2.5307 

10 

•1745 

56 

•9774 

101 

1.7628 

146 

2.5482 

11 

.I92 

57 

•9948 

102 

1.7802 

147 

2.5656 

12 

.2094 

.2269 

58 

1.0123 

103 

1.7977 

148 

2-5831 

L 3 

59 

1.0297 

104 

1.8151 

149 

2.6005 

14 

15 

•2443 

.2618 

60 

1.0472 

x'ol 

1.8326 

1.85 

1.8675 

1.8849 

1.9024 

150 

2.618 

16 

17 

.2792 

.2967 

61 

62 

1.0646 

1.0821 

10 7 

108 

151 

152 

2-6354 

2.6529 

18 

*9 

•3141 

• 33 l6 

63 

64 

1.0995 

1.117 

109 

153 

154 

2 6703 
2.6878 

20 

21 

• 349 i 

•3665 

•384 

.4014 

.4189 

65 

66 

i -1345 

1.1519 

no 

in 

I - 9 I 99 

1-9373 

155 

156 

2.7053 

2.7227 

22 

67 

1.1694 

112 

1.9548 

157 

2.7402 

2^ 

68 

1.1868 

1 13 

1.9722 

158 

2.7576 

O 

24 

69 

1.2043 

114 

1.9897 

159 

2-7751 

25 

26 

27 

•4363 

•4538 

.4712 

70 

7 1 

72 

1.2217 
1.2392 
1.2566 

11 5 

116 

117 

118 

2.0071 

2.0246 

2.042 

160 

161 

162 

2.7925 

2.81 

2.8274 

28 

.4887 

73 

1.2741 

2-0595 

2.0769 

163 

2.8449 

29 

.5061 

74 

1-2915 

n 9 

164 

2.8623 

3 ° 

•5236 

75 

1.309 

120 

2.0944 

165 

2.8798 

3 i 

• 54 i 

76 

1.3264 

121 

2.1118 

166 

2.8972 

32 

•5585 

77 

1-3439 

122 

2.1293 

167 

2.9147 

33 

•5759 

78 

1-3613 

123 

2.1467 

168 

2.9321 

34 

•5934 

79 

1.3788 

124 

2.1642 

169 

2.9496 

35 

36 

37 

38 

39 

.6109 

.6283 

.6458 

.6632 

.6807 

80 

81 

82 

83 

84 

i-39 6 3 

I- 4 I 37 

1 - 43 1 2 
1.4486 
1.4661 

125 

126 

127 

128 

129 

2.1817 

2.1991 
2.2166 
2.2304 
2.2515 

170 

1 7 1 

172 

173 

174 

2.967 

2.9845 

3 002 

3 OI 94 

3-0369 

40 

.6981 

85 

1-4835 

130 

2.2689 

175 

3-0543 

41 

•7156 

86 

i-5°i 

131 

2.2804 

176 

3 0718 

42 

•733 

87 

1-5184 

132 

2.3038 

177 

3.0892 

43 

•7505 

88 

1-5359 

133 

2.3213 

178 

3.1067 

44 

.7679 

89 

1-5533 

134 

2-3387 

179 

3-1241 

45 

•7854 

90 

1.5708 

135 

2-3562 || 

180 

3,1416 


To Ascertain Length of a Circular Arc by Table. 

Rule.—F rom column opposite to degrees of arc, take length, and multi¬ 
ply it by radius of circle. 

Example. —Number of degrees in an arc are 45 0 , and diameter of circle 5 feet. 
Then .7854 tab. length x 5 A 2 = 1.9635 feet. 




































LENGTHS OF ELLIPTIC ABCS, 


263 


Lengths of Elliptic .A-rcs. 

Up to a Semi-ellipse. 

Transverse Diameter =. 1, and divided into 1000 equal Parts. 


H’ght. 

Length. | 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

.1 

1.041 62 

•15 

1-0933 

.2 

I-I 50 I 4 

•25 

1.21136 

•3 

I.27669 

.IOI 

1.042 62 

•151 

1.094 48 

.201 

I-I 5 I 3 I 

.251 

I.21263 

•301 

I.2780.5 

.102 

I.O43 62 

.152 

1.09558 

.202 

I.15248 

.252 

I- 2 I 39 

.302 

I.27937 

.103 

1.044 62 

•153 

1.09669 

•203 

I.15366 

•253 

I- 2 I 5 17 

•303 

I.280 71 

.IO4 

I.O45 62 

•154 

1.097 8 

.204 

I.15484 

•254 

I.21644 

•304 

I.28205 

.105 

I.O46 62 

•155 

i.oqSqi 

.205 

I.I56 02 

•255 

1.217 72 

•305 

I.28339 

.106 

I.O47 62 

.156 

1.100 02 

.206 

I - I 57 2 

.256 

1.219 

•306 

1.284 74 

.IO7 

I.O48 62 

•157 

1.101 13 

.207 

1.15838 

•257 

1.220 28 

•307 

1.28609 

.IOS 

I.O49 62 

.158 

1.102 24 

.208 

1-15957 

•258 

1.221 56 

•308 

1.28744 

.IO9 

I.05063 

•159 

1-10335 

.209 

1.160 76 

•259 

1.222 84 

•309 

1.288 79 

.11 

I.05164 

.16 

1.104 47 

.21 

1.16196 

.26 

I.224 I 2 

• 3 i 

1.29014 

.III 

I.05265 

.161 

1.1056 

.211 

1.163 16 

.261 

1.2254I 

•311 

1.29149 

.112 

I.05366 

.162 

1.106 72 

.212 

1.16436 

.262 

1.226 7 

• 3 12 

1.292 85 

■113 

I.05467 

.163 

1.107 84 

•213 

1-16557 

.263 

I.22799 

• 3 i 3 

1.294 21 

.114 

I.05568 

.164 

1.10896 

.214 

1.166 78 

.264 

I.22928 

• 3 i 4 

1-295 57 

•US 

I.05669 

-165 

1.110 08 

.215 

1.16799 

.265 

I -230 57 

• 3 i 5 

1.29603 

.Il6 

i-o 57 7 

.166 

1.hi 2 

.2l6 

1.1692 

.266 

i.231 86 

.316 

1.298 29 

.117 

1.058 72 

.167 

1.112 32 

.217 

1.17041 

.267 

1-23315 

• 3 i 7 

1.29965 

.Il8 

1.059 74 

.168 

1.H344 

.218 

I - I 7 1 63 

.268 

1-23445 

.318 

1.30102 

.HQ 

1.060 76 

.169 

1.11456 

.219 

1.17285 

.269 

1-235 75 

• 3 i 9 

1.30239 

.12 

1.061 78 

•17 

1.11569 

.22 

1.17407 

•27 

1-23705 

•32 

1.30376 

.121 

1.062 8 

.171 

1.11682 

.221 

1.17529 

.271 

1-23835 

•321 

1-30513 

.122 

1.06382 

.172 

1 • 11 7 95 

.222 

1.17651 

.272 

1.23966 

.322 

1-3065 

.123 

1.064 84 

•173 

1.11908 

.223 

1.17774 

•273 

1.24097 

•323 

1.30787 

.124 

1.065 86 

.174 

1.12021 

.224 

1.17897 

•274 

1.24228 

•324 

1.309 24 

.125 

1.06689 

•175 

1.12134 

.225 

1.1802 

•275 

1-24359 

•325 

1.31061 

.126 

1.06792 

.176 

1.12247 

.226 

1.18143 

.276 

1.2448 

•326 

1.31198 

.127 

1.068 95 

.177 

1.1236 

.227 

1.18266 

.277 

1.24612 

•327 

1-31335 

.128 

1.069 98 

.178 

1-124 73 

.228 

1.1839 

.278 

1.24744 

.328 

1-31472 

.129 

1.07001 

.179 

1.125 86 

.229 

1.185 14 

•279 

1.248 76 

•329 

1.3161 

•13 

1.07204 

.18 

1.12699 

•23 

1.18638 

.28 

1.250 1 

•33 

1.31748 

•131 

1.07308 

.l8l 

1.128 13 

.231 

1.18762 

.281 

1.25142 

• 33 i 

1.31886 

.132 

1.074 12 

.182 

1.12927 

.232 

1.1S886 

.282 

1-25274 

•332 

1.320 24 

• I 33 

1.075 16 

.183 

1-13041 

•233 

1.1901 

•283 

1.254 06 

•333 

1.321 62 

•134 

1.07621 

.184 

I - I 3 I 55 

•234 

i- I 9 I 34 

.284 

1-25538 

•334 

1-323 

•135 

1.077 26 

.185 

1.132 6q 

•235 

1.19258 

.285 

1.2567 

•335 

1.32438 

.136 

1.07831 

.186 

I - I 33 83 

.236 

1.19382 

.286 

1.25803 

•336 

1.325 76 

•137 

1.07937 

.187 

I - I 3497 

-237 

i-i 95 °6 

.287 

1.25936 

337 

1-32715 

.138 

1.080 43 

.188 

1.136 11 

•238 

1.1963 

.288 

1.26069 

•338 

1.32854 

•139 

1.08149 

.189 

1.13726 

•239 

i-i 97 55 

.289 

1.262 02 

•339 

I -32993 

.14 

1.08255 

.19 

1.13841 

.24 

1.1988 

.29 

1-26335 

•34 

1-33132 

.141 

1.083 62 

.191 

I-I 3956 

.24I 

1.200 05 

.291 

1.26468 

• 34 i 

1.332 72 

.142 

1.08469 

.192 

1.140 71 

.242 

1.2013 

.292 

1.26601 

•342 

1 - 334 12 

.143 

1.085 76 

•193 

1.141 86 

•243 

1.20255 

•293 

1.26734 

343 

1-335 52 

.144 

1.086 84 

.194 

1.14301 

•244 

1.2038 

.294 

1.26867 

•344 

1.33692 

.145 

1.08792 

•195 

1.14416 

•245 

1.20506 

•295 

1.27 

•345 

I -33833 

.146 

1.08901 

.196 

I-I 453 I 

.246 

1.206 32 

.296 

1-27133 

•346 

1-33974 

.147 

1.0901 

.197 

1.14646 

•247 

1.207 58 

•297 

1.27267 

•347 

I- 34 II 5 

.148 

1.09119 

.198 

1.14762 

.248 

1.208 84 

.298 

1.27401 

•348 

1.342 56 

.149 

1.092 28 

.199 

1.14888 

.249 

1.2101 

•299 

1-275 35 

•349 

1-34397 
























264 LENGTHS OF ELLIPTIC ARCS. 


H’ght. 

Length. 

H’ght 

Length. 

H’ght 

. Length. 

H’ght 

Length. 

[ H’ght.| Length. 

•35 

1-345 39 

•405 

I -425 33 

.46 

I.508 42 

•515 

I.59408 

•57 

1.68036 

• 35 i 

1.34681 

.406 

1.42681 

.461 

I.50996 

• 5 X 6 

x -595 64 

• 57 i 

1.68195 

•352 

1.348 23 

.407 

1.428 29 

.462 

I- 5 H 5 

•517 

1.5972 

•572 

[ 1-68354 

•353 

1- 349 6 5 

.408 

1.42977 

•463 

1-51304 

- 5 X 8 

1.598 76 

•573 

1.68513 

•354 

1.35108 

.409 

1- 43 1 25 

•464 

1-51458 

•519 

1.60032 

•574 

1.68672 

•355 

I- 3525 I 

.41 

i-432 73 

•465 

1.516 12 

•52 

1.601 88 

•575 

1.68831 

•356 

1-35394 

.411 

1.434 21 

1 .466 

1.51766 

.521 

1.60344 

•576 

1.6899 

•357 

1-35537 

.412 

1 -435 69 

.467 

1-5192 

.522 

1.605 

•577 

1.69149 

.358 

1-3568 

• 4 i 3 

1.437 J 8 

.468 

1.520 74 

•523 

1.60656 

•578 

1.69308 

•359 

i -358 23 

.414 

1.43867 

•469 

1.522 29 

•524 

1.60812 

•579 

1.69467 

•36 

I -35967 

• 4 i 5 

1.440 16 

•47 

1.523 84 

•525 

1.60968 

.58 

1.69626 

.361 

1.361 IX 

.416 

1.44165 

.471 

1-525 39 

•526 

1.611 24 

.581 

1.697 85 

.362 

1-36255 

.417 

1.443 14 

•472 

1.52691 

•527 

1.6128 

.582 

1.69945 

•363 

I -36399 

.418 

I -444 63 

•473 

1.52849 

.528 

1.61436 

•583 

1-70105 

•364 

I -36543 

.419 

1.44613 

•474 

1.53004 

•529 

1.61592 

•584 

1.702 64 

•365 

1.36688 

.42 

1.44763 

•475 

I - 53 I 59 

•53 

1.61748 

•585 

1.70424 

.366 

1-36833 

.421 

T - 449 I 3 

.476 

i -533 J 4 

• 53 i 

1.61904 

.586 

1.705 84 

•367 

1.36978 

.422 

1.45064 

■477 

i -534 69 

•532 

1.6206 

•587 

1-70745 

.368 

1.37123 

•423 

1.452 14 

•478 

1-53625 

•533 

1.622 16 

•588 

1.70905 

•369 

1.37268 

•424 

I -45364 

•479 

i- 5378 i 

•534 

1.623 72 

•5S9 

1.71065 

•37 

I- 374 I 4 

•425 

I- 455 I 5 

.48 

1-53937 

•535 

1.625 28 

•59 

1.71225 

• 37 i 

1.376 62 

.426 

1.45665 

.481 

I -540 93 

•536 

1.626 84 

I -591 

1.71286 

•372 

1.37708 

•427 

i- 458 i 5 

.482 

i -542 49 1 

•537 

1.6284 

•592 

1.71546 

•373 

I -37854 

.428 

1.45966 

•4S3 

I -54405 

•538 

1.62996 

j -593 

1.71707 

•374 

1.38 

•429 

1.461 67 

•484 

i -545 6i 

•539 

1.63152 

•594 

1.71868 

•375 

1-38146 

•43 

1.462 68 

•485 

1.54718 

•54 

1.633 °9 

•595 

1.720 29 

•376 

1.38292; 

• 43 i 

1.464 iq 

.486 

i -548 75 

• 54 i 

1.634 65 

•596 

1.7219 

•377 

1-38439 

•432 

1 -465 7 

•487 

1-55032 

•542 

1.636 23 

•597 

1-7235 

•378 

1-385 85 

•433 

1.467 21 

.488 

1-55189 

•543 

1.6378 

•598 

1.72511 

•379 

1.38732 

•4 FT 

1.468 72 

.489 

I -55346 

•544 

1-639 37 

•599 

1.726 72 

•38 

1.388 79 

•435 

1.47023 

•49 

1-555 03 

•545 

1.640 94 

.6 

1-72833 

.381 

1.390 24 

•436 

1.47174 

.491 

1.5566 

•546 

1.64251 

.601 

1.72994 

.382 

1.39169 

•437 

1.473 26 

•492 

1-55817 

•547 

1.644 °8 

.602 

1-73155 

•383 

I- 393 I 4 

•438 ; 

1.474 78 

•493 

x -559 74 

•548 

1-645 65 

.603 

1-73316 

•384 

1-394 59 

•439 

^476 3 

•494 

1-56131 

•549 

1.647 22 

.604 

x -734 77 

•385 

1.39605 

•44 

1.47782 

•495 

1.562 89 

•55 

1.648 79 

•605 

1-73638 

•386 

1 -397 5 1 

.441 

1 -479 34 

•496 

1.56447 

• 55 i 

1.650 36 

.606 

x -73799 

•387 

1.39897 

•442 

1.480 86 

•497 

1.56605 

•552 

1.651 93 

.607 

i- 739 6 

•388 

1.40043 

•443 

1.482 38 

•498 

1.56763 

•553 

I -653 5 

.608 

1.741 21 

•389 

1.401 89 

•444 

1.48391 

•499 

1.56921 

•554 

1-65507 

.609 

1.742 83 

•39 

1-403 35 

•445 

1 -485 44 

•5 

1.57089 

•555 

1.65665 

.61 

1.744 44 

• 39 i 

1.40481 

•446 

1.48697 

.501 

x- 57 2 34 

•556 

1.65S23 

..611 

1.74605 

•392 

1.406 27 

•447 

1.4885 

• 5°2 

x -573 89 

•557 

1.65981 

.612 

1.74767 

•393 

I -407 73 

•448 

1.49003 

•503 

x -575 44 

•558 

1.661 39 

.613 

1.749 29 

•394 

1.409 19 

■449 

i- 49 i 57 

•504 

1.57699 

•559 

1.66297 

.614 

1.75091 

•395 

1.41065 

•45 

1.493 11 

•505 

i -578 54 

•56 

1.664 55 

.615 

1-75252 

•396 

1.412 11 

•451 

1.49465 

.506 

1.58009 

.561 

1.666 ii 

.616 

I- 754 I 4 

•397 

I - 4 I 3 57 

•452 

1.496 18 

•507 

1.58164 

.562 j 

1.667 7 1 

.617 

x -755 76 

•398 

1.4x504 

•453 

1.497 71 

•508 

x- 5§3 19 

•ks i 

1.66q 2 q 

.618 

I -75738 

•399 

I - 4 I 6 51 

•454 

1.49924 

•509 i 

x- 5§4 74 

•564 

1.67087 

.619 

x -759 

•4 

1.4179S 

•455 

1.50077 

• 5 1 1 

1.586 29 

•565 j 

1.67245 

.620 

1.76062 

.401 

I- 4 I 945 

■456 

1-5023 

•511 

1.58784 

'.566! 

1.674 03! 

.621 

1.762 24 

.402 

1.420 92 

•457 

1-50383 

.512 

I- 5 S 94 

•567 

1.67561 

.622 

1.763 86 

•403 

1.422 39 

• 45 s 

1-505 36 

• 5 L 3 

1.59096 

.568 1 

1.677 i 9 | 

.623 

1.765 48 

.404 

1.423 86, 

•459 

1.50689 j| 

•514! 

I -592 52 1, 

•569! 

1.678 77 | 

.624 ! 

1.767 1 






























































LENGTHS OF ELLIPTIC ARCS. 


265 


H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

.625 

I.768 72 

.68 

I.858 74 

•735 

1-950 59 

•79 

2.044 62 

•845 

2.14155 

.626 

I.77034 

.681 

I.86039 

•736 

I.952 28 

.791 

2.04635 

.846 

2.14334 

.627 

1.771 97 

.682 

1.862 05 

•737 

1-95397 

•792 

2.04809 

•847 

2-I45I3 

.628 

1-773 59 

.683 

I.863 7 

•738 

1.95566 

•793 

2.04983 

.848 

2.146 92 

.629 

1.775 21 

.684 

1 -865 35 

•739 

1 -957 35 

•794 

2.05157 

.849 

2.148 71 

•63 

1.776 84 

.685 

1.867 

•74 

1-95994 

•795 

2.053 3 1 

•85 

2-I505 

.631 

1.77847 

.686 

1.86866 

.741 

i.q6o 74 

.796 

2-05505 

.851 

2 . 1^2 2Q 

.632 

1.780 09 

.687 

1.87031 

•742 

1.062 44 

•797 

2.056 79 

.852 

2.I54 09 

•633 

1.781 72 

.688 

1.871 96 

-743 

1.964 14 

•798 

2-05853 

•853 

2.I5589 

•634 

11 783 35 

.689 

1.87362 

•744 

1.965 83 

•799 

2 060 27 

•854 

2-1577 

•635 

1.784 98 

.69 

1.87527 

•745 

I-967 53 

.8 

2.062 02 

.855 

2-1595 

.636 

1.7866 

.691 

1.87693 

.746 

1.96923 

.801 

2.063 77 

.856 

2.l6l 3 

•637 

1.788 23 

.692 

1.878 59 

•747 

1.97093 

.802 

2.06552 

•857 

2.163OQ 

.638 

1.789 86 

•693 

1.880 24 

•748 

1.972 62 

•803 

2.067 27 

.858 

2.16489 

•639 

1.791 49 

.694 

1.881 9 

•749 

1.97432 

.804 

0 

& 

q 

d 

•859 

2.166 68 

.64 

i-793 12 

•695 

1.88356 

•75 

1.976 02 

•805 

2.070 76 

.86 

2.168 48 

.641 

1-794 75 

.696 

1.885 22 

•75i 

1.97772 

.806 

2.07251 

.861 

2.17028 

.642 

1.70038 

.697 

1.886 88 

•752 

1-97943 

.807 

2.07427 

.862 

2.172 09 

•643 

1.79801 

.698 

1.888 54 

•753 

1.98113 

.808 

2.076 02 

.863 

2.173 89 

.644 

1.79964 

.699 

1.8902 

•754 

1.982 83 

.809 

2.07777 

.864 

2-1757 

•645 

1.801 27 

•7 

1.891 86 

•755 

I-984 53 

.81 

2-07953 

.865 

2.17751 

.646 

1.8029 

.701 

1-89352 

•756 

1.986 23 

.811 

2.081 28 

.866 

2.17932 

.647 

1.80454 

.702 

1.895 19 

•757 

1.98794 

.812 

2.083 04 

.867 

2.18113 

.648 

1.806 17 

•70.3 

1.89685 

•758 

i.q8q 64 

.813 

2.0848 

.868 

2.18294 

.649 

1.807 8 

•704 

1.89851 

•759 

I-99I34 

.814 

2.08656 

.869 

2.18475 

•65 

1.80943 

•705 

1.900 17 

.76 

1-99305 

I..815 

2.088 32 

•87 

2.18656 

.651 

1.811 07 

.706 

1.901 84 

.761 

1.994 76 

.816 

2.090 08 

.871 

2.18837 

.652 

1.812 71 

•707 

1-9035 

.762 

1.99647 

.817 

2.091 98 

.872 

2.19018 

•653 

i-8i4 35 

.708 

1.905 17 

•763 

i.qqS 18 

.818 

2.0936 

•873 

2.192 

•654 

1.81599 

•709 

1.906 84 

.764 

1.99989 

.819 

2-095 36 

•874 

2.193 82 

.655 

1.8x763 

•7i 

1.908 52 

•765 

2.001 6 

.82 

2.097 12 

•875 

2.19564 

.656 

1.81928 

.711 

1.910 19 

.766 

2.00331 

.821 

2.098 88 1 

.876 

2.IQ7 46 

•657 

1.82091 

.712 

I-9H 87 

.767 

2.005 02 

.822 

2.100 65 

•877 

2.19928 

.658 

1.82255 

.713 

I -9 I 3 55 

.768 

2.006 73 

.823 

2.102 42 

.878 

2.201 I 

•659 

1.824 19 

•7 I 4 

I -9 I 5 23 

.769 

2.008 44 

.824 

2.104 19 

•879 

2.202 92 

.66 

1.82583 

•7 I 5 

1.91691 

•77 

2.010 16 

1 -825 

2.105 96 

.88 

2.204 74 

.661 

1.82747 

.716 

1.91859 

.771 

2.on 87 

.826 

2.10773 

.881 

2.20656 

.662 

1.829 11 

.717 

1.92027 

•772 

2.01359 

1 -827 

2.1095 

.882 

2.20839 

.663 

1-830 75 

•718 

1-92195 

•773 

2.01531 

.828 

2.in 27 

.883 

2.21022 

.664 

1.8324 

.719 

1.92363 

•774 

2.01702 

.829 

2.113 04 

.884 

2 212 05 

.665 

1.83404 

.72 

I -9 2 5 3 1 

•775 

2.018 74 

•83 

2.11481 

.885 

2.213 88 

.666 

1.835 68 

.721 

1.927 

.776 

2.02045 

.831 

2.11659 

.886 

2.215 7 1 

.667 

1-837 33 

.722 

1.92868 

•777 

2.022 17 

^832 

2.11837 

.887 

2.21754 

.668 

1.83897 

•723 

1.93036 

•778 

2.023 89 

•833 

2.120 15 

.888 

2.21937 

.669 

1.84061 

.724 

1.93204 

•779 

2.025 61 

•834 

2.121 93 

.889 

2.221 2 

.67 

1.842 26 

•725 

1-933 73 

.78 

2.02733 

•835 

2.123 71 

.89 

2.22303 

.671 

1.84391 

.726 

i-935 4i 

.781 

2.02907 

L83 6 

2.125 49 

.891 

2.224 86 

.672 

1.845 56 

•727 

1-937 1 

.7S2 

2.0308 

•837 

2.127 2 7 

.892 

2,226 7 

•67 3 

1.8472 

.728 

1.938 78 

.783 

2.03252 

1 -838 

2.12905 

•893 

2.228 54 

.674 

1.848 85 

•729 

1.940 46 

•784 

2.03425 

•839 

2.13083 

•894 

2.23038 

•67S 

1-8505 

•73 

1.942 15 

•785 

2.035 98 

•84 

2.132 61 

•895 

2.232 22 

.676 

1.85215 

•73i 

I-943 83 

.786 

2.037 71 

| .841 

2-134 39 

.896 

2.23406 

'•677 

1-853 79 

•732 

i-945 52 

.787 

2.03944 

.842 

2.136 18 

•897 

2-2359 

.678 

1-855 44 

•733 

1.947 21 

.788 

2.041 17 

1-843 

2.13797 

.898 

2.23774 

.679 

1.85709 

•734 

1.9489 

.789 

2.0429 

1 -844 

2.139 76! 

•899 

2.239 58 


Z 




































266 


LENGTHS OF ELLIPTIC ARCS. 


H’ght. 

Length. | 

H’ght. 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

j H’ght. 

Length. 

•9 

2.241 42 

.92 

2.27803 

•94 

2 - 3 I 4 79 

.96 

2 352 41 

.98 

2.390 55 

.901 

2.24325 

.921 

2.27987 

.941 

2.31666 

.961 

2-354 3 1 

.981 

2.392 47 

.902 

2.245 08 

.922 

2.28l 7 

•942 

2.31852 

.962 

2.35621 

.982 

2-39439 

•903 

2.24691 

•923 

2.283 54 

•943 

2.32038 

.•963 

2.358 1 

•983 

2.39631 

.904 

2.248 74 

.924 

2.28537 

•944 

2.322 24 

.964 

2.36 

.984 

2.39823 

• 9°5 

2 25057 

•925 

2.287 2 

•945 

2.324 11 

• 9 6 5 

2.36191 

.985 

2.4OO 16 

.906 

2.2524 

.926 

2.28903 

.946 

2.325 98 

.966 

2.36381 

.986 

2.402 08 

• 9°7 

2.254 23 

.927 

2.29086 

•947 

2.32785 

.967 

2.365 71 

•987 

2.404 

.908 

2.25606 

.928 

2.292 7 

.948 

2.329 72 

.968 

2.367 62 

.988 

2.405 92 

.909 

2.257 89, 

.929 

2.29453 

•949 

2.3316 

.969 

2.36952 

.989 

2.407 84 









•99 

2.409 76 

.91 

2.259 72 

•93 

2.29636 

•95 

2.33348 

•97 

2.371 43 

.991 

2.4II 69 

.911 

2.26155 

• 93 i 

2.298 2 

•95 x 

2-335 37 

.971 

2-37334 

•992 

2.41362 

.912 

2-26338 1 

•932 

2.30004 

•952 

2.337 26 

.972 

2.37525 

•993 

2.41556 

•913 

2.265 21 

•933 

2.301 88 

•953 

2-339 15 

•973 

2.377 16 

•994 

2.41749 

.914 

2.26704 

•934 

2-303 73 

•954 

2.34104 

•974 

2.37908 

•995 

2.41943 

• 9 T 5 

2.268 S8 

•935 

2-305 57 

•955 

2-342 93 

•975 

2.381 

.996 

2.421 36 

.916 

2.270 71 

•936 

2.30741 

•956 

2.34483 

.976 

2.38291 

•997 

2.423 29 

• 9 X 7 

2.27254 

•937 

2.309 26 

•957 

2-346 73 

•977 

2.384 82 

.998 

2.425 22 

.9x8 

2.27437 

•938 

2.31111 

•958 

2.34862 

•978 

2.386 73 

•999 

2.427 15 

.919 

2.2762 

•939 

2.31295 

•959 

2 -350 5 1 

•979 

2.38864 

1. 

2.429 08 


To Ascertain I jengtli of an Elliptic Arc (riglit Semi- 
Ellipse) Toy preceding Table. 

Rule. — Divide height by base, lind quotient in column of heights, and 
take length for that height from next right-hand column. Multiply length 
thus obtained by base of arc, and product will give length. 

Example.— -What is length of arc of a semi-ellipse, base being 70 feet, and height 
30.10 feet ? 

30.10 - 4 - 70 = .43; and 43 ,per table, = 1.46268. 

Then 1.46268 X 70 = 102.3876 feet. 

When Curve is not that of a right Semi-Ellipse , Height being half of Trans¬ 
verse Diameter. 

Rule. —Divide half base by twice height, then proceed as in preceding 
example; multiply tabular length by twice height, and product will give 
length. 

Example.— What is length of arc of a semi-ellipse, height being 33 feet, and base 
60 feet? ^ ’ 

60-^2 = 30 , and 3 0-7-35 X 2 = .428, tabular length of which— 1.45966. 

Then 1.45966 X 35 X 2 = 102.1762 feet. 

When , in Division of a Height by Base , Quotient has a Remainder after 
third Place of Decimals , and great A ccuraey is required , 

Rule. —Take length for first three figures, subtract it from next following 
length; multiply remainder by this fractional remainder, add product to 
first length, and sum will give length for whole quotient. 

Example. — What is length of an arc of a semi-ellipse, base being 171 3 feet and 
height 125 feet? 

I 7 J - 3 ~ 2 — 85-65, and 125X2 = 250. 171.3 = 250 = .3426 ; tabular length for 

.342= 1.334 12, and i° r -343 = 1-335 52, the difference between which is .0014. 

Then .6 X -0014 = .0084. „ 

Hence, .342 =1.33412 
.0006= .0084 

1-34252 ■, the sum by which base of arc 
is to be multiplied; and 1.34252 X 171.3 = 229.973 676 feet. 























AREAS OF SEGMENTS OF A CIRCLE 


267 


Areas of Segments of a Circle. 


The Diameter of a Circle — 1, and divided into 1000 equal Parts. 


Versed 
Sine. 

Seg. Area. 

Versed 

Sine. 

Seg. Area. 

Versed 

Sine. 

Seg. Area. 

] Versed 
| Sine. 

Seg. Area. 

Versed 

Sine. 

Seg. Area. 

•OOI 

.OOOO4 

•052 

•015 56 

.103 

.042 69 

•154 

.07675 

.205 

.11584 

.002 

.OOO 12 

•053 

.01601 

.IO4 

.0431 

•155 

•077 47 

.206 

.11665 

.003 

.000 22 

•054 

.01646 

.105 

.04391 

.156 

.0782 

.207 

.11746 

.004 

.OOO34 

•055 

.01691 

.106 

•044 52 

•157 

.07892 

.208 

.118 27 

•005 

.OOO47 

.056 

•017 37 

.107 

•04514 

.158 

.07965 

.209 

.11908 

.006 

.OO062 

•057 

.01783 

.108 

•045 75 

•159 

.080 38 

.21 

• II 99 

.OO7 

.OOO 78 

.058 

•0183 

.109 

.04638 

.16 

.081 II 

.211 

.120 71 

.OOS 

.OOO95 

•059 

.Ol8 77 

.11 

•047 

• l6l 

.081 85 

.212 

• I2 i 53 

.009 

.OOI 13 

.06 

.OI924 

• III 

•047 63 

.162 

.082 58 

.213 

• 122 35 

.OI 

.OOI 33 

.061 

.01972 

.112 

.048 26 

.163 

.08332 

.214 

.12317 

.Oil 

•001 53 

.062 

.020 2 

•113 

.048 89 

.164 

.084 06 

.215 

.123 99 

.012 

•001 75 

•063 

.020 68 

.114 

•04953 

•165 

.0848 

.2l6 

.12481 

.013 

.001 97 

.064 

.021 17 

• 1 X 5 

.05016 

.166 

•085 54 

.217 

• 125 63 

.OI4 

.002 2 

•065 

.021 65 

.Il6 

.0508 

.167 

.086 29 

.218 

.126 46 

.015 

.002 44 

.066 

.022 15 

.117 

•05145 

.168 

.08704 

.219 

.127 28 

.Ol6 

.002 68 

.067 

.022 65 

.Il8 

.05209 

.169 

.087 79 

.22 

.128 11 

.017 

.002 94 

.068 

•023 15 

.119 

•052 74 

•17 

.088 53 

.221 

.128 94 

•Ol8 

.0032 

.069 

.023 36 

.12 

•05338 

.171 

.089 29 

.222 

.12977 

.OI9 

•00347 

•07 

.02417 

.121 

.05404 

.172 

.090 04 

.223 

.1306 

.02 

■00375 

.071 

.024 68 

.122 

.05469 

•173 

.0908 

.224 

•13144 

.021 

.00403 

.072 

.025 19 

.123 

•055 34 

.174 

•091 55 

.225 

.13227 

.022 

•00432 

•073 

•025 71 

.124 

.056 

•175 

.09231 

.226 

•i 33 ii 

.023 

.004 62 

•074 

.026 24 

.125 

.05666 

.176 

•09307 

.227 

•13394 

.024 

.004 92 

•075 

.026 76 

.126 

•057 33 

.177 

.09384 

.228 

•i 34 78 

.025 

•005 23 

.076 

.027 29 

.127 

•05799 

.178 

.0946 

.229 

• i35 62 

.026 

•00555 

•077 

.027 82 

.128 

.05866 

.179 

•095 37 

•23 

.136 46 

.027 

.005 87 

.078 

•02835 

.129 

•05933 

.18 

.09613 

.231 

• I 37 3 1 

.028 

.00619 

•079 

.028 89 

•13 

.06 

.l8l 

.oq6q 

.232 

•13815 

.O29 

•00653 

.08’ 

.02943 

•131 

.060 67 

.182 

.09767 

•233 

•139 

•°3 

.006 86 

.081 

.02997 

.132 

•061 35 

.183 

.09845 

•234 

.139 84 

•03I 

.00721 

.082 

•03052 

•133 

.062 03 

.184 

.099 22 

•235 

.14069 

.032 

•007 56 

•083 

.031 07 

•134 

.062 71 

.185 

.1 

.236 

•I 4 I 54 

•033 

.007 91 

.084 

.031 62 

•135 

•06339 

.186 

.100 77 

•237 

.142 39 

•034 

.00827 

.085 

.03218 

.136 

.06407 

.187 

•10155 

.238 

.143 24 

•035 

.008 64 

.086 

.032 74 

•137 

.064 76 

.188 

.102 33 

•239 

.14409 

.036 

.00901 

.087 

•0333 

.138 

•065 45 

.189 

.10312 

.24 

.14494 

•037 

.00938 

.088 

•033 87 

•139 

.066 14 

.19 

.1039 

.241 

.1458 

.038 

.009 76 

.089 

•034 44 

.14 

.066 83 

.191 

.10468 

.242 

.14665 

•039 

.01015 

.09 

•035 01 

.141 

.06753 

.192 

•105 47 

•243 

•i 475 i 

.04 

.01054 

.091 

•035 58 

.142 

.068 22 

•193 

.106 26 

•244 

•14837 

.041 

.01093 

.092 

.03616 

•143 

.068 92 

.194 

.10705 

•245 

•14923 

.042 

•0x133 

•093 

.036 74 

.144 

.069 62 

•195 

.107 84 

.246 

.15009 

•043 

.011 73 

•094 

•037 32 

•145 

•07033 

.196 

.10864 

•247 

•15095 

.044 

.01214 

•095 

•037 9 

.146 

.07103 

.197 

.10943 

.248 

.151 82 

•045 

•01255 

.096 

•03849 

.147 

.071 74 

.198 

.11023 

•249 

.152 68 

.046 

.012 97 

•097 

.03908 

.148 

•07245 

•X 99 

. 111 02 

•25 

•153 55 

.047 

•01339 

.098 

.039 68 

.149 

•073 16 

.2 

.111 82 

.251 

•i 544 i 

.048 

.01382 

•099 

.04027 

•15 

•07387 

.201 

.112 62 

.252 

• 15528 

.O49 

•01425 

.1 

.040 87 

•151 

•074 59 

.202 

• 1 X 343 

•253 

•15615 

•05 

.01468 

.101 

.041 48 

.152 

•07531 

.203 

.11423 

•254 

.15702 

.051 

.015 12 

.102 

.042 08 

•iS 3 

.07603 

.204 

•11503 

•255 

•15789 



























268 AREAS OF SEGMENTS OF A CIRCLE. 


Versed 

Sine. 

Seg. Area. 

Vessed 

Sine. 

Seg. Area. 

Versed 

Sine. 

Seg. Area. 

|V ersed 
Sine. 

Seg. Area. 

Versed 

Sine. 

Seg. Area. 

.256 

.158 76 

•305 

.202 76 

•354 

.2488 ’ 

.•403 

.29631 

•452 

•344 77 

•257 

.15964 

.306 

.20368 

•355 

.249 76 

.404 

.297 29 

•453 

•345 77 

.258 

.160 51 

• 3 ° 7 

.2046 

•356 

.25071 

•405 

.298 27 

•454 

.346 76 

•259 

.16139 

.308 

•205 53 

•357 

.25167 

.406 

.29925 

•455 

•347 76 

.26 

.162 26 

•309 

.20645 

•358 

•25263 

.407 

.300 24 

•456 

•348 75 

.261 

.16314 

• 3 i 

•207 38 

•359 

•25359 

.408 

.3OI 22 

•457 

•349 75 

.262 

.16402 

• 3 ii 

.2083 

•36 

•254 55 

.409 

.302 2 

•458 

•350 75 

.263 

.1649 

.312 

.209 23 

.361 

•255 51 

.41 

•30319 

•459 

• 35 i 74 

.264 

.16578 

• 3 i 3 

.21015 

.362 

.25647 

.411 

•30417 

.46 

•352 74 

.265 

.166 66 

• 3 i 4 

.211 08 

•363 

•257 43 

.412 

• 3°515 

.461 

•353 74 

.266 

•16755 

• 3 i 5 

.21201 

•364 

•25839 

•413 

.306 14 

.462 

•354 74 

.267 

.168 44 

.316 

.21294 

•365 

•25936 

.414 

.30712 

•463 

•355 73 

.268 

.16931 

• 3 i 7 

.21387 

•366 

.26032 

•415 

.308 11 

•464 

•356 73 

.269 

.1702 

.318 

.2148 

•367 

.261 28 

.416 

.30909 

•465 

•357 73 

.27 

.171 09 

• 3 i 9 

•215 73 

.368 

.262 25 

.417 

.31008 

.466 

•358 72 

.271 

.17197 

•32 

.21667 

•369 

.263 21 

.418 

.3IIO7 

.467 

•359 72 

.272 

.172 87 

.321 

.2176 

•37 

.264 18 

.419 

• 3 i2 05 

.468 

.360 72 

•273 

•173 76 

.322 

•21853 

•371 

.265 14 

.42 

•31304 

•469 

•361 72 

.274 

•i 74 6 5 

•323 

.21947 

•372 

.266 11 

.421 

•31403 

•47 

•362 72 

•275 

•175 54 

•324 

.2204 

•373 

.26708 

.422 

■31502 

• 47 i 

•363 71 

.276 

•17643 

•325 

.221 34 

•374 

.268 04 

•423 

• 3!6 

.472 

•36471 

.277 

•177 33 

•326 

.222 28 

•375 

.26901 

•424 

• 3 i6 99 

•473 

•36571 

.278 

.178 22 

•327 

.22321 

•376 

.269 98 

•425 

• 3 x 798 

•474 

•36671 

.279 

.179 12 

.328 

.22415 

•377 

.27095 

.426 

•31897 

•475 

.36771 

.28 

.180 02 

•329 

.225 09 

•378 

.271 92 

•42 7 

.31996 

•476 

•36871 

.281 

. 180 92 

•33 

.226 03 

•379 

.272 89 

.428 

•32095 

•477 

.36971 

.282 

.181 82 

• 33 i 

.226 97 

.38 

•273 86 

•429 

•32194 

.478 

•37071 

.283 

.182 72 

•332 

.227 91 

.381 

•27483 

•43 

•32293 

•479 

• 37 i 7 

.284 

.18361 

•333 

.22886 

•3S2 

•275 80 

• 43 i 

•32391 

.48 

•372 7 

.285 

.18452 

•334 

.2298 

•383 

.27677 

•432 

•3249 

.4S1 

•373 7 

.286 

.185 42 

•335 

.230 74 

•384 

•277 75 

•433 

•325 9 

.482 

•374 7 

.287 

•18633 

•336 

.231 69 

•385 

.278 72 

•434 

.32689 

•4S3 

•375 7 

.288 

.18723 

•337 

•232 63 

.386 

.279 69 

•435 

.32788 

•484 

•3767 

.289 

.18814 

•338 

•233 58 

•387 

.28067 

•436 

.32887 

•485 

•377 7 

.29 

.18905 

•339 

•234 53 

.388 

.281 64 

•437 

.32987 

.486 

•378 7 

.291 

.18995 

•34 

•235 47 

•389 

.28262 

•438 

.33086 

•487 

•379 7 

.292 

.19086 

• 34 i 

.23642 

•39 

•2S3 59 

•439 

• 33 i 85 

.488 

•3807 

•293 

.191 77 

•342 

•237 37 

• 39 1 

.28457 

•44 

•332 S4 

•489 

•381 7 

•294 

.192 68 

•343 

.23832 

•392 

•285 54 

.441 

•333 84 

•49 

.3827 

•295 

.1936 

•344 

.23927 

•393 

.28652 

•442 

•334 83 

• 49 i 

•3837 

.296 

•i 945 i 

•345 

.24022 

•394 

•2875 

•443 

•335 82 

.492 

•384 7 

.297 

.195 42 

•346 

.24117 

•395 

.28848 

•444 

.33682 

•493 

•385 7 

.298 

•19634 

•347 

.242 12 

•396 

.28945 

•445 

•337 Si 

•494 

•3867 

.299 

•19725 

•348 

.24307 

.•397 

.29043 

•446 

.3388 

•495 

•3877 

•3 

.198 17 

•349 

.24403 

•398 

.29141 

•447 

•3398 

•496 

.3887 

.301 

.19908 

•35 

.24498 

•399 

•292 39 

.448 

•340 79 

•497 

•389 7 

.302 

.2 

• 35 i 

•245 93 

•4 

•29337 

•449 

• 34 i 79 

.498 

•390 7 

•303 

.200 92 

•352 

.24689 

.401 

•29435 

•45 

•342 78 

•499 

• 39 i 7 

• 3°4 

.201 84 

•353 

.24784 

.402 

•29533 

451 

■343 78 

•5 

•392 7 


To Compute Area, of a Segment of a Circle by preceding- 

Table. 

Rule. —Divide height or versed sine by diameter of circle; find quotient in 
column of versed sines. Take area for versed sine opposite to it in next col¬ 
umn on right hand, multiply it by square of diameter, and it will give area. 


































AREAS OF ZONES OF A CIRCLE. 


269 

Example. —Required area of a segment of a circle, its height being 10 feet and 
diameter of circle 50. 

10-f- 50 = .2, and .2, per table , =.hi 82; then ,11182 X 50 2 = 279.55 feet. 

When, in Division of a Height hy Base, Quotient has a Remainder after 
Third Place of Decimals , and great Accuracy is required. 

Rule. —Take area for first three figures, subtract it from next following 
area, multiply remainder by said fraction, add product to first area, and 
sum will give area for whole quotient. 

Example. —What is area of a segment of a circle, diameter of which is 10 feet, and 
height of it 1.575? 

1.575 4-10 = .1575; tabular area for .157 = .078 92. and for . 158 = .079 65, the dif¬ 
ference behveen which is ■ 00073. 

Then .5 X -ooo 73 = .000 365. 

Hence, .157 =.07892 

.0005 = .000 365 

.079 285, sum by which square of diameter 
of circle is to be multiplied ; and .079 285 X io 2 = 7.9285 feet. 


Areas of' Zones of a Circle. 

The Diameter of a Circle = 1, tend divided into 1000 equal Parts. 


H’ght. 

Area. 

H’ght. 

Area. ! 

H’ght. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

.OOI 

.OOI 

.035 

•034 97 

.069 

.068 78 

.103 

.102 27 

•137 

•135 27 

.002 

.002 

.036 

•035 97 

•07 

.o6q 77 

.104 

.IO325 

.138 

•I 3 6 23 

.003 

.003 

•037 

.03697 

.071 

.070 76 

.105 

.IO4 22 

•139 

•13719 

.004 

.004 

.038 

.03796 

.072 

•071 75 

.106 

.105 2 

.14 

•13815 

.005 

.005 

•039 

.038 96 

•073 

.072 74 

.107 

.106 l8 

.141 

.139II 

.006 

.006 

.04 

.03996 

•074 

•073 73 

.108 

.IO715 

.142 

.I4OO7 

.007 

.007 

.041 

•04095 

•075 

.07472 

.109 

.10813 

•143 

• I 4 I °3 

.008 

.008 

.042 

.041 95 

.076 

•075 5 

.11 

.109 II 

.144 

.141 98 

.009 

.009 

•043 

.04295 

•077 

.07669 

.III 

.IIO08 

•145 

.142 94 

.OI 

.OI 

.044 

•04394 

.078 

.077 68 

.112 

. 111 06 

.146 

•1439 

.on 

.on 

•045 

•044 94 

•079 

.07867 

•113 

.112 03 

.147 

.14485 

.012 

.012 

.046 

•045 93 

.08 

.079 66 

.114 

•113 

.148 

.14581 

.013 

.013 

.047 

.04693 

.081 

.080 64 

•115 

.II398 

.149 

.14677 

.014 

.014 

.048 

•04793 

.082 

.081 63 

.Il6 

•II 49 S 

•15 

.14772 

.015 

.015 

.049 

.048 92 

.083 

.082 62 

.117 

.II592 

•151 

.14867 

.0x6 

.016 

•05 

.04QQ2 

.084 

.0836 

.Il8 

.1169 

.152 

.149 62 

.017 

.017 

.051 

.05091 

•085 

•08459 

.119 

• H7 87 

•153 

.15058 

.018 

.018 

.052 

.0519 

.086 

•085 57 

.12 

.11884 

•154 

.15153 

.019 

.019 

.053 

.0529 

.087 

.086 56 

.121 

.11981 

•155 

.15248 

.02 

.02 

•054 

•05389 

.088 

.087 54 

.122 

.120 78 

.156 

.15343 

.021 

.021 

•055 

.05489 

.089 

•088 53 

.123 

.121 75 

•157 

•15438 

.022 

.022 

.056 

.05588 

.09 

.08951 

.124 

.122 72 

.158 

•155 33 

.023 

.023 

•057 

.05688 

.091 

.0905 

.125 

.12369 

•159 

.15628 

.024 

.024 

.058 

.05787 

.092 

.091 48 

.126 

.12469 

.16 

•157 23 

.025 

.025 

•059 

.058 86 

•093 

.092 46 

.127 

.12562 

.161 

.15817 

.026 

.02599 

.06 

.059 86 

•094 

.09344 

.128 

.12659 

.162 

.15912 

.027 

.02699 

.061 

.060 85 

•095 

.09443 

.129 

•12755 

.163 

.160 06 

.028 

.02799 

.062 

.061 84 

.096 

•0954 

•13 

.128 52 

.164 

.16101 

.029 

.028 98 

.063 

.062 83 

.097 

•09639 

•131 

.12949 

.165 

.16195 

•03 

.029^8 

.064 

.063 82 

.098 

•097 37 

.132 

•13045 

.166 

.1629 

.031 

.03098 

.065 

.064 82 

.099 

•09835 

•133 

• I 3 1 4 1 

.167 

.16384 

.032 

•031 98 

.066 

.0658 

.1 

•09933 

•134 

•1323 8 

.168 

.164 78 

•°33 

.032 98 

.067 

.0668 

.IOI 

.10031 

1 -i 35 

•13334 

.169 

.165 72 

.034 

•033 97 

| 068 

.0678 

.102 | .IOI 29 

, .136 

• I 343 

•17 

.16667 


Z* 




























270 


AREAS OF ZONES OF A CIRCLE. 


H'ght. 

Area. j 

H’ght. j 

Area. 

H’ght. | 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

.171 

.16761 

.226 

.21805 

.281 

.26541 . 

•33 6 

.308 64 

•391 

•34632 

.172 

•16855 

.227 

.21894 

.282 

.26624 

'•337 

•30938 

•392 

.34694 

.173 

.16948 

.228 

.21983 

•283 

.26706 

•338 

.3IO 12 

•393 

•347 56 

.174 

.17042 

.229 

.220 72 

.284 

.267 89 

•339 

•31085 

•394 

.348 18 

•175 

.17136 

•23 

.221 6l 

•285 

.268 71 

•34 

• 3 i i 59 

•395 

•348 79 

.176 

.1723 

.231 

.222 5 

.286 

•26953 

• 34 i 

• 3 12 32 

•396 

•3494 

.177 

• I 73 2 3 

.232 

•223 35 

.287 

•270 35 

•342 

•31305 

•397 

•350 01 

.178 

.17417 

•233 

.224 27 

.288 

.271 17 

•343 

•313 78 

•398 

.35062 

.179 

•175 1 

•234 

•225 15 

.289 

• 27 L 99 

•344 

• 3 I 4 5 

•399 

•35i 22 

.18 

.17603 

•235 

.22604 

.29 

.2728 

•345 

•31523 

•4 

•35182 

.181 

.17697 

.236 

.226 92 

.291 

.273 62 

, -346 

•315 95 

.401 

•352 42 

.182 

.1779 

•237 

.227 8 

.292 

•274 43 

: - 347 J 

.3x667 

.402 

•35302 

.183 

.17883 

•238 

.22868 

•293 

.275 24 

•348 

•317 39 

•403 

•353 61 

.184 

.179 76 

•239 

.22956 

•294 

.27605 

•349 

.31811 

•404 

•354 2 

•185 

.180 69 

.24 

.23044 

•295 

.27686 

•35 

.31882 

•405 

•354 79 

.186 

.181 62 

.241 

•231 3 1 

.296 

.277 66 

• 35 i 

•31954 

.406 

•355 38 

.187 

.182 54 

.242 

.232 19 

.297 

•27847 

•352 

•32025 

•407 

•35596 

.188 

•i 83 47 

•243 

•233 06 

.298 

.279 27 

•353 

.320 96 

.408 

•35654 

.189 

.1844 

.244 

•233 94 

.299 

.28007 

•354 

.32167 

•409 

•357 11 

.19 

.185 3 2 

•245 

.23481 

•3 

.28088 

•355 

.32237 

.41 

•357 69 

.191 

.18625 

.246 

•235 68 

•30 1 

.281 67 

•356 

•32307 

.411 

•35826 

.192 

.18717 

.247 

•23655 

^302 

.282 47 

•357 

•323 77 

.412 

•358 83 

•193 

.188 09 

.248 

•237 42 

I 4 P 3 

.28327 

•358 

•32447 

• 4 i 3 

•35939 

.194 

.189 02 

.249 

.238 29 

• 3°4 

.284 06 

•359 

•32517 

.414 

•35995 

•195 

.18994 

•25 

•239 I 5 

•305 

.284 86 

•36 

•32587 

• 4 i 5 

•36051 

.196 

.190 86 

.251 

.240 02 

.306 

•285 65 

.361 

•32656 

.416 

•361 07 

.197 

.191 78 

.252 

.240 89 

1 -307 

.28644 

.362 

•32725 

.417 

.361 62 

.198 

.1927 

•253 

•241 75 

.308 

.28723 

•363 

•32794 

.418 

.362 17 

.199 

.19361 

•254 

.242 61 

•309 

.28801 

•364 

.32862 

.419 

.362 72 

.2 

•194 53 

•255 

•243 47 

• 3 i 

.2888 

•365 

•329 3 1 

.42 

•363 26 

.201 

•195 45 

.256 

•244 33 

• 3 i 1 

.28958 

•366 

•32999 

.421 

•3638 

.202 

.19636 

•257 

.245 19 

• 3 12 

.29036 

•367 

•33067 

.422 

•36434 

.203 

.197 28 

.258 

.246 04 

• 3 T 3 

.29115 

•368 

•33135 

•423 

.36488 

.204 

.198 19 

•259 

.246 9 

• 3 T 4 

.291 92 

•369 

•332 03 

•424 

•36541 

.205 

.1991 

.26 

•247 75 

• 3 i 5 

.292 7 

•37 

•332 7 

•425 

•365 94 

.206 

.20001 

.261 

.248 61 

.316 

.29348 

• 37 i 

•333 37 

.426 

.36646 

.207 

.20092 

.262 

.249 46 

• 3 I 7 

•294 25 

•372 

•33404 

.427 

.36698 

.208 

.201 83 

.263 

.250 21 

.318 

.295 02 

•373 

•334 7 1 

.428 

•3675 

.209 

.202 74 

.264 

.25116 

• 3 i 9 

.2958 

•374 

•335 37 

•429 

.368 02 

.21 

.20365 

.265 

.25201 

•32 

.29656 

•375 

•33604 

•43 

•36853 

.211 

.204 56 

.266 

•252 85 

.321 

.29733 

•376 

•336 7 

• 43 i 

•36904 

.212 

.205 46 

.267 

•253 7 

.322 

.298 1 

•377 

•337 35 

•432 

•369 54 

.213 

.20637 

.268 

•254 55 

^323 

.29S 86 

•378 

•33801 

•433 

•37005 

. 2 X 4 

.207 27 

.269 

•255 39 

•324 

.299 62 

•379 

•338 66 

•434 

•370 54 

.215 

.20818 

.27 

.25623 

•325 

.30039 

•38 

•339 3 1 

■435 

•37104 

.2l6 

.209 08 

.271 

.25707 

.326 

.301 14 

.381 

•33996 

•436 

•37153 

.217 

.20998 

.272 

• 2579 1 

•327 

.3019 

.382 

•34061 

•437 

.37202 

.218 

.21088 

•273 

•25875 

.328 

.302 66 

•3S3 

• 34 i 25 

■438 

•372 5 

.219 

.21178 

.274 

•25959 

•329 

•30341 

v 384 

•3419 

•439 

.37298 

.22 

.21268 

•275 

.26043 

•33 

•304 16 

•385 

•342 53 

•44 

•373 46 

.221 

•21358 

.276 

.261 26 

• 33 i 

.30491 

.386 

•343 1 7 

.441 

•37393 

.222 

.21447 

.277 

.26209 

•332 

•305 66 

•387 

•343 8 

•442 

■3744 

.223 

•21537 

.278 

.26293 

•333 

.30641 

.388 

•344 44 

•443 

•374 87 

.224 

.2x626 

.279 

.26376 

•334 

•307 15 

•389 

•345 07 

•444 

•37533 

.225 

.21716 

.28 

.26459 

•335 

•307 9 

•39 

•345 69 

•445 

•375 79 










































AREAS OF ZONES OF A CIRCLE. 2 J I 


Tight. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

.446 

.37624 

•457 

.38096 

.468 

•38514 

•479 

.38867 

•49 

• 39 1 37 

•447 

.37669 

•458 

•381 37 

.469 

•385 49 

•48 

• 38 S 95 

• 49 1 

• 39 1 56 

.448 

•37714 

•459 

•381 77 

•47 

•385 83 

.481 

•389 23 

.492 

• 39 i 75 

•449 

•377 58 

.46 

.382 16 

.471 

.38617 

.482 

•3895 

•493 

• 39 1 92 

•45 

.37802 

.461 

•382 55 

•472 

•3865 

•483 

•389 76 

•494 

.392 08 

• 45 i 

•378 45 

.462 

.382 94 

•473 

.38683 

•484 

.39OOI 

•495 

•392 23 

•452 

.37888 

•463 

•383 32 

•474 

•38715 

•485 

.39026 

.496 

.392 36 

•453 

•37931 

.464 

•383 69 

•475 

•38747 

.486 

•390 5 

•497 

.392 48 

•454 

•379 73 

•465 

.38406 

.476 

.387 78 

•487 

• 39 ° 73 

•498 

•392 58 

•455 

•380 14 

.466 

•384 43 

•477 

.388 08 

.488 

•390 95 

•499 

.392 66 

•456 

•38056 

.467 

•384 79 

•478 

•38838 

.489 

• 39 1 17 

•5 

•392 7 


This Table is computed only for Zones , longest Chord of which is Diam¬ 
eter. 


To Compute Area of a Zone by preceding TaLle. 

When Zone is Less than a Semicircle. 

Rule.—D ivide height by diameter, find quotient in column of heights. 
Take area for height opposite to it in next column on right hand, multiply 
it by square of longest chord, and product will give area of zone. 

Example. —Required area of a Zone, diameter of which is 50, and its height 15. 

15 -r- 50 =. 3; and . 3, as per table, = . 280 88. 

Hence . 280 88 X 50 2 = 702.2 area. 

When Zone is Greater than a Semicircle. 

Rule.— Take height on each side of diameter of circle, and ascertain, by 
preceding Rule, their respective areas; add areas of these two portions to¬ 
gether, and sum will give area. 

Example. —Required area of a zone, diameter of circle being 50, and heights of 
zone on each side of diameter of circle 20 and 15. 

20-4- 50 = .4; .4, as per table, = .351 82; and .351 82 X so 2 = 879.55. 

15 - 4 - 50 = . 3 ; .3 , as per table, — . 280 88; and . 280 88 X 50 2 = 702.2. 

Hence 879.554-702.2= 1581.75 area. 

When , in Division of a Height by Chord , Quotient has a Remainder after 
Third Place of Decimals , and great Accuracy is required. 

Rule. —Take area for first three figures, subtract it from the next follow¬ 
ing area, multiply remainder by said fraction, and add product to first area; 
sum will give area for whole quotient. 

Example. —What is area of a zone of a circle, greater chord being 100 feet, and 
breadth of it 14 feet 3 ins.? 

14 feet 3 ins. = 14-25, and 14.25- 4 -100 = .1425; tabular length for .142 = .140 07, 
and for . 143 = .141 03. difference between which is .000 96. 

Then .5 X .00096 = .00048. Hence .142 =.14007 

.0005 = .000 48 

. 140 55, sum by which square of greater 
chord is to be multiplied ; and . 140 55 X 100 2 = 1405.5 feet. 





























272 SQUARES, CUBES, AND ROOTS. 

Squares, Cnbes, and Square and Cube Roots, 

From 1 to 1600. 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

I 

I 

I 

I 

I 

2 

4 

8 

1.4142136 

1.259 9 21 

3 

9 

27 

I.732 0508 

1.442 249 6 

4 

16 

64 

2 

1.587401 1 

5 

25 

125 

2.236 06S 

1.7099759 

6 

36 

216 

2.449 489 7 

1.817 1206 

7 

49 

343 

2-645 75 i 3 

1.912 931 2 

8 

64 

512 

2.828 427 1 

2 

9 

81 

729 

3 

2.080 083 7 

10 

1 00 

1 000 

3.162 277 7 

2.154 434 7 

11 

1 21 

1 33 1 

33166248 

2.223 980 1 

12 

1 44 

1 728 

3.464 101 6 

2.289 428 6 

13 

1 69 

2 197 

3 605 5513 

2.351 334 7 

14 

1 96 

2 744 

3.7416574 

2.410 142 2 

15 

225 

3 375 

3.8729833 

2.466 212 1 

16 

256 

4096 

4 

2.519 842 1 

17 

2 89 

49 i 3 

4.123 1056 

2.571 281 6 

18 

324 

5832 

4.242 640 7 

2.620 741 4 

19 

361 

6859 

4-3585989 

2.668 401 6 

20 

400 

8 000 

4.472 136 

2 - 7 I 4 4 I 7 7 

21 

441 

9 261 

4-5825757 

2.75S9243 

22 

484 

10 648 

4.6904158 

2.8020393 

23 

529 

12 167 

4-795 831 5 

2.843 867 

24 

5 76 

13824 

4.8989795 

2.884499 1 

25 

625 

15625 

5 

2.924 017 7 

26 

6 76 

17 576 

5.0990195 

2.962496 

27 

729 

196S3 

5 1961524 

3 

28 

784 

21 952 

5 291 5026 

3.0365889 

29 

841 

24 389 

5.385 1648 

3.0723168 

30 

900 

27 000 

5 4772256 

3.107 2325 

3 i 

961 

29 79i 

5.567 7644 

3-i4i 3806 

32 

1024 

32 768 

5.656 8542 

3.174 802 1 

33 

1089 

35 937 

5.744 5626 

3-207 534 3 

34 

11 56 

39 304 

5.8309519 

3.239611 8 

35 

1225 

42875 

5.9160798 

3.271 0663 

36 

1296 

46 656 

6 

3-301 Q 27 2 

37 

1369 

50653 

6.082 762 5 

3.332 221 8 

3 S 

1444 

54872 

6.164 414 

3 - 36 i 975 4 

39 

15 21 

59 319 

6.244 998 

3.391 2114 

40 

1600 

64 000 

6-324 555 3 

3.419 951 9 

4 i 

16 81 

68 921 

6 403 124 2 

3.4482172 

42 

1764 

74088 

6.480 740 7 

3 476 026 6 

43 

1849 

79 507 

6.557 438 5 

3-503 398 1 

44 

1936 

85 184 

6.6332496 

3-5303483 

45 

20 25 

91 125 

6.708 203 9 

3-55689 33 

46 

21 16 

97 336 

6.782 33 

3.5830479 

47 

22 09 

103 823 

6.855 654 6 

3.608 826 1 

48 

23 04 

110 592 

6.928 203 2 

3.634 241 1 

49 

2401 

117649 

7 

3-659305 7 

50 

2500 

125 000 

7.071 0678 

3.684 0314 

5 i 

2601 

132 651 

7.141 4284 

3.708 4298 

52 

2704 

140 608 

7.211 1026 

3-732 5 ii 1 

53 

28 09 

148 877 

7.280 1099 

3.7562858 

54 

2916 

157464 

7.348 469 2 

3.779 763 1 












SQUARES, CUBES, AND ROOTS. 


273 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

55 

30 25 

166375 

7.416 198 5 

3.802 952 5 

56 

3136 

175616 

7.4833148 

3.825 862 4 

57 

32 49 

185 193 

7.549 8344 

3.848 501 I 

58 

33 64 

195 112 

7 - 6 i 5 773 1 

3.8708766 

59 

34 8i 

205 379 

7.681 145 7 

3.892 996 5 

60 

3600 

216000 

7.745 966 7 

3.9148676 

61 

3721 

226 981 

7.810249 7 

3.9364972 

62 

3844 

238 328 

7.8740079 

3-957 891 5 

63 

3969 

250 047 

7-937 253 9 

3-979057 1 

64 

4096 

262 144 

8 

4 

65 

42 25 

274 625 

80622577 

4.020 725 6 

66 

43 56 

287 496 

8.124038 4 

4.041 240 1 

67 

44 89 

3 °° 763 

8.185 352 8 

4.061 548 

68 

4624 

314 432 

8.246211 3 

4.081 655 1 

69 

4761 

328 509 

8.3066239 

4.101 566 1 

70 

4900 

343000 

8.366 600 3 

4.1212853 

7 i 

5041 

357 9 i 1 

8.426 149 8 

4.1408x7 8 

7 2 

5184 

373 248 

8.485 281 4 

4.160 167 6 

73 

53 29 

389017 

8.5440037 

4-179339 

74 

54 76 

405 224 

8.602 325 3 

4.1983364 

75 

5625 

421 875 

8.660 254 

4.2171633 

76 

57 76 

438 976 

8.7177979 

4.235 823 6 

77 

59 29 

456 533 

8.7749644 

4-254 321 

78 

60 84 

474 552 

8.831 7609 

4.272 6586 

79 

62 41 

493 039 

8.888 194 4 

4.290 840 4 

80 

6400 

512 000 

, 8.9442719 

4.308 869 5 

81 

6561 

53 i 44 i 

9 

4.326 748 7 

82 

6724 

55 i 368 

9-055 385 1 

4.3444815 

83 

68 89 

571 787 

9.1104336 

4.362 070 7 

84 

70 56 

592 704 

9.1651514 

4-379 5 i 9 1 

85 

72 23 

6x4125 

9.2195445 

4.396 829 6 

86 

73 96 

636056 

9.2736185 

4.4140049 

87 

7569 

658 503 

9-327 379 1 

4.431 047 6 

88 

77 44 

681 472 

9.3808315 

4.447 960 2 

89 

7921 

704969 

9.433981 1 

4.464 745 1 

90 

81 00 

729 000 

9.486 833 

4.481 404 7 

9 1 

82 81 

753 571 

9-539392 

4.497 941 4 

92 

8464 

778 688 

9.591 663 

4 - 5 I 43574 

93 

86 49 

804 357 

9.6436508 

4.5306549 

94 

8836 

830 584 

9-695 359 7 

4.546 835 9 

95 

9025 

857 375 

9.746 794 3 

4.562 902 6 

96 

92 16 

884736 

9-797 959 

4-578 857 

97 

9409 

912 673 

9.848 8578 

4.594 7009 

98 

9604 

941 192 

9.8994949 

4.6104363 

99 

9801 

970 299 

9 949 874 4 

4.626 065 

100 

10000 

1 000000 

10 

4.641 588 8 

IOI 

102 01 

1 030 301 

10.0498756 

4.657 009 5 

102 

104 04 

x 061 208 

10.0995049 

4.672 328 7 

103 

106 09 

1 092 727 

10.148 891 6 

4.687 548 2 

104 

108 16 

1 124864 

10.198039 

4.702 6694 

.105 

11025 

1 157625 

10.246 950 8 

4.7x7694 

106 

11236 

1 191 016 

10.295 630 1 

4.732 6235 

107 

1 1449 

1 225 043 

10.344 °8o 4 

4-747 4594 

108 

11664 

x 259712 

10.392 304 8 

4.762 203 2 

109 

11881 

1295 029 

10.440 306 5 

4.776 8562 

no 

1 21 00 

1331000 

10.488 088 5 

4.79x4199 














Ill 

±12 

113 

114 

115 

116 

II 7 

Il8 

11 9 

120 

121 

122 

I2 3 

I2 4 

125 

126 

I2 7 

128 

I29 

130 

131 

132 

133 

134 

135 

136 

137 

138 

139 

I40 

I4I 

142 

143 

I44 

145 

146 

I47 

148 

I49 

150 

151 

152 

153 

154 

155 

156 

157 

158 

*59 

160 

161 

162 

163 

164 

165 

166 


SQUARES, CUBES, AND ROOTS. 


Square. 

Cube. 

Square Root. 

Cube Root. 

I 2321 

I 367 631 

*o -535 653 8 

4.805 895 5 

12544 

I 404 928 

10.583 005 2 

4.820 284 5 

I 2769 

I 442 897 

10.630 145 8 

4.834 588 I 

I 2996 

I 481 544 

10.6770783 

4.848 807 6 

132 25 

I 520 875 

10.723 805 3 

4.862 944 2 

i 34 56 

I 560 896 

10.770 3296 

4.87.6999 

136 89 

I 601 613 

10.8166538 

4.8909732 

1 39 24 

1643032 

10.862 780 5 

4.904 868 1 

1 41 61 

I 685 159 

% 10.908 712 1 

4.918 684 7 

1 44 00 

I 728000 

10.9544512 

4.932 424 2 

1 4641 

I 771 561 

11 

4.946 087 4 

1 48 84 

I 815 848 

11.045 361 

4.9596757 

15129 

I 860867 

11,0905365 

4.973 189 8 

153 76 

I 906 624 

11.135 5287 

4.986631 

. 1 56 25 

i 953 125 

11.1803399 

5 

1 58 76 

2000376 

11.2249722 

5.013 297 9 

1 61 29 

2 048 383 

11.269427 7 

5.0265257 

1 63 84 

2097 152 

ii-3i3 7085 

5.039 684 2 

1 66 41 

2 146 689 

11.3578167 

5.052 774 3 

1 6900 

2 197000 

11.401 7543 

5.065 797 

1 71 61 

2 248 091 

n-445 5231 

5-078 753 1 

1 7424 

2 299 968 

11.4891253 

5.091 643 4 

1 76 89 

2 352 637 

11.532 5626 

5.104468 7 

1 79 56 

2 406104 

11-5758369 

5.1172299 

1 82 25 

2460375 

11.61895 

5.1299278 

1 8496 

2 5i5,45 6 

11.661 903 8 

5.142 5632 

1 8769 

2 57 1 353 

11.7046999 

5-155 1367 

19044 

_ ^2 628 072 

IT -747 34° 1 

5.1676493 

1 93 21 

2 685 619 

11.789826 1 

5.180101 5 

1 9600 

2 744000 

11.832 1596 

5.192 494 1 

1 98 81 

2 803 221 

11.8743421 

5.204 827 9 

201 64 

2 863 288 

11.9163753 

5.217 103 4 

20449 

2 924 207 

11.958 2607 

5.2293215 

20736 

2 985 984 

12 

5.241 482 8 

2 1025 

3 048 625 

12.041 5946 

5-253 587 9 

2 13 16 

3112136 

12.083046 

5.265 637 4 

2 1609 

3 1 76 523 

12.1243557 

5.277632 1 

2 1904 

3 24 1 792 

12.165 525 1 

5.2895725 

2 22 01 

3 307 949 

12.2065556 

5-3 01 4592 

2 25 00 

3375000 

i2 -247 448 7 

5.313 2928 

2 28 01 

3442 951 

12.288 205 7 

5.325 074 

23104 

3511008 

12.328 828 

5-3368033 

23409 

3 581577 

12.3693169 

5.3484812 

237 16 

3 652 264 

12.4096736 

5.3601084 

24025 

3 723 875 

12.4498996 

5-37 1 685 4 

2 43 36 

3796416 

12.489996 

5.3832126 

24649 

3869893 

12.5299641 

5.3946907 

24964 

3944312 

12.569805 1 

5.406 120 2 

25281 

4019679 

12.609 520 2 

5.417 501 5 

.2 5600 

4 096 000 

.. 12.6491106 

5.4288352 

25921 

4173 281 

12,688 5775 

5.440 121 8 

2 62 44 

4251528 

12.727922 1 

5.4513618 

26569 

433° 747 

12.767 1453 

5.462 5556 

2 68 96 

4410944 

12.806248 5 

' 5-473 703 7 

272 25 

4492125 

12.845 232 6 

5.4848066 

2 75 56 

4 574 296 J 

12.884 098 7 

5-495 8647 










SQUARES, CUuES, AND ROOTS. 


Square. 

Cube. 

Square Root. 

2 7889 

4657463 

12.922 848 

2 82 24 

4 74i 632 

12.961 48X 4 

28561 

4826809 

13 

2 89OO 

4913OOO 

13.038 404 8 

2 92 41 

5 000211 

13.076 6968 

29584 

5 088 448 

13.114877 

29929 

51777 i 7 

13.1529464 

302 76 

5 268 024 

13.190906 

30625 

5 359375 

13.228 7566 

30976 

5 451 776 

13.2664992 

3 *3 2 9 

5 545 233 

13.3041347 

3 16 84 

5 639 752 

13.341 664 1 

32041 

5 735 339 

*3-379 0882 

32400 

5832000 

i 3 - 4 i6 4 Q 7 9 

32761 

5 929 74 i 

13.453 624 

3 3 1 2 4 

6 028 568 

I 3 - 49 ° 737 6 

3 34 89 

6 128487 

I 3-527 7493 

3 38 56 

6 229 504 

13.564 66 

3 42 25 

6331625 

13.601 470 5 

3 45 96 

6 434 856 

13.638 181 7 

3 4969 

6 539 203 

13.674 7943 

3 53 44 

6 644 672 

I 3 - 7 11 3°9 2 

35721 

6 751 269 

13.747 727 1 

361 00 

6859000 

13.784 0488 

36481 

6967 871 

13.820275 

36864 

7 077 888 

13.856 4065 

3 72 49 

7 189057 

13.892 44 

3 76 36 

7 301 384 

13.928 388 3 

38025 

7414875 

13.964 24 

384x6 

7 529 536 

14 

38809' 

7 645 373 

14.035 668 8 

39204 

7 762 392 

14.071 2473 

3 96 01 

7 880 599 

14.106736 

40000 

8000000 

14.142 1356 

40401 

8 120601 

14.1774469 

40804 

8 242 408 

14.2126704 

41209 

8 365 427 

14.247 8068 

4 16 16 

8 489 664 

14.282 8569 

420 25 

8615 125 

14.317821 1 

42436 

8 741 816 

14.352 700 1 

42849 

8 869 743 

14.387 494 6 

43264 

8998912 

14.422 205 1 

43681 

9129329 

14.456 832 3 

441 00 

9 261 000 

I 4 - 49 I 376 7 

4 45 21 

9393 931 

14.525839 

4 4944 

9528 128 

14.560 219 8 

4 5369 

9663597 

14.5945195 

4 57 96 

9800344 

14.628 738 8 

462 25 

9938 375 

14.662 878 3 

46656 

10077 696 

14.696938 5 

4 70 89 

10218313 

14.7309199 

4 75 24 

10 360 232 

14.764 823 1 

47961 

10 503 459 

14.798 648 6 

4 8400 

10 648 000 

14.832 397 

4 88 4 1 

10 793 861 

14.866068 7 

492 84 

10 941 048 

14.899664 4 


275 


Cube Root. 


5.5068784 
5.5178484 
5.5287748 
S-539658 3 
5.550499 I 
5.561 297 8 
5.572 0546 
5.5827702 

5-593 444 7 
5.604078 7 
5.6146724 
5.625 2263 
5.635 740 8 
5.646 216 2 
5.6566528 
5.667051 1 
5.6774114 
5.687 734 
5.6980192 
5.708 2675 
5.7184791 
5.728 6543 
5-738 7936 
5.7488971 
5.7589652 
5.7689982 
5.7789966 
5.7889604 
5.798 89 
5.808 785 7 
5.8186479 
5.8284767 
5.838 272 5 
5.8480355 
5-857 766 
5.867 4643 
5.877 130 7 
5.886 765 3 
5.8963685 
5.9059406 
5.915481 7 
5.924 992 1 
5.9344721 
5.943 922 
5-953 34 1 8 
5.962 732 
5.972 092 6 
5.981 424 
5.990 7264 
6 

6.009 245 
6.018 461 7 
6.027 650 2 
6.036 8x0 7 
6 045 943 5 
6.0550489 









276 SQUARES, CUBES, AND ROOTS. 


N UMBER, 

Square. 

Cube. 

Square Root. 

Cube Root. 

223 

4 97 29 

IIO89 567 

14.9331845 

6.064 127 

224 

501 76 

II 239424 

14.966 629 5 

6.0731779 

225 

50625 

II 390625 

i 5 ' 

6.082 202 

226 

5 10 76 

II543176 

15-033 2964 

6.O9I 1994 

227 

51529 

II 697 083 

15.0665192 

6.100 1702 

228 

5 i 9 84 

II 852352 

15.0996689 

6.109 114 7 

229 

52441 

12 008 989 

15.132 746 

6.1180332 

230 

52900 

12 167 OOO 

15.165 7509 

6.126 925 7 

231 

5 33 6i 

I232639I 

„ 15.1986842 

6135 792 4 

232 

5 38 24 

12 487 168 

15-2315462 

6.1446337 

233 

54289 

I2649337 

15.2643375 

6-153 445 5 

2.34 

5 47 56 

12 8l2 904 

15.297 0585 

6.162 240 1 

235 

5 52 25 

I2977875 

15.329 709 7 

6.171 005 8 

236 

55696 

I3I44256 

15.362 291 5 

6.179 746 6 

237 

5 61 69 

13 3 12 053 

15.394 804 3 

6.188 462 8 

238 

56644 

13 481272 

15.4272486 

6.197 1544 

239 

5 71 21 

13651919 

15.4596248 

6.205 821 8 

240 

5 7600 

13824000 

I 5 - 49 I 933 4 

6.214465 

241 

5 8081 

13 997 521 

15.5241747 

6.223 084 3 

242 

58564 

14 172 488 

I 5-556 349 2 

6.231 679 7 

243 

5 90 49 

14 348 907 

15.588 4573 

6.240 251 5 

244 

5 95 36 

14 526 784 

15.6204994 

6 248 7998 

245 

60025 

14 706 125 

15.6524758 

6.2573248 

246 

605 16 

14886 936 

15.684387 1 

6.265 8266 

247 

6 1009 

15069 223 

15.7162336 

6.274 3054 

248 

6 1504 

15252992 

15.7480157 

6.282 761 3 

249 

62001 

15438 249 

15-779 733 8 

6.291 1946 

250 

6 25 00 

15 625 OOO 

15.811 3883 

6.2996053 

251 

63001 

15813251 

15.842 9795 

6.307 993 5 

252 

63504 

16 003 008 

15.874 5079 

6.3163596 

253 

64009 

16194277 

15 905 973 7 

6 324 703 5 

254 

645 16 

16 387 064 

15-937 377 5 

6-333 025 6 

255 

650 25 

16581375 

15.9687194 

6-341 325 7 

256 

6 55 36 

16 777 216 

16 

6.349 604 2 

257 

66049 

16974 593 

16.031 2195 

6.357 861 1 

258 

66564 

17173512 

16.062 378 4 

6.366 096 8 

259 

6 7081 

17 373 979 

16.093 476 9 

6-374 3 H 1 

260 

6 7600 

17576000 

16.1245155 

6.382 5043 

26l 

6 81 21 

17779581 

16.1554944 

6.390 676 5 

262 

686 44 

17 984 728 

16.186 414 1 

6.398 8279 

263 

691 69 

18 191 447 

16.217 274 7 

6.4069585 

264 

69696 

18399 744 

16.248 076 8 

6.415 068 7 

265 

702 25 

18609625 

16.278 8206 

64231583 

266 

707 56 

18 821 096 

16.3095064 

6.431 2276 

267 

71289 

19034 163 

16.340 1346 

6.4392767 

268 

7 1824 

19 248 832 

16.370 705 5 

6 447 3°5 7 

269 

7 2361 

19465 109 

16.401 2195 

6-455 3 I 4 8 

270 

7 29 00 

19 683 OOO ' 

16.431 676 7 

6.463 304 1 

271 

7 34 41 

19902 511 

16 462 077 6 

6.471 2736 

272 

7 3984 

20 I23 648 

„ 164924225 

6479 223 6 

273 

7 45 29 

20346417 

16.522 711 6 

6.487 154 1 

274 

750 76 

20570 824 

16 552 945 4 

6-495 065 3 

275 

7 56 25 

20 796 875 

16 583 124 

6.502 957 2 

276 

761 76 

21024576 

16 613 247 7 

* 6.51083 

277 

7 67 29 

21253 933 

16.643317 

6518 683 9 

278 

77284 

21 484952 

16.678332 | 

6.5265189 













UMBER. 

279 

280 

281 

282 

283 

284 

285 

2S6 

287 

288 

289 

290 

291 

292 

293 

294 

295 

296 

297 

298 

299 

3 °° 

3 ° 1 

302 

303 

3°4 

305 

306 

307 

3 °S 

309 

31° 

. 3 “ 

312 

3 i 3 

3 T 4 

3 X 5 

316 

317 

313 

3 i 9 

320 

32 1 

322 

323 

324 

325 

326 

327 

328 

329 

330 

33 i 

332 

333 

334 


SQUARES, CUBES, AND ROOTS. 2 77 


Square. | 

7 7841 
7 84 00 
7 89 61 

7 95 24 

8 00 89 
806 56 
8 12 25 
8 17 96 
82369 
8 29 44 
83521 
8 41 00 
8 46 81 
8 52 64 

85849 

864 36 
8 70 25 
8 76 16 
8 82 09 
8 88 04 

8 9401 

9 00 00 
90601 

9 12 04 
9 18 09 
9 24 16 
9 3°25 
93636 
942 49 
948 64. 

9 54 8i 
961 00 
9 67 21 

9 73 44 
97969 
98596 
992 25 
9 98 56 
10 04 89 
10 11 24 

10 17 61 
10 24 00 
1030 41 
10 36 84 
10 43 29 
10 49 76 
1056 25 
10 62 76 
1069 29 

10 75 84 | 

10 82 41 
108900 
1095 61 

11 02 24 
11 08 89 

11 1556 


Cube. 

21 717639 

21 952 OOO 

22 l88 04I 
22 425 768 
22 663 187 

22 906 304 

23 149 125 
23 393 656 
23 639 903 

23 887 872 

24 137 569 
24 389 OOO 
24 642 171 

24 897 088 

25 153 757 

25412 184 

25672375 

25 934 336 

26 198 073 
26 463 592 

26 730 899 

27 000000 
27 270901 

27 543 608 

27 818 127 

28 094 464 
28372 625 
28 632 616 

28 934 443 

29 218 112 

29 503629 

29 791 OOO 

30 080 231 
30371328 
30 664 297 

30 959144 
31255 87s 

3 1 554 496 

31 855013 

32 157432 
32 461 759 

32 768 OOO 
33076161 

33 386 248 

33 698 267 

34 012 224 
34328125 
34 645 976 

34 965 783 

35 287552 
35 611 289 

35 937 000 

36 264691 

36 594 368 

36 926037 

37 259 704 

A A 


Square Root. 

16.703 293 I 
16.733 200 5 
16.763 0546 
16.792 8556 
16.822 603 8 
16.8522995 
16.881 943 
16.911 5345 
16.941 0743 
16.970 562 7 

17 

17.029 3864 
17.058 722 1 
17.088 007 5 
17.117 242 8 
17.146 428 2 
I 7 -I 75 564 
17.204 6505 
17.233 6879 
17.262 676 5 
17.291 616 5 
17.320 508 1 
17-349 35i6 
I7-378 1472 
17.406 895 2 

17-435 595 8 
17.464 249 2 
17.492 855 7 
17.521 415 5 
17-549 9288 
I7-578 395 8 
17.6068169 
I7-635 192 1 

17.663521 7 
17.691 806 
17.720045 1 
17.748 2393 
17.776 388 8 
17.8044938 
i7- 8 32 554 5 
17.860 571 1 
17.888 5438 
17.916 472 9 

17- 944 358 4 
17.972 200 8 

18 

18.027 7564 

18- 0554701 
18.083 141 3 
18.no 770 3 

18.138357 1 

18.165 902 1 
18.193 4054 
18.220 867 2 
18.248 287 6 
18.275 666 9 


Cube Root. 

6.534 335 1 
6.542 132 6 
6.549911 6 
6.5576722 

6.5654144 

6.573 1385 
6.580 844 3 
6-588 5323 
6.596 202 3 
6.603 854 5 
6.611 489 
6.619 !°6 
6.626 705 4 
6.634 287 4 
6.641 852 2 
6.649 399 8 
6.656 930 2 
6.664 443 7 
6.671 9403 
6.67942 
6.686 883 1 
6.694 329 5 
6.701 759 3 
6.709 172 9 
6.71657 

6.723 95o8 
6.731 3 i 5 5 

6.738 664 1 
6-745 9967 
6-753 3 T 3 4 
6.760 614 3 
6.767899 5 
6-775 169 
6.782 422 9 
6.7S9 661 3 
6.796 884 4 
6.804 092 1 
6.811 284 7 
6.818 462 
6.825 624 2 
6.832 771 4 
6.839 903 7 
6 847 021 3 
6.854 124 
6.861 212 
6.868 285 5 
6.875 344 3 
6.882 388 8 
6.889418 8 
6.8964345 
6.903 435 9 
6 910423 2 
6.9173964 
6.9243556 
6.931 3088 
6.938 232 1 


















UMBER. 

335 

336 

337 

338 

339 

340 

34i 

342 

343 

344 

345 

346 

347 

348 

349 

350 

35i 

352 

353 

354 

355 

356 

357 

358 

359 

360 

361 

362 

363 

364 

3 6 5 

366 

367 

368 

369 

370 

37i 

372 

373 

374 

375 

376 

377 

378 

379 

380 

381 

382 

383 

384 

385 

386 

387 

388 

389 

390 


SQUARES, CUBES, AND ROOTS. 


Square. | 

Cube. 

Square Root. 

Cube Root. 

XI 22 25 

37 595 375 

‘18.303005 2 

6.945 149 6 

11 28 96 

37 933 056 

18.330 302 8 

6.952 053 3 

II 35 69 

38 272 753 

18.357 559 8 

6.9589434 

11 42 44 

38 614472 

18.3847763 

6.965 8198 

II 49 21 

38 958219 

18.411 952 6 

6.972 682 6 

11 56 OO 

39 3°4 000 

18.439 °88 9 

6-979 532 1 

II 62 8l 

39 651 821 

18.466 185 3 

6.986368 1 

11 69 64 

40 001 688 

18.493242 

6.993 1906 

II 7649 

40 353 607 

18.520 259 2 

7 

II 8336 

40 707 584 

i 8.547 237 

7.006 796 2 

1190 25 

41 063 625 

18.574 1756 

7.013 579 1 

ii 97 16 

41 421 736 

18.601 075 2 

7.020 349 

12 0409 

41 781 923 

18.627 936 

7.027 105 8 

12 11 04 

42 144 192' 

18.654 758 1 

7-033 849 7 

12 18 01 

42 50S 549 

18.681 541 7 

7.0405806 

12 25 00 

42 875 000 

18.708 2869 

7.047 298 7 

12 32 01 

43 243 551 

18.734 994 

7.054 004 1 

1239 °4 

43 614 208 

18.761 663 

7.060 696 7 

12 4609 

43 986977 

18.788 294 2 

7.0673767 

125316 

44 361 864 

18.814 887 7 

7-074 044 

12 60 25 

44 738 875 

18.841 443 7 

7.080 698 8 

12 67 36 

45 118016 

18.867 962 3 

7.087341 1 

12 74 49 

45 499 293 

18.894 443 6 

7.093 9709 

12 81 64 

45 882 712 

18.920 887 9 

7.1005885 

12 8881 

46 268 279 

18.9472953 

7.107 1937 

12 9600 

46 656 000 

18.973 666 

7.113 7866 

13 03 21 

47 045 831 

19 

7.1203674 

13 i°44 

47 437 928 

19.026 297 6 

7.126936 

13 17 6 9 

47 832 147 

19.0525589 

7-133 492 5 

13 24 96 

48 228 544 

19.078 784 

7.140037 

13 32 25 

48 627 125 

19.1049732 

7.146 5695 

13 39 56 

49 027 896 

19-131 126 5 

7.153 0901 

13 46 89 

49 430 863 

19.157244 1 

7.159 5988 

13 54 24 

49 836 032 

19.183 3261 

7.166 0957 

13 61 61 

50 243 409 

19.209372 7 

7.1725809 

136900 

50 653 000 

I9-235 384 1 

7.1790544 

137641 

51 064 811 

19.261 360 3 

7.1855162 

13 83 84 

5147884S 

19.287301 5 

7.191 9663 

1391 29 

5i 895 117 

19.3132079 

7.198 405 

13 98 76 

52313624 

I9-339079 6 

7.204 832 2 

1406 25 

52 734 375 

19.364 9 i6 7 

7.211 2479 

14 13 76 

53I57 37 6 

19.3907194 

7.217 652 2 

14 21 29 

53 582 633 

19.416487 8 

7.224045 

14 28 84 

54010 152 

19.442 222 1 

7.2304268 

143641 

54 439 939 

19.467 922 3 

7.236 797 2 

14 44 00 

54 872 000 

19-493 588 7 

7.243 1565 

1451 61 

55 306 341 

19.519 221 3 

7.249 5045 

14 59 24 

55 742 968 

19.5448203 

7.255 841 5 

14 66 89 

56 181 8S7 

19-5703858 

7.262 167 5 

14 74 56 

56 623 104 

* I9-595 9I7 9 

7.268 482 4 

14 82 25 

57066 625 

19.621 416 9 

7.274 7864 

14 89 96 

57512456 

19.646 882 7 

7.281 0794 

14 97 69 

57960 603 

19.6723156 

7.287361 7 

150544 

58411 072 

19.697 7156 

7-293633 

15 1321 

58 863 869 

19.723 082 9 

7.299 8936 

15 21 00 

59319000 

19 74S417 7 

7.3061436 













SQUAKES, CUBES, A]SD BOOTS. 


279 


Square. | Cube. 


152881 
15 3664 
I 5 4449 
15 52 36 
15 6025 
15 68 16 
15 7609 
15 8404 

15 92 01 
160000 

16 08 01 
16 1604 
162409 
1632 16 
1640 25 
16 48 36 
16 56 49 
16 64 64 
16 72 81 

16 81 00 
j 168921 

1697 44 
170569 
171396 

17 22 25 
17 30 56 
1738 89 
174724 
i7 55 6i 
17 6400 
17 72 41' 

17 80 84 
178929 
1797 76 

18 06 25 
18 14 76 
18 23 29 
1831 84 
18 4041 
184900 
185761 
18 66 24 
18 74 89 
18 83 56 

18 92 25 
190096 
190969 

19 18 44 
19 27 21 
193600 
1944 81 
19 53 64 
1962 49 
19 71 36 
19 80 25 
19 89 16 


59 776 47i 

60 236 288 
60698 457 

61 162 984 

61 629 875 

62 099 136 
62 570 773 
63044 79 2 
63521 199 
64000000 
64 481 201 

64 964 808 

65 450 827 

65 939 264 

66 430125 
66923416 

67 4i9 T 43 

67 9 I 7 3 12 

68 417 929 

68 921 000 

69 426 53 1 
69934 528 
70444997 
70957944 

7M73 375 
71 991 296 
72511 713 
73034632 

73 560059 

74 088 000 
74618461 

75 I5M48 

75 686 967 

76 225 024 
76 765 625 
77308776 
77854483 
78402 752 
78953 589 
79507.000 
80062991 
80621 568 
81182 737 

81 746504 
82312875 

82 881 856 

83 453 453 

84 027672 
84604519 

85 184000 

85 766 121 

86 350 888 
86 938 307 
87528384 
88 121 125 
88 716536 


Square Root. 


I 9-773 7 I 9 9 
19.7989899 
19.824 227 6 
19.8494332 
19.874 6069 
19.899 748 7 
19.924 858 8 

19-949 937 3 
19.9749844 

20 

20.024 984 4 
20.0499377 
20.074 859 9 
20.099 75 1 2 
20.124611 8 
20.1494417 
20.174 241 
20.1990099 
20.223 748 4 
20.248 456 7 
20.273 1349 
20.297 783 1 
20.322 401 4 
20.346 989 9 
20.371 548 8 
20.396 078 1 
20.4205779 
20.445 048 3 
20.469 489 5 
20.493 901 5 
20.5182845 
20.542 638 6 
20.566 963 8 
20.591 260 3 
20.615 528 1 
20.639 7674 
20.663 978 3 
20.688 1609 
20.712 315 2 
20.7364414 
20.7605395 
20.784609 7 
20.808 652 
20.832 666 7 
20.856 653 6 
20.880 613 
20.904 545 
20.928 449 5 
20.952 326 8 
20.976 177 

21 

21.023 796 
21.047 5 6 5 2 
21.0713075 
21.095 0231 
21.118 712 1 


Cube Root. 


7.312 382 8 
7.318611 4 
7.3248295 
7.3310369 
7-337 233 9 
7.3434205 
7-349 596 6 
7-355 762 4 
7.3619178 
7.368 063 
7.3741979 
7.3803227 
7.3864373 
7.392 541 8 
7.3986363 
7.404 7206 
7.410 795 
7.4168595- 
7.4229142 
7.428 9589 

7-434 993 8 
7.441 018 9 
7.447 034 2 
7.4530399 

7-459 °35 9 
7.465 022 3 
7.4709991 
7.476 966 4 
7.482 924 2 
7.488 872 4 
7.494811 3 
7.500 740 6 
7.506660 7 
7.512 571 5 
7-5i8 473 
7.5243652 
7.5302482 
7.536 122 1 
7.541 986 7 
7.5478423 
7.553 688 8 
7-559 526 3 
7-565 354 8 
7.571 1743 
7.5769849 
7.582 786 5 
7-588 579 3 
7.5943633 
7.600 138 5 
7.6059049 
7.611 662 6 
7.617 411 6 
7.623 1519 
7.628 883 7 
7.634 606 7 
7.6403213 














280 squares, cubes, and roots. 


Number. 

Square. 

Cube. 

[ Square Root. 

Cube Root. 

447 

199809 

89314623 

31.1423745 

7.646 027 2 

44S 

20 07 04 

899 I 5 39 2 

21.1660105 

7.651 724 7 

449 

20 l6oi 

90518849 

2I1189 620 I 

7.657 413 8 

450 

20 25 00 

QI 12^000 

21.213 203 4 

7.663 094 3 

45 i 

20 34 OI 

91 733 851 

21.236 7606 

7.668 766 5 

452 

20 43 04 

92 345 40S 

21.260 291 6 

7.674 430 3 

453 

20 5209 

92 959677 

21.283 79 6 7 

7.680085 7 

454 

20 61 l6 

93576664 

21.3072758 

7.685 732 8 

455 

20 70 25 

94196375 

> 21.330 729 

7.691 371 7 

456 

20 79 36 

94818816 

21.3541565 

7.697 002 3 

457 

20 88 49 

95 443 993 

2 I -377 5583 

7.702 6246 

458 

20 97 64 

96071 912 

21.4009346 

7.708 238 8 

459 

21 0681 

96 702 579 

21.424 2853 

7.7138448 

460 

21 1600 

97 336 000 

21.447 6106 

7.7194426 

461 

21 25 21 

97972 181 

21.4709106 

7.7250325 

462 

21 34 44 

98611 128 

21.4941853 

7.7306141 

4 6 3 

214369 

99252 847 

21.5174348 

7.736 187 7 

.464 

21 52 96 

99 897 344 

21.5406592 

7.741 7532 

465 

21 62 25 

100 544 625 

21.5638587 

7-747 3 io 9 

466 

21 71 56 

101194 696 

21.587033 1 

7.752 8606 

467 

21 80 89 

101847 563 

21.610 182 8 

7.7584023 

468 

21 90 24 

102 503 232 

21.6333077 

7.7639361 

469 

21 9961 

103 161 709 

21.656407 8 

7.769462 

470 

22 O9OO 

103 823 000 

21.6794834 

7.7749801 

47 i 

22 1841 

104487 hi 

21.7025344 

7.7804904 

472 

22 27 84 

105 154048 

21.725 561 

7.7859928 

473 

223729 

105 823817 

21.7485632 

7.791 487 5 

474 

22 46 76 

106 496 424 

21.771541 1 

7.7969745 

475 

22 56 25 

10717187 5 

21.7944947 

7.802 453 8 

476 

22 65 76 

107 850 176 

21.8174242 

7.807 925 4 

477 

22 75 29 

108 531 333 

21.8403297 

7.813 3892 

478 

22 84 84 

109 215 352 t 

21.863 211 1 

7.818 845 6 

479 

22944I 

109 902 239 

21.886 068 6 

7.824 294 2 

480 

23 04 OO. 

no 592 000 

21.9089023 

7-829 735 3 

481 

23 13 6l 

II [ 284 641 

21.931 7122 

7.835 1688 

482 

23 23 24 

III 980 168 

21.9544984 . 

7.840 594 9 

483 

233289 

112 678 587 

21.977 261 

7.8460134 

484 

234256 

113 379 904 

22 

7.8514244 

485 

23 52 25 

II4084 125 

22.022 715 5 

7.856 828 1 

486 

23 6l 96 * 

114 79I 256 

. 22.045 407 7 

7.862 224 2 

487 

237169 

II55OI303 

22.068076 5 

7.867 613 

488 

23 8144 

Il6 214 272 

22.090 722 

7.8729944 

489 

23 91 21 

I1693O 169 

22 .H 3 344 4 

7.878 3684 

490 

24OI OO 

117 649OOO 

22.1359436 

7-883 735 2 

49 1 

24 I0 8l 

I1837077I 

22.1585198 

7.8890946 

492 

24 20 64 

II9095488 

22.181 073 

7.8944468 

493 

243049 

II9823 157 

22.203 603 3 

7.899 791 7 

494 

244036 

!20 553 784 

22.226 1108 

7.905 1294 

495 

24 50 25 

121287375 

22.248 595 5 

7.9104599 

496 

24 60 l6 

122023936 

„ 22.2710575 

7.9157832 

497 

24 7OO9 

122 763473 

22.293 496 8 

7.9210994 

498 

24 80 04 

123 505 992 

22.3159136 

7.926 408 5 

499 

249OOI 

124251499 

22.3383079 

7.931 7104 

500 

25 OOOO 

125000000 

22.3606798 

■ 7-937 005 3 

501 

25 IO OI 

125 751 501 

22.3830293 

7.942 293 1 

502 

25 20 04 

126506008 

22.405 356 5 

7-947 5739 












UM-BEK. 

503 

504 

505 

506 

507 

508 

509 

510 

511 

512 

513 

514 

515 

Sl6 

517 

518 

519 

520 

521 

522 

523 

524 

525 

526 

527 

528 

529 

530 

531 

532 

533 

534 

535 

536 

537 

538 

539 

540 

54 i 

542 

543 

544 

545 

546 

547 

548 

549 

550 

55 i 

552 

553 

554 

555 

556 

557 

558 


SQUARES, CUBES, AND ROOTS. 


28l 


Square. 

253009 
25 40 16 
'25 50 25 
25 60 36 

25 70 49 

25 80 64 

25 90 81 

26 01 00 
26 11 21 
26 21 44 
2631 69 
26 41 96 
26 52 25 
26 62 56 
26 72 89 
26 83 24 

26 9361 
270400 

27 1441 
27 24 84 
27 35 29 
27 45 76 
275625 
27 66 76 

I 277729 
27 87 84 

27 98 41 

28 09 00 
28 19 61 
28 30 24 
28 40 89 
2851 56 
28 62 25 
28 72 96 
28 83 69 

28 94 44 

29 05 21 
29 1600 
29 26 81 
29 37 64 
29 48 49 
29 59 3 6 
29 70 25 
29 81 16 

29 92 09 
300304 

30 1401 
30 25 00 
30 3601 

304704 

305809 
30 69 16 

30 80 25 
3091 36 

3 1 02 49 

3 1 *3 64 


Cube. 

127263527 
128 024 064 

128 787 625 

129 554216 
130.323 843 
131 096512 

131 872 229 

132 651 000 

133 432 831 

134 217 728 

135 005 697 

135 79 6 744 

136 590875 

137 388 096 

138 188413 
138991 832 

139 798 359 
140608 000 

141 420 761 

142 236 648 

143 055 667 
143877824 

144 703 125 

145 53i 576 

146 363 183 

147 i97 952 

148 035 889 

148 877 000 

149 721 291 

150 568 768 

15 1 419 437 
152273304 

153 U30 375 
153990656 

154 854153 

155 720872 

156 590819 

157 464000 
158340 421 

159 220 088 

160 103 007 

160 989 184 

161 878 625 

162 771 336 

163 667 323 

164 566 592 

165 469 149 

166 375 000 

167 284 151 

168 196608 

169 112 377 

170 031 464 

170 953 875 

171 879616 

172 808 693 

1 73 74i 112 

A a* 


Square Root. 

22.427 661 5 
22.449 9443 
22.472 205 1 
22.494 4438 
22.516 660 5 
22.538 8553 

22.561 028 3 
22.583 1796 
22.605 309 1 
22.627 417 
22.649 503 3 
22.671 568 1 
22.693611 4 

22.7156334 

22.737634 

22.7596134 

22.781 571 5 
22.8035085 
22.825 4244 
22.8473193 
22.869 193 3 
22.891 046 3 
22.912 878 5 
22.934 6899 
22.956 4806 
22.978 2506 

23 

23.021 728 9 
23-043 437 2 
23.065 125 2 
23.086 792 8 
23.108 44 
23.130 067 
23-151 6738 
23.173 2605 
23.194 827 
23.2163735 
23.237 900 1 
23.2594067 
23.280 8935 
23 302 360 4 

23 323 807 6 
23-345 235 1 
23.3666429 
23.388 031 1 
23-409 399 8 
23-430 749 
23.452 078 8 
23-473 389 2 
23.494 680 2 
23 - 5 I 5 952 
23-537 2046 
23558 438 
23-5796522 
23.600 847 4 
23.622 023 6 


Cube Root. 


7.952 847 7 
7.958 1144 

7-963 374 3 
7.968 627 1 
7-973 873 1 
7.979 112 2 
7.9843444 
7.989 5697 
7.994 788 3 
8 

8.005 204 9 
8.010 403 2 
8.015 5946 
8.020 7794 
8.025 9574 
8.031 128 7 
8.036 293 5 
8.041451.5 
8.046 603 
8.051 7479 
8.056 886 2 
8.062 018 
8.067 143 2 
8.072 262 
8.077 374 3 
8.082 48 
8.0875794 
8.092 672 3 

8.097 7589 

8.102 839 
8.107 912 8 
8.112 980 3 
8.118 041 4 
8.1230962 
8.128 144 7 
8.133187 
8.138 223 
8.1432529 
8.148 276 5 
8.153 2939 

8.158305 1 

8.1633102 
8.168 309 2 
8.173 302 
8.178 288 8 
8.183 2695 
8.188 244 1 
8.193 212 7 
8.1981753 
8.203 131 9 
8.208 082 5 
8.213027 x 
8.2179657 
8.222 898 5 
8.227 825 4 
8.232 7463 
















56i 

562 

563 

564 

565 

566 

S67 

568 

569 

570 

57i 

572 

573 

574 

575 

576 

577 

578 

579 

580 

58i 

582 

583 

584 

585 

586 

587 

588 

589 

590 

59i 

592 

593 

594 

595 

596 

597 

598 

599 

600 

601 

602 

603 

604 

605 

606 

607 

608 

609 

610 

611 

612 

613 

614 


SQUARES, CUBES, AND ROOTS. 


Square. 

Cube. 

Square Root. 

| Cube Root. 

31 2481 

174 676879 

23.643 1808 

8.237 66l 4 

313600 

175 616000 

23.664 319 I 

8.242 5706 

3i 47 21 

176558481 

23.685 438 6 

8.247 474 

3158 44 

177504328 

23.706 539 2 

8.2523715 

316969 

178453 547 

23.727 621 

8.2572633 

31 8096 

179406 144 

23.748 6842 

8.262 I49 2 

31 92 25 

180362 125 

23.769 7286 

8.267 O294 

32 03 56 

181 321 496 

23.790 754 5 

8.271 903 9 

32 14 89 

182 284 263 

x 23.8117618 

8.276 7726 

32 26 24 

183 250432 

23.832 750 6 

8.28l 625 5 

32 37 61 

184220009 

23.853 7209 

8.286492 8 

32 49 °° 

185 193000 

23.874 672 8 

8.2913444 

32 60 41 

186 169411 

23.8956063 

8.296 190 3 

32 71 84 

187 149 248 

23.916521 5 

8.3010304 

32 83 29 

188132517 

23.9374184 

8.305 865 1 

32 94 76 

189 119 224 

23.958 297 1 

8.310694 1 

33 °6 25 

190 109375 

23-979 J 57 6 

8.315 517 5 

33 17 76 

191 102976 

24 

8.3203353 

33 29 29 

192 100033 

24.020 824 3 

8.325 147 5 

334084 

193 100552 

24.041 630 6 

8.329954 2 

33 52 41 

194 104 539 

24.062 418 8 

8-334 755 3 

336400 

195 112 000 

24.083 189 1 

8-339 550 9 

33 75 61 

196 122 941 

24.103941 6 

8.344341 

33 87 24 

197 137 368 

24.124 6762 

8.349 125 6 

339889 

198 155 287 

24.145 3929 

8-353 904 7 

34 10 56 

199 176 704 

24.166091 9 

8.3586784 

34 22 25 

200 201 625 

24.1867732 

8.3634466 

34 33 96 

201 230 056 

24.207 436 9 

8.368 209 5 

3445 69 

202 262 003 

24.228 082 9 

8 372 9668 

34 57 44 

203297472 

24.2487113 

8.3777188 

346921 

204336469 

24.269 322 2 

8.382 465 3 

34 81 00 

205 379 000 

24.289915 6 

8.387 2065 

3492 8i 

206425 071 

24.3104916 

8.3919423 

35 04 64 

207 474 688 

24-33I 050 1 

8.396 672 9 

35 1649 

208527857 

24-351 59 1 3 

8.401 398 I 

35 28 36 

209 584 584 

24.3721152 

8.406 118 

35 40 25 

210644875 

24.392 621 8 

8.4108326 

35 52 16 

211 708 736 

24-413 111 2 

8.415 54i 9 

356409 

212776173 

24-433 583 4 

8.420 246 

35 7604 

213 847 192 

244540385 

8.424 944 8 

35 88 01 

214 921 799 

24-474 476 5 

8.429 638 3 

360000 

216000000 

24.494 897 4 

8.434 326 7 

36 1201 

217 081 801 

24-5153013 

8.439 009 8 

36 24 04 

218 167 208 

24-5356883 

8-443 687 7 

36 3609 

219256 227 

24-5560583 

8.448 360 5 

3648 16 

220 348 864 

24.576411 5 

8.453 028 1 

36 60 25 

221 445 125 

24.596 747 8 

8.4576906 

36 72 36 

222545016 ■ 

24.6170673 

8.462 347 9 

368449 

223648 543 

24-637 37 

8.467 

36 96 64 

224 755 712 

24.657 656 

8.471 647 1 

37 08 81 

225 866 529 

24.677 925 4 

8.476 289 2 

37 21 00 

226 981000 

24.698 178 1 

8.480926 1 

37 33 21 

228 099 131 

24.718414 2 

8.485 5579 

37 45 44 

229 220 928 

24-7386338 

8.490 184 8 

37 57 69 

230346397 

24-7588368 

8.494 806 5 

376996 

231*475 544 

24.7790234 

8.4994233 
















SQUARES, CUBES, AND ROOTS. 


Square. 

Cube. 

Square Root. 

37 82 25 

232 608 375 

24.799 *93 5 

37 94 56 

233 744 896 

24.8193473 

38 06 89 

234885 113 

24.839 484 7 

38 19 24 

236 029 032 

24.859 605 8 

3831 61 

237 176659 

24.879 7106 

38 44 00 

238 328 000 

24.899 799 2 

38 5641 

239 483061 

24.919871 6 

38 68 84 

240641 848 

24.9399278 

3881 29 

24 x 804 367 

24.9599679 

38 93 76 

242 970 624 

24.979992 

390625 

244 140625 

25 

39 18 76 

245 134 376 

25.019992 

393129 

246491 883 

25.039968 1 

3943 84 

247 673152 

25.059 9282 

39 5641 

248 858 189 

25.0798724 

39 69 00 

250 047 000 

25.099 8008 

398161 

251239 591 

25- II 9 7 I 34 

3994 24 

252 435 968 

25.1396102 

400689 

253636 137 

25.159491 3 

40 19 56 

254 840 104 

25.179 356 6 

4032 25 

256047875 

25.199 2063 

404496 

257 259456 

25.2190404 

405769 

258 474853 

25.238 8589 

407044 

259694072 

25.258661 9 

4083 21 

260917 119 

25.278 4493 

40 96 00 

262 144 000 

25.298 221 3 

41 0881 

263374721 

2 5-3 I 7 977 8 

41 21 64 

264 609 288 

25.3377189 

4i 3449 

265 847 707 

25-357 444 7 

4i 47 36 

267 089 984 

25.377I55I 

41 60 25' 

26S336125 

25.3968502 

41 73 16 

269 585 136 

25.4165301 

41 86 09 

270 840 023 

25436 194 7 

419904 

272 097 792 

25.455 8441 

42 12 01 

2.73 359 549 

25.475 478 4 

42 25 00 

274 625 000 

25.495 0976 

42 3801 

275 894 45i 

25.5147016 

425104 

277 167 808 

25.534 290 7 

42 64 09 

278445 077 

25-553 864 7 

42 77 16 

279 726 264 

25.573423 7 

429025 

281 on 375 

25.5929678 

43 03 36 

282 300416 

25.6124969 

43 1649 

283 593393 

25.632011 2 

432964 

284 890312 

25.6515107 

434281 

286 191 179 

25.6709953 

43 56oo 

287 496 000 

25.6904652 

436921 

288 804 781 

25.7099203 

43 82 44 

290 117 528. 

25.729 3607 

43 95 69 

291 434 247 

25.748 7864 

44 08 96 

292 754 944 

25.768 197 5 

44 22 25 

294079625 

25.787 5939 

44 35 56 

295 408 296 

25.8069758 

44 48 89 

296740963 

25.8263431 

4462 24 

298077632 

25.845696 

44 75 61 

299418309 

I 25.8650343 

448900 

300 763 000 

| 25.8843582 


283 


Cube Root. 


8.504035 
8.508 641 7 
8-5I3 243 5 
8.5178403 
8.522 432 I 
8.5270189 
8.5316009 
8.536178 
8.540 750 I 
8.545 3173 
8.549 879 7 
8.554 437 2 

8.558 9899 
8.563 537 7 
8.568080 7 
8.5726189 
8.577152 3 
8.581 6809 
8.586 204 7 
8.590 723 8 
8.595 238 
8-599 747 6 
8,604 252 5 
8.608 752 6 
8.613 248 
8.6177388 
8.622 224 8 
8.626 706 3 
8.631183 
8.635 655 1 
8.640 1226 

8.644 585 5 

8.6490437 
8.653497 4 

8.6579465 

8.662 391 1 
8.666831 
8.671 2665 

8.6756974 

8.680 123 7 
8.684 545 6 
8.688 963 
8.693 375 9 
8.697 7843 
8.702 188 2 
8.706 5877 
8.710982 7 
8.715 3734 
8.7197596 
8.724 141 4 
8.7285187 
8.732 891 8 

8.7372604 

8.741 6246 
8-745 9846 
8.7503401 













671 

672 

673 

674 

675 

6 7 6 

677 

678 

679 

680 

681 

682 

683 

684 

685 

686 

687 

688 

689 

690 

691 

692 

693 

694 

695 

696 

697 

698 

699 

700 

701 

702 

703 

704 

705 

706 

707 

708 

709 

710 

711 

712 

7i3 

714 

7i5 

716 

717 

718 

719 

720 

721 

722 

723 

724 

725 

726 


SQUARES, CUBES, AND ROOTS. 


Square. 

Cube. 

45 02 41 

302 III 711 

45 15 84 

303 464 448 

45 29 29 

304821 217 

45 42 76 

306 182 024 

45 56 25 

307 546875 

456976 

308915776 

45 83 29 

310288 733 

459684 

311665752 

46 1041 

313046839 

46 24 00 

314432000 

463761 

315 821 241 

4651 24 

317214568 

46 64 89 

318611987 

46 78 56 

320013504 

4692 25 

321 419 125 

4705 96 

322 828 856 

47 1969 

324 242 703 

47 3344 

325 660 672 

47 47 21 

327082 769 

47 6,i 00 

328 509 OOO 

47 748 i 

329939371 

47 88 64 

33U873 888 

48 02 49 

332812557 

48 1636 

334 255 384 

48 3° 25 

335 7°2 375 

48 44 16 

337153 536 

485809 

338 608 873 

48 72 04 

340 068 392 

48 86 01 

34i 532 099 

490000 

343 OOO OOO 

49 14 01 

344472 IOI 

49 28 04 

345948408 

494209 

347 428 927 

49 56 16 

348913664 

49 70 25 

350 402 625 

498436 

351 895 816 . 

4998 49 

353 393 243 

50 1264 

354 894912 

50 2681 

356400829 

5041 00 

357 911 OOO 

505521 . 

359425 431 

506944 

360944 128 

50 83 69 

362 467 097 

509796 

363994 344 

51 1225 

365 525 875 

51 2656 

367 061 696 

514089 

368 601 813 

5i 55 24 

370 146 232 

51 6961 

371694 959 

51 8400 

373 248000 

51 98 41 

374 805 361 

52 1284 

376367048 

52 27 29 

377 933 067 

52 41 76 

379503 424 

52 56 25 

381078125 

S 2 7° 76 

382657176 | 


Square Root. 

Cube Root. 

25.9036677 

8.7546913 

25.922 962 8 

8.759 038 3 

25:942 243 5 

8.7633809 

25.961 51 

8.767 7192 

25.980 762 1 

8.772 053 2 

26 

8.776383 

26.019 223 7 

8.780 7084 

26.038 433 1 

8.7850296 

x 26.0576284 

8.789 3466 

26.076 809 6 

8.793 6593 

26.095 976 7 

8.7979679 

26.115 129 7 

8.802 272 I 

26.134 268 7 

8.806 572 2 

26.1533937 

8.810868 I 

26.172 5047 

8.815 1598 

26.191601 7 

8.8194474 

26.2106848 

8.823 73O 7 

26.229 754 1 

8.8280099 

26.248 8095 

8.832 285 

26.267 851 1 

8.836 555 9 

26.2868789 

8.840822 7 

26.305 892 9 

8.845 085 4 

26.324 893 2 

8.849344 

26.343 879 7 

8.853 598 5 

26.362 852 7 

8.8578489 

26.381 811 9 

8.862 095 2 

26.400 757 6 

8.8663375 

26.419 689 6 

8.8705757 

26.438 608 1 

8.874 809 9 

26.4575131 

8.87904 

26.476 404 6 

8.883 266 1 

26.495 282 6 

8.887 488 2 

26.5141472 

8.891 7063 

26.532 998 3 

8.895 920 4 

26.5518361 

8.900 1304 

26.5706605 

8.9043366 

26.5894716 

8.908 538 7 

26.608 269 4 

8.912 7369 

26.6270539 

8.9169311 

26.645 825 2 

8.921 121 4 

26.664 583 3 

8.925 307 8 

26.683 328 1 

8.929 490 2 

26.702 059 8 

8.933 668 7 

26.720 7784 

8.937 8433 

26.7394839 

8.942 014 

26.7581763 

8.946 1809 

26.776855 7 

8.950 3438 

26.795 522 

8.954 502 9 

26.8141754 

8.958 658 1 

« 26.8328157 

8.962 809 5 

26.851 443 2 

8.966957 

26.870057 7 

8.9711007 

26.888 659 3 

8.975 2406 

26.907 248 1 

• 8.9793766 

26.925 824 

8.983 508 9 

26.944 387 2 

8.9876373 












UM-UA.lt., 

727 

728 

7 2 9 

73° 

73 i 

732 

733 

734 

735 

73 6 

737 

73§ 

739 

740 

74 i 

742 

743 

744 

745 

746 

747 

748 

749 

750 

75i 

752 

753 

754 

755 

756 

757 

758 

759 

760 

761 

762 

763 

764 

7 6 5 

766 

767 

768 

769 

770 

771 

772 

773 

774 

775 

776 

.777 

778 

779 

780 

781 

782 


SQUARES, CUBES, AND ROOTS. 


Square. t Cube. ] Square Root. 


52 85 29 

52 99 84 

53 14 41 
532900 
53 43 61 
53 58 24 
53 72 89 

53 87 56 

54 0225 
54 16 96 
54 3 1 69 
54 4644 
5461 21 

54 76 00 
549081 

55 05 64 
55 20 49 
55 35 36 
55 5025 
55 6 5 16 
55 80 09 

55 95 04 

56 10 01 
562500 
564001 
56 55 04 
z 6 70 09 

56 85 16 
570025 
5715 36 

57 30 49 
57 45 64 
576081 
57 7600 

57 9 1 21 

58 06 44 
58 21 69 
58 36 96 
585225 
5S 67 56 
58 82 89 

58 98 24 

59 x 3 61 
59 29 00 
59 44 4i 
59 59 84 

59 75 29 
599076 

60 06 25 
60 21 76 
603729 
60 52 84 
6068 41 
60 84 00 

60 99 61 

61 15 24 


384 240 583 

385 828 352 
387 420 489 
389017 000 
390617 891 

392 223 168 

393 832 837 
395446904 

397 065 375 

398 688 256 

400315553 

401 947 272 

403 583 4i9 
405 224000 
406869 021 
408 518 488 

410 172 407 

411 830 784 
413493625 
4*5 160936 
416832 723 
418 508 992 

420 189 749 

421 875 000 
423 564 75i 

425 259 008 

426 957 777 
428 661 064 
430 368 875 

432 081 216 

433 798 093 
435 5 I 9 5 12 

437 245 479 

438 976 000 
440 711 081 
442 450 728 
444194947 
445 943 744 
447 697 I2 5 
449 455 096 

451 217663 

452 984 832 
454 756609 
456 533 000 
458 3 : 4 011 

460 099 648 

461 889917 
463 684 824 
465 484 375 
467 288 576 

469097 433 
470910952 
472 729 i39 
474 552 000 
476 379 54i 
478 211 768 


26.962 937 5 
26.981 475 1 
27 

27.018 512 2 
27.037 on 7 
27.055 498 5 
27.073 972 7 
27.092 4344 
27.1108834 
2 7- I2 93 1 99 
2 7- I 47 7439 
27.166 1554 

2 7- l8 4 5544 
27.202 941 
27.221 3152 
27.239 676 9 
27.2580263 
27.276 3634 
27.294 688 1 
27.313 0006 
27-33I 300 7 
27-3495887 
27.367 8644 
27.386 127 9 
27-404 379 2 
27.422 618 4 
27.440 845 5 
27.4590604 
27.4772633 
27-495 454 2 

27-513633 

27-531 799 8 
27-549 954 6 
27.568 0975 
27.586 228 4 
27.604 347 5 
27.622 4546 
27.6405499 
27.6586334 
27.676 705 
27.694 764 8 
27.712 812 9 
27.730 8492 
27.7488739 
27.766 886 8 
27.784 888 
27.802 877 5 
27.820 855 5 
27.838 821 8 
27.856 7766 
27.8747197 
27.892 651 4 

2 7-9 IQ 57 1 5 
27.928 480 1 
27-9463772 
27.964 262 9 


285 

Cube Root. 

8.991 762 
8.995 882 9 
9 

9.004 1134 
9.008 222 9 
9.012 328 8 
9.0164309 
9.020 529 3 
9.024 623 9 
9.028 7149 
9.032 802 1 
9.036 885 7 
9.040 965 5 
9.045 041 7 
9.049 114 2 
9.053 183 1 
9.0572482 
9.061 309 8 
9.065 367 7 
9.069 422 
9.0734726 
9.0775197 
9.081 563 1 
9.085 603 
9.089 639 2 
9093671 9 
9 097 701 
9.101 7265 

9- io 5 748 5 
9.109 7669 
9.113 7818 

9 -H 7 793 1 
9.121 801 
9.125 8053 
9.129 806 1 
9-133 8034 
9- I 37 7971 
9- I 4 I 7874 
9- I 45 774 2 
9- I 49 757 6 
9- J 53 737 5 
9- I 57 7 I 39 
9.161 6869 
9.165 6565 
9.169 622 5 
9- I 73 5852 
9- I 77 544 5 
9.181 5003 
9.185 4527 
9.189 401 8 

9- I 93 347 4 
9.197 289 7 
9.201 2286 
9.205 164 1 
9.209 0962 
9.213025 












286 


SQUARES, CUBES, AND ROOTS. 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

783 

61 30 89 

480 048 687 

2^.982 137 2 

9.2169505 

784 

6l 46 56 

481 890 304 

28 

9.2208726 

785 

61 62 25 

483 736625 

28.617851 5 

9.224 791 4 

786 

6l 77 96 

485 587656 

28.035 691 5 

9.228 706 8 

787 

6l 93 69 

487 443 403 

28.0535203 

9.2326189 

788 

620944 

489303872 

28.071 337 7 

9.2365277 

789 

62 25 21 

491169069 

28.089 I 43 8 

9.2404333 

790 

62 41 OO 

493039000 

28.1069386 

9.2443355 

791 

62 56 8l 

494 9 I 3 671 

' 28.1247222 

9.2482344 

79 2 

62 72 64 

496 793 088 

28.142 4946 

9.252 13 

793 

62 88 49 

498677257 

28.1602557 

9.2560224 . 

794 

630436 

500 566 184 

28.1780056 

9 - 2599 TI 4 

795 

63 20 25 

502 459875 

28.195 7444 

9.2637973 

796 

63 36 16 

504358336 

28.213472 

9.267 679 8 

797 

63 52 09 

506 261 573 

28.231 1884 

9.2715592 

798 

63 68 04 

508169 592 

28.248 893 8 

9-275 435 2 

799 

63 84 01 

510082399 

28.266 588 1 

9.2793081 

800 

640000 

512000000 

28.284 27 1 2 

9.2831777 

801 

64 1601 

513 922 401 

28.3019434 

9.287 044 

802 

64 32 04 

515849608 

28.3196045 

9.290 907 2 

803 

64 48 09 

517781627 

28.337 254 6 

9.294 767 1 

804 

6464 16 

519718464 

28.354 893 8 

9.298 623 9 

805 

64 80 23 

521 660 125 

28.372521 9 

9.302 477 5 

806 

649636 

523606616 

28.390 139 1 

9.306 327 8 

807 

65 12 49 

525 557 943 

28.407 745 4 

9 - 3 IOI 75 

808 

65 28 64 

527514112 

28.425 340 8 

9.314 019 

809 

654481 

529475129 

28.442 9253 

9.3178599 

810 

65 61 00 

531 441 000 

28.460498 9 

9.321 697 5 

811 

65 77 21 

533 4 11 73 i 

28.478 061 7 

9-325 532 

812 

65 93 44 

535 387 328 

28.495 613 7 

9-3293634 

813 

660969 

537 367 797 

28.5131549 

9-333 191 6 

814 

66 25 96 

539353 144 

28.530 685 2 

9.337 016 7 

815 

66 42 25 

54 i 343 375 

28.548 2048 

9.3408386 

816 

66 58 56 

543 338 496 

28.565 713 7 

9.3446575 

817 

66 74 89 

545 338513 

28.583211 9 

9-348 473 1 

818 

66 91 24 

547 343 432 

28.6006993 

9.352 285 7 

819 

67 0761 

549353 259 

28.618 176 

9-3560952 

820 

67 2400 

551368000 

28.635 642 1 

9.3599016 

821 

67 40 41 

553387 66 i 

28.653 097 6 

9.363 704 9 

822 

67 56 84 

555 412 248 

28.6705424 

9-367 505 1 

823 

67 73 29 

557 44 1 767 

28.6879766 

9.3713022 

824 

67 89 76 

559476 224 

28.705 400 2 

9.3750963 

825 

68 06 25 

561 515 625 

28.722 813 2 

9-378 8873 

826 

68 22 76 

563 559976 

28.740215 7 

9.382 675 2 

827 

6839 29 

565 609 283 

28.7576077 

9.38646 

828 

68 55 84 

567 663 552 

28.7749891 

9.3902419 

829 

68 7241 

569722 789 

28.792 360 1 

9.3940206 

830 

68 89 00 

571 787000 

28.809 7206 

9-397 7964 

831 

69 05 61 

573856 191 

28.8270706 

9.401 5691 

832 

69 22 24 

575 93 ° 368 

-28.8444102 

9 . 4 0 5 338 7 

833 

69 38 89 

578009537 

28.861 7394 

9 409 105 4 

834 

6955 56 

580 093 704 

288790582 

9.412 869 

835 

69 72 25 

582 182 875 

28.896 366 6 

9.416 629 7 

836 

69 88 96 

584 277 056 

28.913 664 6 

9.4203873 

837 

700569 

586376 253 

28.9309523 

9.424 142 

838 

70 22 44 

588 480 472 

28.948 229 7 | 

9.4278936 









SQUARES, CUBES, AND ROOTS. 


28 7 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

839 

703921 

590589719 

28.965 496 7 

9.431 642 3 

840 

70 5600 

592 704 OOO 

28.982 753 5 

9-435 38 

841 

70 72 8l 

594 823 321 

29 

9.439 130 7 

842 

70 89 64 

•596947 688 

29.0172363 

9.442 870 4 

843 

71 0649 

599077107 

29.034 462 3 

9.446 607 2 

844 

71 2336 

601 211 584 

29 051 678 I 

9.45034I 

845 

71 40 25 

603351125 

29.068 883 7 

9.4540719 

846 

71 5716 

605 495 736 

29.086 079 I 

9-457 7999 

847 

71 7409 

607 645 423 

29.1032644 

9.461 5249 

848 

71 91 04 

609 800 192 

29.120 4396 

9.465 247 

849 

72 08 OI 

611 960049 

29.1376046 

9.468 966 1 

850 

72 25 OO 

614 125000 

29-154 759 5 

9.472 682 4 

851 

72 4201 

616295051 

29.1719043 

9.4763957 

852 

72 5904 

618470 208 

29.189039 

9.480 106 1 

853 

72 76 09 

620650477 

29.206 163 7 

9.4838136 

854 

7 2 93 16 

622 835 864 

29.223 278 4 

9.4875182 

855 

73 10 2 5 

625026375 

29.240383 

9.491 22 

856 

. 73 27 36 

627222016 

29.257 477 7 

9.4949188 

857 

73 44 49 

629 422 793 

29.274 562 3 

9.4986147 

8 5 8 

73 61 64 

631 628 712 

29.291 637 

9.502 307 8 

859 

73 7881 

633 839 779 

29.308 701 8 

9.505 998 

860 

739600 

636 056 OOO 

29-325 7566 

9.5096854 

861 

74 1321 

638 277 381 

29.342 801 5 

9.5133699 

862 

74 3 ° 44 

640 503 928 

29-3598365 

9.517 051 5 

863 

74 4769 

642 735 647 

29.376 861 6 

9.520 7303 

864 

746496 

644972 544 

29-3938769 

9.5244063 

865 

74 82 25 

647214625 

29.410882 3 

9.5280794 

866 

7499 56 

649 461 896 

29.427 877 9 

9 - 53 I 749 7 

867 

75 16 89 

651 714363 

29.4448637 

9 - 5354 I 7 2 

868 

75 34 24 

653 972 032 

29.461 839 7 

9.5390818 

869 

75 5 i 6 i 

656 234909 

29.478 805 9 

9.542 743 7 

870 

756900 

658 503 OOO 

29.495 762 4 

9.5464027 

871 

758641 

660776311 

29.512 709 1 

9-5500589 

872 

760384 

663054848 

29.529646 1 

9-553 7 12 3 

873 

76 21 29 

665 338617 

29-546 573 4 

9-557 363 

874 

763876 

667 627 624 

29563491 

9.561 0108 

875 

76 56 25 

669921 875 

29.580 3989 

9.5646559 

876 

76 73 76 

672 221 376 

29.597 297 2 

9.568 298 2 

877 

7691 29 

674 526 133 

29 614 185 8 

9-571 937 7 

878 

77 08 84 

676 836152 

29.631 064 8 

9-575 5745 

879 

77 2641 

679151439 

29.647 934 2 

9.579 2085 

880 

774400 

681 472 OOO 

29.664 793 9 

9.582 839 7 

881 

77 61 61 

683 797 841 

29.681 644 2 

9.586 468 2 

882 

77 79 24 

686 128 968 

29.698 484 8 

9.5900937 

883 

77 96 89 

688 465 387 

29 - 7 I 5 3 I 5 9 

9'593 7 i 69 

884 

78 14 56 

690 807 104 

29.7321375 

9-597 337 3 

885 

78 32 25 

693 154 125 

29.7489496 

9.600 954 8 

886 

784996 

695 506456 

29 765 752 1 

9.604 569 6 

887 

78 67 69 

697 864 103 

29.782 5452 

9.608 181 7 

888 

78 85 44 

700227072 

29.7993289 

9.6x1 7911 

.889 

790321 

702 595 369 

29.816 103 

9 - 6 r 5 397 7 

890 

7921 00 

704969000 

29.832 867 8 

9.619001 7 

891 

793881 

707 347 971 

29.849623 1 

9.622 603 

892 

79 5664 

7°9 732 288 

29.866369 

9.626 2016 

893 

79 7449 

712 121 957 

29.883 105 6 

9.629 797 5 

894 

7992 36 

1 714516984 

29.899 832 8 

9-633390 7 












288 

Numbe: 

895 

896 

897 

898 

899 

900 

901 

902 

9°3 

9°4 

9°5 

906 

9°7 

908 

909 

910 

911 

912 

9 i 3 

9 r 4 

9 i 5 

916 

917 

918 

919 

920 

921 

922 

933 

924 

9 2 5 

926 

927 

928 

929 

930 

93 i 

93 2 

933 

934 

935 

936 

937 

938 

939 

940 

941 

942 

943 

944 

945 

946 

947 

948 

949 

950 


SQUARES, CUBES, AND ROOTS. 


Square. 

80 IO 25 
80 28 16 
80 4609 
80 64 04 

80 82 OI 

81 OOOO 

8l 1801 
8l 3604 
8l 5409 
8l 72 16 

81 90 25 

82 08 36 
82 26 49 
82 44 64 
82 62 8l 
82 8l OO 

82 99 21 

83 17 44 
83 35 69 
83 539 6 
83 72 25 

83 9 ° 56 

84 08 89 
84 27 24 
84 45 61 
84 64 00 

84 8241 
850084 

85 1929 
85 37 76 
85 56 25 
85 74 76 

85 93 29 

86 11 84 
863041; 
86 49 00 
86 67 61 

86 86 24 

87 04 89 

87 23 56 
87 42 25 
87 60 96 
877969 

87 98 44 

88 17 21 
88 3600 
88 54 81 
88 73 64 

88 92 49 
891136 
8930 25 

89 49 16 
89 68 09 

89 87 04 

90 06 01 
902500 


Cube. 


716917375 
7^ 323 136 
7 2I 734273 
724150792 
726 572 699 
729 OOO OOO 

73 1 432 701 
733 870 808 
736314327 
738 763 264 
741 217625 
743 677 4i6 
746142643 
748613312 
751 089 429 
753 57i 000 
756058031 
758550528 
761 048 497 
763 551944 
766060 875 
768 575 296 

771095213 

773 620632 
776151 559 
778688000 
781 229961 
783 777 448 
786330467 
788 889 024 

79 1 453 I2 5 
794022 776 
796597 983 
799178 752 
801 765 089 
804357000 
806 954491 
809557568 
812 166 237 
814 780 504 
817400375 
820025 856 
822 656 953 
825 293 672 
827936019 
830 584000 
833237621. 

835 896 888 
838 561 807 
841 232 384 
843 908 625 
846 590 536 

849 278 123 

851971392 

854670349 

857375 000 


Square Root. 


29.9165506 

29- 933 259 I 
29.9499583 
29.966 648 I 
29.983 328 7 
30 

30.016 662 
30.033 314 8 
30.0499584 
30.066 592 8 
30.0832179 
30.099 833 9 
30.1164407 
30.1330383 
30.1496269 
30.166 2063 
30.182 7765 
30.1993377 
30.2158899 
30.232 432 9 
30.2489669 
30.265 491 9 
30.282 007 9 
30.2985148 
30.315 0 12 8 
30.33 1 501 8 
30.347 981 8 
30.3644529 
30.380915 1 
30.397 368 3 
30.413812 7 
30.430 2481 
30.4466747 
30.463 092 4 
30.479 501 3 
30.495 901 4 
30.512 2926 
30.528675 
30.545 048 7 
30.5614136 
30.577 769 7 
30.594117 1 
30.610455 7 
30.626 785 7 
30.643 106 9 
30.6594194 
30.675 723 3 
30.692 018 5 
30.708 305 1 
30.724583 
30.740 8523 

30- 757113 
30.773 365 1 
30.7896086 
30.805 843 6 
30.822 07 


Cube Root. 


9.636 981 2 
9.640 569 
9.6441542 
9.647 736 7 
9.651 3166 
9.654 893 8 
9.658 468 4 
9.662 040 3 
9.665 6096 
9.669 176 2 
9.672 7403 
9.676 301 7 
9.679 860 4 
9.683 4166 
9.686 970 1 
9.690 521 1 
9.694 069 4 

9.697615 1 

9.701 1583 
9.7046989 
9.7082369 
9.711 7723 
9-7I5 305 1 
9.7188354 
9.722 363 1 
9.725 888 3 
9.7294109 
9-732 930 9 
9.736 4484 
9-739 9634 

9-743 475 8 
9.746985 7 
9.750493 
9-753 997 9 

9-757 500 2 
9.761 000 1 
9.7644974 
9.767 992 2 
9.7714845 

9-774974 3 
9.778461 6 
9.781 9466 
9.785 4288 
9.788 908 7 
9.792 386 1 
9.795 8611 
9-799333 6 
9.802 8036 
9.806271 1 
9.809 736 2 
9.813 198 9 
9.816659 1 
9.820 1169 
'9.8235723 
9.827 025 2 
9.8304757 











SQUARES, CUBES, AND ROOTS. 


289 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

951 

90 44 OI 

860 085 351 

30.838 287 9 

9.8339238 

952 

906304 

862 801 408 

30.854 497 2 

9.837 369 5 

953 

908209 

865 523 177 

30.870 698 I 

9.840812 7 

954 

91 OI 16 

868 250 664 

30.886 89P 4 

9.844 253 6 

955 

91 20 25 

870983S75 

30.903 074 3 

9.847 692 

956 

9 1 3936 

873 722 816 

30.9192477 

9.851128 

957 

91 58 49 

876467493 

30.9354166 

9.854561 7 

958 

9 I 77 64 

879217912 

30.951 575 1 

9.857 992 9 

959 

91 9681 

881 974079 

30.967 725 1 

9.8614218 

960 

92 1600 

884 736 000 

30.983 866 8 

9.864 848 3 

961 

92 35 21 

887503 681 

3 i 

9.868 272 4 

962 

9 2 54 44 

890277 128 

31.016 124 8 

9.871 694 1 

963 

92 7369 

893 056 347 

31.032 241 3 

9.875 1135 

964 

92 92 96 

895 841 344 

31.0483494 

9.8785305 

965 

931225 

898 632 125 

31.0644491 

9.881945 1 

966 

93 3 1 56 

901 428 696 

31.080 5405 

9-885 357 4 

967 

93 5089 

904 231 063 

31.0966236 

9.888 767 3 

968 

93 70 24 

907 039 232 

31.1126984 

9.8921749 

969 

938961 

909853 209 

31.128 7648 

9.895 580 1 

970 

94 09 00 

912 673 000 

31.144 823 

9.898 983 

971 

94 28 41 

9*5 498 611 

31.1608729 

9.902 383 5 

972 

94 47 84 

918330048 

31.1769145 

9.905 781 7 

973 

94 67 29 

921 167317 

31.1929479 

9.909 1776 

974 

94 86 76 

924010424 

31.20S973 1 

9.912 571 2 

975 

95 06 25 

926 859 375 

31.224 99 

9.9159624 

976 

95 25 76 

929 7 J 4 176 

31.2409987 

9 - 9 I 935 I 3 

977 

95 45 29 

932 574 833 

31.2569992 

9.922 737 9 

978 

95 64 84 

935 44 i 352 

| 31.2729915 

9.926 122 2 

979 

95 8441 

938 3 T 3 739 

31.2889757 

9.929 504 2 

980 

96 04 00 

941 192 000 

31.304951 7 

9 932 8839 

981 

96 23 6i 

944 076 141 

31.3209195 

9.936 261 3 

982 

96 43 24 

946 966 168 

3 U 336 879 2 

9.9396363 

983 

96 62 89 

949 862 087 

3 i -352 83 o 8 

9.943 009 2 

• 984 

96 82 56 

952 763 904 

3 i -368 774 3 

9-946379 7 

985 

97 02 25 

955671625 

31.384 7097 

9.949 747 9 

986 

97 21 96 

958 585 256 

31.4006369 

9.9531138 

987 

974169 

961 504803 

31.4165561 

9 956477 5 

988 

9761 44 

964430272 

31.432 4673 

9.9598389 

989 

97 81 21 

967 361 669 

3^.4483704 

9.963 198 1 

990 

9801 00 

970 299 000 

31.4642654 

9.9665549 

99 1 

98 20 81 

973 242271 

31.4801525 

9.9699095 

992 

98 40 64 

976 191 488 

31.496031 5 

9.973 261 9 

993 

98 6049 

979146657 

31.5119025 

9.976612 

994 

98 80 36 

982 107 784 

31.5277655 

9-9799599 

995 

990025 

985 074 875 

3 r -543 6206 

9-983 305 5 

996 

99 20 16 

988 047 936 

31.5594677 

9.986 648 8 

997 

994009 

991 026 973 

31.5753068 

. 9.989 99 

998 

996004 

994 011 992 

31.591 138 

9.993 328 9 

999 

998001 

997 002 999 

31.606961 3 

9.996 665 6 

1000 

1 000000 

1 000 000 000 

31.622 7766 

10 

* 1001 

1 000201 

1 003 003 001 

31.638 584 

10.003322 2 

1002 

1 00 40 04 

1 006012 008 

31.654 3836 

10.006662 2 

1093 

1 006009 

1 009027 027 

31.6701752 

10.009 989 9 

1004 

10080 16 

1 012 048064 

31.685 959 

10.013 315 5 

1005 

1 01 0025 

1015075 125 

31.701 7349 

10.0166389 

1006 

1 01 2036 

1 018 108 216 

1 3 r- 7 i 7 503 

i 10.019 960 1 


B B 


















290 


SQUARES, CUBES, AND ROOTS. 


Number. 

Square. 

Cube. 

Square Root. 

1 Cube Root. 

1007 

I OI 40 49 

1021 147 343 

3 I -733 2633 

IO.023 279 I 

1008 

I OI 60 64 

I O24 192 512 

31.749015 7 

10.026 595 8 

IOO9 

I 01 80 81 

I O27 243 729 

31.764 7603 

IO.029 910 4 

IOIO 

I 02 OI OO 

I 030 30I OOO 

31.780 4972 

10.033 222 8 

IOII 

I 02 21 21 

IO33364331 

31.7962262 

IO.036 533 

1012 

I 02 41 44 

I O36 433 728 

31.8119474 

IO.039 841 

1013 

I 02 6l 69 

I O39 509 I97 

31.827 660 9 

IO.043 1469 

IOI4 

I 02 8l 96 

I O42 590 744 

31.8433666 

10.046 450 6 

1015 

I 03 02 25 

I 045 678 375 

31.8590646 

10.049 752 1 

IOl6 

I 03 22 56 

I 048 772 096 

31.874 7549 

10.053 0514 

IOI7 

I 03 42 89 

1051 871 913 

31.890 4374 

10.056 348 5 

IOl8 

I 03 63 24 

1 °54 977 832 

31.906 1123 

10.0596435 

IOI9 

I 03 8361 

I 058 089 859 

31.921 7794 

10.062 936 4 

1020 

I 04 04 OO 

I o6l 208 OOO 

3I-937 438 8 

10.066 227 1 

1021 

I 04 24 41 

I 064332 261 

3 r -953 0906 

10.069 515 6 

1022 

I 04 44 84 

I 067 462 648 

3 i -9 6 8 734 7 

10.072 802 

IO23 

I 04 65 29 

I 070 599 167 

31.984371 2 

10.076 0863 

IO24 

I 04 85 76 

I O73 74I 824 

32 

10.079 3684 

1025 

I 05 06 25 

I O76 890 625 

32.015 621 2 

10.082 648 4 

1026 

I 05 26 76 

1080045576 

32 031 2348 

10.085 9 2 6 2 

IO27 

I 05 47 29 

I 083 206 683 

32.046 840 7 

10.089 201 9 

1028 

I 05 67 84 

I 086373 952 

32.062 439 1 

10.092 475 5 

IO29 

i 05 88 41 

I 089 547 389 

32.078 029 8 

10.095 746 9 

1030 

1 06 09 00 

I 092 727 OOO 

32.093613 1 

10.0990163 

IO3I 

1 06 29 61 

1095 912 791 

32.109 188 7 

10.102 283 5 

IO32 

1 06 50 24 

I 099 104 768 

32.124 7568 

10.105 5487 

1033 

1 06 70 89 

I 102 302 937 

32.1403173 

10.108 811 7 

1034 

1 0691 56 

I 105 507 304 

32.1558704 

10.112 072 6 

1035 

1 07 12 25 

I IO8 7I7 875 

32.171 4159 

10.1155314 

IO36 

1 07 32 96 

i 111 934 656 

32.1S6 9539 

10.118 588 2 

1037 

1 07 53 69 

1 115 157653 

32.202 484 4 

10.121 842 8 

IO38 

1 07 74 44 

1 1183S6872 

32.218007 4 

10.126095 3 

IO39 

1 07 95 21 

1 121 622 319 

32.2335229 

10.128 345 7 

IO40 

1 08 1600 

1 124 S64 000 

32.249031 

10.131 594 1 

IO4I 

1 08 36 81 

1 128 hi 921 

32.264 531 6 

10.134 8403 

IO42 

1085764 

1 131 366 088 

32.280 0248 

10.138 0845 

1043 

1 08 78 49 

1 134626507 

32.2955105 

10.141 326 6 

IO44 

108 9936 

1 137893 184 

32 310988 8 

10.144 566 7 

1045 

1 09 20 25 

1 141 166 125 

32.3264598 

10.147 8b4 7 

IO46 

1 09 41 16 

■i 144 445 33 6 

32.3419233 

10.151 0406 

IO47 

1 09 62 09 

1 147 730 823 

3 2 -357 379 4 

10.1542744 

IO48 

1 09 83 04 

1 151 022 592 

32.372 828 1 

10.1575062 

IO49 

1 10 04 01 

1 154320649 

32.3882695 

10.160 7359 

1050 

1 10 25 00 

1 157 625 000 

32.403 703 5 

10.163 963 6 

1051 

1 104601 

1 160935651 

32.419 130 1 

10.167 189 3 

IO52 

1 1067 04 

1 164 252 608 

32.434 5495 

10.170412 9 

1053 

1 108809 

1 167575877 

32.449961 5 

10.1736344 

IO54 

1 11 09 16 

1 170905464 

32.465 3662 

10.176 8539 

1055 

1 11 30 25 

1 174241375 

32.480 763 5 

10.180071 4 

IO56 

1 11 51 36 

1 177583616 

32.496 1536 

10.183 286 8 

1057 

1 1172 49 

1 180932 193 

32.511 5364 

10.186 500 2 

IO58 

1 11 93 64 

1 184 287 112 

32.526911 9 

10.189 711 6 

1059 

1 12 14 81 

1 187 648 379 

32.542 280 2 

10.192 9209 

1060 

1 12 3600 

1 191 016000 

32.557641 2 

10.196 1283 

1061 

1 12 57 21 

1 194389981 

32.5729949 

IO - I 993336 

1062 

1 12 78 44 

1 197 770328 

32.5883415 | 

10.202 5369 



















SQUARES, CUBES, AND ROOTS. 


29I 


Number. | 

Square. I 

Cube. 

Square Root. 

Cube Root. 

1063 

I 12 9969 

I 201 157 047 

32.603680 7 

10.205 738 2 

1064 

I 13 20 96 

I 204 550 144 

32.619012 9 

10.208 9375 

1065 

I 13 42 25 

I 207 949 625 

32.634 337 7 

10.212 1347 

1066 

I 13 63 56 

1211355496 

32.6496554 

IO.215 33 

1067 

I 138489 

I 214 767 763 

32.6649659 

IO.2185233 

1068 

I 140624 

I 2l8 186432 

32,680 2693 

10.221 7146 

1069 

I 14 27 6l 

I 221 6l I 509 

32.695 565 4 

IO.224 903 9 

IO7O 

I I449OO 

I 225 O43 OOO 

32.7108544 

IO.22809I 2 

IO7I 

I 14 704I 

I 228 4809II 

32.7261363 

IO.23I 2766 

IO72 

I 14 91 84 

I 23I 925 248 

32.74x4111 

IO.234 4599 

1073 

I 15 I329 

I 235376017 

32.7566787 

IO.237 641 3 

IO74 

I 15 34 76 

I 238 833 224 

32. 77 1 939 2 

IO.24O 820 7 

1075 

I 15 56 25 

I 242 296 875 

32.787 1926 

IO.243 998 I 

1076 

I 15 77 76 

I 245 766 976 

32.8024389 

IO.247 1735 

IO77 

I I 5 99 2 9 

I 249 243 533 

32.817 678 2 

IO.250347 

1078 

I 162084 

I 252 726552 

32.8329x03 

IO.2535186 

IO79 

I 1642 41 

I 256216039 

32.848 135 4 

10.256 688 1 

1080 

I 1664OO 

I 259 712 000 

32.863 353 5 

10.259 8557 

1081 

I l6 85 6l 

I 263 214 44I 

32.878 564 4 

10.263021 3 

1082 

I 1707 24 

I 266 723 368 

32.8937684 

10.266 185 

1083 

i 17 28 89 

I 27O 238 787 

32.908 965 3 

10.269 346 7 

1084 

1 175056 

I 273 760 704 

32.924 155 3 

10.2725065 

1085 

x 1772 25 

I 277 289 125 

32.939338 2 

10.2756644 

1086 

1 17 93 96 

I 280 824 056 

32.9545141 

10.278 8203 

1087 

118 15 69 

I 284 365 503 

32.969 683 

10.2819743 

1088 

1183744 

I 287913472 

32.984 845 

10.285 1264 

1089 

I l8 59 21 

I 291 467 969 

33 

10.288 2765 

1090 

I l8 8l OO 

I 295 O29 OOO 

33.015 148 

10.291 424 7 

I09I 

I 1902 8l 

I 298 596 571 

33.030 289 1 

10.294 570 9 

IO92 

I 19 24 64 

I 302 170688 

33.045 423 3 

10.2977153 

IO93 

x 194649 

1 3 °S 75 i 357 

33.0605505 

10.300 857 7 

IO94 

119 68 36 

1309338584 

33.0756708 

10.303 998 2 

1095 

119 90 25 

1 312932375 

33.090 784 2 

10.307 1368 

IO96 

1 20 12 16 

1 3 l6 53 2 736 

33.105 890 7 

10.3102735 

IO97 

1 203409 

1 320 139 673 

33.1209903 

10.3134083 

IO98 

1 20 56 04 

1 3 2 3 753 x 9 2 

33.136083 

10.3165411 

IO99 

1 20 78 ox 

1 3 2 7 373 2 99 

33.151 1689 

10.319672 1 

IIOO 

I 21 OOOO 

1 331 000000 

33.1662479 

10.322 801 2 

IIOI 

I 21 22 OI 

1 334 633 301 

33 -i 8 i 3 2 

10.325 928 4 

1102 

I 21 4404 

1 338 273 208 

33.1963853 

xo.329 053 7 

1103 

I 21 6609 

1 34 i 9 J 9 727 

33 - 2 II 443 8 

10.332 177 

II04 

I 21 88 16 

1 345 572 864 

33.226 695 5 

10.335 298 5 

1105 

1 22 IO 25 

1 349 232 625 

33,241 5403 

10.338418 1 

II06 

I 22 32 36 

1352899016 

33 -2565783 

10341535 8 

1107 

I 22 5449 

1 356 572 043 

33.2716095 

10.344651 7 

II08 

1,22 7664 

1360 251 712 

33.2866339 

10.347 765 7 

1109 

I 22 98 8l 

1363938029 

33.30 1 651 6 

10.3508778 

iiio 

I 23 21 OO 

1 367 631 000 

33.3x66625 

10.353988 

mi 

I 23 43 2T 

1371 330631 

33-33 1 666 6 

10.3570964 

1112 

I 23 65 44 

1375036928 

33.346 664 

10.300 202 9 

1113 

I 23 87 69 

• 1378 749 897 

33.361 6546 

10.3633076 

1114 

I 24 09 96 

1 382 469 544 

33'37 66 3&5 

10.3664x03 

1115 

I 243225 

1 386 195 875 

33 - 39 1 615 7 

*0.369 5 11 3 

1116 

I 24 54 56 

1 389 928 896 

33.406 586 3 

*0.372 610 3 

1117 

1247689 

x 393 668 613 

33.4215499 

?o -375 7076 

1118 

1 I 24, £9 24 

1 397 4 I 5 033 

33-436 5°7 

10.378 803 













292 


SQUARES, CUBES, AND ROOTS. 


Number. 

Square. 

Cube.. 

Square Root. 

Cube Root. 

III 9 

I 25 21 6l 

I 401 168 159 

33 - 45 I 457 3 

IO.381 8965 

1120 

I 25 44OO 

I 404 928 OOO 

33.466401 I 

IO.3849882 

II2I 

i 25 66 41 

I 408 694 561 

33.481 338 I 

IO.388 078 I 

1122 

1 25 88 84 

I 412467 848 

33.4962684 

IO.391 166 I 

1123 

1 2611 29 

I 416247 867 

33.511 1921 

IO.3942523 

1124 

12633 76 

I 420 034 624 

33.5261092 

10.397 3366 

1125 

1 26 56 25 

1423 828 125 

33.5410196 

IO.4OO 419 2 

1126 

1 26 78 76 

I 427 628 376 

33-555 9 2 3 4 

IO.4034999 

1127 

1 2701 29 

1 43 1 435 383 

33.570 8206 

IO.406 578 7 

1128 

1 27 23 84 

1 435 249 152 

33.5857112 

IO.409 655 7 

II29 

1 27 4641 

1 439 069 689 

33.600 595 2 

IO.412 731 

1130 

1 276900 

1 442 8q7 000 

33.6154726 

IO.415 8044 

II3I 

1 2791 61 

1 446 731 091 

33.6303434 

IO.418 876 

1132 

1 28 14 24 

1 450 57 1 968 

33.645 207 7 

10.421 945 8 

1133 

1 28 36 89 

1454419637 

33.660 0653 

10.4250138 

1134 

1 28 59 56 

1 458 274 104 

33.674 916 5 

10.428 08 

1135 

1 28 82 25 

1462 135375 

33.689 761 

10.431 1443 

II36 

1 29 04 96 

1 466 003 456 

33 - 7°4 599 1 

10.434 206 9 

1137 

1 29 27 69 

1 469 878 353 

33 - 7 i 943 o 6 

10.4372677 

II38 

129 5044 

1473 760072 

33 734255 6 

10.4403267 

1139 

I 29 73 21 

1 477 648 619 

33 - 749 074 1 

10.4433839 

II40 

I 29 96 OO 

1 481 544000 

33.7638860 

10.4464393 

II4I 

I 30 l8 8l 

1 485 446221 

33.7786915 

10.4494929 

II42 

I304I 64 

1 489 355 288 

33-793 490 5 

10.452 544 8 

1143 

1306449 

1 493 271 207 

33.808 283 

10.455 5948 

II44 

I 30 87 36 

1497 193984 

33.823 069 1 

10.458 643 1 

1145 

131 10 25 

1501123625 

33.837 848 6 

10.461 6896 

II46 

1 31 33 16 

1 505 060 136 

33.8526218 

10.4647343 

II47 

1 31 5609 

1509603523 

33.867 3884 

10.467 7773 

II48 

1 317904 

1512953 792 

33.882 148 7 

10.4708185 

II49 

1 32 02 01 

1 516910949 

33.8969025 

10.473 8579 

1150 

1 32 25 00 

1 520 875 000 

33.9116499 

10.476 895 5 

IISI 

1 32 48 01 

1524 845 95 X 

33.9263909 

10.479 93 1 4 

1152 

1 32 71 04 

1 528 823 808 

33.941 125 5 

10.482 965 6 

“S 3 

1329409 

x 532 808 577 

33-955 853 7 

10.485 998 

1154 

1331716 

1 536 800 264 

33-970 575 5 

10.489 028 6 

H 55 

1 33 40 25 

1 540 798 875 

33.985 291 

10.4920575 

II56 

1336336 

1 544 804416 

34 

10.495 084 7 

1157 

1 33 86 49 

1 548 816 893 

34.014 702 7 

10.498 no 1 

II58 

1340964 

1552 836312 

34.029399 

10.501133 7 

1159 

1343281 

1 556 862 679 

34.044 089 

10.504 155 6 

Il6o 

134 5600 

1 560 896 000 

34.058 772 7 

10.507 175 7 

Il6l 

134 79 21 

1564936 281 

34-073 450 1 

10.510 1942 

Il62 

135 02 44 

1 568 983 528 

34.088 121 1 

10.513 210 9 

H63 

135 25 69 

1 573 037 747 

34.102 7858 

10.5162259 

H64 

1 35 48 96 

1577098944 

34.1174442 

19.5192391 

H65 

1 35 72 25 

1 581167 125 

34.1320963 

10.522 2506 

Il66 

1 35 95 56 

x 585 242 296 

34-1467422 

10.525 2604 

I167 

1361889 

1589324463 

34.161381 7 

10.528 268 5 

Il68 

1364224 

1 593 413 632 

34.176015 

10.5312749 

U69 

1 36 65 61 

1 597 509 809 

34.190642 

IO -534 279 5 

II70 

1 36 89 00 

1 601 613 000 

34.205 262 7 

10.537 282 5 

II7I 

1 37 12 41 

1 605 723 211 

34.2198773 

10.540 283 7 

II72 

1 37 35 84 

1 609 840 448 

34.2344855 

10.543 283 2 

1173 

1 37 59 29 

1613964717 

34.2490875 

10.546 281 

II74 

x 37 82 76 

1 618 096 024 

34.263 683 4 

10.549 277 1 









SQUARES, CUBES, AND ROOTS. 


293 


Number. 1 

Square. 

Cube. 

Square Root. 

Cube Root. 

H 75 

I 38 06 25 

I 622 234 375 

34.278 273 

IO.552 2715 

1176 

I 38 29 76 

I 626379776 

34.292 8564 

IO.555 264 2 

II77 

i 38 53 29 

I 630 532 233 

34.307 4336 

IO.5582552 

1178 

1 38 76 84 

I 634 691 752 

34.322 004 6 

10561 2445 

II79 

1 390041 

I 638 858 339 

34.336 5694 

IO.5642322 

Il8o 

1 39 24 00 

I 643 O32 OOO 

34.351 128 I 

10,567 218 X 

1181 s 

1394761 

I 647 212 74I 

34.365 680 5 

IO 57O 202 4 

1182 

1 39 71 24 

I 651 4OO 568 

34.380 226 8 

IO.573 1849 

1183 j 

1399489 

I 655 595 487 

34-394 767 

IO 576 1658 

1184 

1 40 18 56 

i 659 797 504 

34.409301 1 

IO5791449 

1x85 

14042 25 

1 664 006 625 

34.423 828.9 

IO 582 122 5 

1186 

1 40 65 96 

1 668 222 856 

34 438 350 7 

IO.5850983 

1187 

x 40 89 69 

1 672 446 203 

344528663 

IO.588 O725 

1188 

1 41 1344 

1 676 676 672 

34-467 375 9 

IO.59I O45 

1189 

1 41 37 21 

1 680 914 269 

344818793 

IO.594OI58 

1190 

1 41 61 00 

1 685 159000 

34 4963766 

IO.596 985 

1191 

1 41 8481 

1 689410871 

34.5108678 

IO.599952 5 

1192 

1 42 08 64 

1 693 669 888 

34 525 353 

10.602 9184 

1193 

1 42 32 49 

1697936057 

34.539 832 1 

10.605 882 6 

1194 

142 5636 

1 702 209 384 

34-554 305 1 

10.608 845 1 

1195 

1 42 80 25 

1 706 489 875 

34.568 772 

10.611 806 

1196 

1430416 

1 7 io 777 536 

34.583 232 9 

10.614 765 2 

1197 

1 43 28 09 

1 715072373 

34.5976879 

10.617 722 8 

1198 

1 43 52 04 

1 719 374392 

34.612 1366 

10 620 678 8 

1199 

1 43 76 01 

1 723 683 599 

34.626 579 4 

10.623 633 1 

1200 

144 00 00 

1 728 000 000 

34.641 016 2 

10 626 585 7 

1201 

1 44 24 01 

1 732 323 601 

34.655 446 9 

10.629 5367 

1202 

1 44 48 04 

1 73 6 6544 0 8 

34669871 6 

10 632 486 

1203 

1 44 72 09 

1 740992 427 

34.684 290 4 

10.635 433 8 

1204 

1 4496 16 

1 745 337 664 

34-698 703 1 

10.6383799 

1205 

1 45 20 25 

1 749690125 

34.713 1099 

x 0.641 3244 

1206 

1 45 44 36 

1 754049816 

34.7275107 

10.644 267 2 

1207 

1 45 68 49 

1 758416743 

34.741 905 5 

10.647 208 5 

1208 

1 45 92 64 

1 762 790912 

34.756 2944 

10.650 148 

1209 

1461681 

1 767 172329 

34.7706773 

10.653 086 

1210 

1 4641 00 

1 771 561 000 

34.785 0543 

xo 656 0223 

1211 

1 4665 21 

1 775 956931 

' 34-799425 3 

10.658957 

1212 

I 1 46 89 44 

1 780360 128 

34.813 7904 

10.661 8902 

1213 

1471369 

1 784 770 597 

34.828 149 5 

10.664 821 7 

1214 

1 47 37 96 

1 789188344 

34.842 502 8 

10.667 7516 

1215 

1 47 62 25 

1 793 613 375 

34.856 8501 

10 670 6799 

I2l6 

1 47 86 56 

1 798 045 696 

34.871191 5 

10.673 6066 

1217 

1 48 10 89 

1 802 485 313 

34.885 527 1 

10.676 531 7 

12X8 

1 48 35 24 

1 806 932 232 

34.8998567 

10.6794552 

1219 

148 5961 

1 811 386459 

34.914 1805 

10.682 377 1 

1220 

1 48 84 00 

1 815 848 000 

34.928 498 4 

10.685 2973 

1221 

1 490841 

1 820316861 

34.9428104 

10.688 216 

1222 

14932 84 

1 824 793 048 

34.957 1166 

10.691133 1 

1223 

1495729 

1 829 276 567 

34 - 97 1 4 i6 9 

10.694 048 6 

1224 

1 4981 76 

1 833 767 424 

34.985 71x4 

10.696 962 5 

1225 

1 50 06 25 

1 838 265 625 

35 

10.6998748 

1226 

1 5 ° 3 ° 76 

1 842 771 176 

35.0142828 

10.702 785 5 

1227 

1 5 ° 55 29 

1 847 284 083 

35.028 5598 

10.7056947 

1228 

1 50 79 84 

1851 804352 

35.042 8309 

10.708 602 3 

1229 

1 51 0441 

1 856 331 989 

35.0570963 

10.711 5083 

1230 

1 51 2900 

1 860 867 000 

35-071 355 8 

10.714412 7 


B B* 













294 

SQUARES, CUBES, 

AND ROOTS, 

Number. 

Square. 

Cube. 

| Square Root. 

1231 

1515361 

I 86540939X 

35.0856096 

I23 2 

I 51 78 24 

I 869 959 168 

35.099 857 5 

1233 

I 52 02 89 

1874516337 

35.1140997 

1234 

I 52 27 56 

I 879 080 904 

35.128 3361 

1235 

I 52 52 25 

I 883 652 875 

35.1425568 

1236 

I 52 76 96 

i 888 232 256 

35- i 56 79 1 7 

1237 

1530169 

1892819053 

35.1710108 

1238 

153 2644 

1 897413272 

35.185 2242 

1239 

i 53 5 1 21 

1 902 014 919 

35-199 43 1 8 

1240 

153 7600 

1 906 624 000 

35.2136337 

1241 

1 540081 

1 911 240521 

35.2278299 

1242 

x 54 25 64 

1 915 864 488 

35.242 0204 

1243 

154 5049 

1920495 907 

35.256 205 1 

1244 

1 54 75 36 

1 925 134 784 

35.2703842 

1245 

1 55 00 25 

1 929 781 125 

35.2845575 

1246 

1 55 25 16 

1934434 936 

35.298 725 2 

1247 

155 5009 

1 939 096 223 

35.3128872 

1248 

1 55 75 04 

1943 764 992 

35.3270435 

1249 

1 56 00 01 

1948441249 

35-341 194 1 

1250 

1 56 25 00 

1 953 I2 5 000 

35-355 339 * 

1251 

1 56 50 01 

1957816251 

35-3694784 

1252 

1 56 75 04 

1 962 515008 

35-383612 

1253 

1 570009 

1 967 221 277 

35-397 74 

1254 

1 57 25 16 

197x935064 

35.4118624 

1255 

1 57 50 25 

1976 656375 

35.425 9792 

1256 

1 57 75 36 

1 981 385 216 

35.4400903 

1257 

1580049 

1 986 121 593 

35-454I95 8 

1258 

1 58 25 64 

1990865512 

35.468 295 7 

1259 

1 58 5081 

1995616979 

35.48239 

1260 

1 58 76 00 

2 000 376 000 

35.4964787 

1261 

1 5901 21 

2005 142 581 

35.510 5618 

1262 

159 2644 

2 009 916 728 

35.5246393 

1263 

1595169 

2 014 698 447 

35.5387113 

1264 

159 7696 

2019487 744 

35-552 777 7 

1265 

1 60 02 25 

2 024 284 62 5 

35.566 838 5 

1266 

1 60 27 56 

2 029 089 096 

35.5808937 

1267 

1 60 52 89 

2 033 901 163 

35-594 9434 

1268 

1 60 78 24 

2 038 720 832 

35.608 9876 

1269 

1 61 03 61 

2 043 548 109 

35.623 0262 

1270 

1 61 2900 

2 048 383 0O0 

35-637 0593 

1271 

1 61 5441 

2053 225 511 

35.6510869 

1272 

1 61 79 84 

2 058 075 648 

35.665 109 

1273 

1 62 05 29 

2062 933417 

35.6791255 

1274 

1 62 30 76 

2 067 798 824 

35.693 136 6 

1275 

1 62 56 25 

2072671 875 

35.707 142 1 

J276 

1 62 81 76 

2 077 552 576 

35.721142 2 

I2 77 

16307 29 

2082440933 

35-735 136 7 

1278 

1 63 32 84 

2087336952 

35-749 125 8 

1279 

1635841 

2 092 240 639 

35.763 1095 

1280 

1 63 84 00 

2097 152000 

35-7770876 

1281 

1 64 09 61 

2 102 07I 04I 

35.7910603 

1282 

16435 24 

2 106 997 768 

35.805 027 6 

1283 

164 60 89 

2 III 932 187 

35.8189894 

1284 

1 64 86 56 

2Il6 874304 

35.832 9457 

1285 

1 65 12 25 

2 121 824 125 

35.8468966 

1286 

1653796 

2 126 781 656 

35.860 842 1 


Cube Root. 


10.717 315 5 
10.7202168 
10.723 1165 
10.7260146 
10.728911 2 
xo.731 8062 
10.7346997 
io-737 59i6 
10.740481 9 

10.7433707 

10.746 2579 
10.7491436 
10.752027 7 
10.7549103 
10.757 79 1 3 
10.760 6708 
10.7635488 
10.766 425 2 
10.7693001 
10.7721735 
10.7750453 
10.7779156 
10.7807843 
10.783 6516 
10.7865173 

10.7893815 

10.792 244 1 
10.795 1053 
10.7979649 
10.800 823 
10.803 679 7 
10.806 534 8 
10.809 388 4 
10.812 2404 
10.8150909 
10.81794 
10.820 787 6 
10.8236336 
10.826 478 2 
10.8293213 
10.832 162 9 
10.835 003 
10.837 841 6 
10.8406788 
10.8435144 
10.846 348 5 
10.849 181 2 
10.852 012 5 
10.854 842 2 
10.8576704 
10.860 497 2 
10.863 3 22 5 
10.866 1464 
10.868 968 7 
10.871 789 7 
10.874609 1 












1287 

1288 

1289 

1290 

1291 

1292 

i2 9 3 

i2 9 4 

I2 95 

1296 

1297 

1298 

I2 99 

1300 

I 3° I 

1302 

1303 

1304 

1305 

1306 

1307 

1308 

1309 

1310 

1311 

1312 

1313 

1314 

131S 

1316 

1317 

1318 

1319 

1320 

1321 

1322 

1323 

1324 

132S 

1326 

1327 

1328 

1329 

1330 

i33i 

1332 

1333 

1334 

1335 

1336 

1337 

1338 

1339 

1340 

i34i 


SQUARES, CUBES, AND ROOTS. 


295 


Square. 

Cube. 

Square Root. j 

Cube Root. 

I 65 63 69 

2 131 746903 

35.874 782 2 

IO.877 427 I 

I 65 89 44 

2 136 719 872 

35.8887169 

10.880 243 6 

i 66 15 21 

2 I4I 7OO 569 

35.902 646 I 

IO.884 048 7 

1 6641 00 

2 146 689 OOO 

35.9x65699 

IO.8858723 

1 666681 

2 151 685 171 

35.9304884 

10.888 6845 

1 66 92 64 

2 156 689 088 

35-944401 5 

10.891 495 2 

1 67 1849 

2 l6l 7OO 757 

35-958 309 2 

10.894 3°4 4 

1 67 44 36 

2 l66 72O 184 

35.972211 5 

10.897 112 3 

1 67 70 25 

2171 747375 

35.986 108 4 

10.8999186 

1 67 96 16 

2 176 782 336 

36 

10.902 723 5 

1 68 22 09 

2 l8l 825 O73 

36.0x3 886 2 

10.9055269 

1 68 48 04 

2 186875 592 

36.027 767 1 

10.908 329 

1 68 7401 

2 19 1 933 899 

36.041 642 6 

10.9111296 

1 69 00 00 

2 I97 OOO OOO 

36.0555128 

10.913928 7 

1 69 2601 

2 202 073 90I 

36.069 377 6 

10.916 726 5 

1 69 52 04 

2 207 155 608 

36.083 237 1 

10.919 522 8 

1 69 78 09 

2 2X2 245 I27 

36.097 091 3 

10.922 317 7 

1 7004 16 

2 217 342 464 

36.1109402 

10.925 111 1 

1 70 30 25 

2 222 447 625 

36.124 783 7 

10.927 903 1 

1 70 5636 

2 227 560 6l6 

36.138 622 

10.9306937 

1 70 82 49 

2232681 443 

36.152 455 

io-933 482 9 

1 71 0864 

2 237 8lO 112 

36.166 282 6 

10.9362706 

1 713481 

2 242 946 629 

36.180105 

10.939 0569 

1 71 61 00 

2 248 09I OOO 

36.193 922 1 

10.941 841 8 

1 7187 21 

2 253 243 23I 

36.207 734 

10.944 625 3 

1 72 13 44 

2 258 403 328 

36.221 540 6 

10.947 507 4 

1 72 39 69 

2 263 57I 297 

36.235 341 9 

10.950 188 

1 72 65 96 

2 268 747 I44 

36.249137 9 

10.952 967 3 

1 72 92 25 

2 273 93O 875 

36.262 928 7 

10.955 745 1 

1 73 18 56 

2 279 122 496 

36.2767143 

10.95S 521 5 

1 73 44 89 

2 284 322 OI3 

36.2904946 

10.961 296 5 

1 73 71 24 

2 289 529 432 

36.304 269 7 

10.964 070 1 

1 739761 

2 294 744 759 

36.3180396 

10.966 842 3 

1 74 24 00 

2 299 968 OOO 

36.331 804 2 

10.969 613 1 

1 74 5041 

2305 I99 l6l 

36.345 563 7 

10.972 382 5 

1 74 76 84 

2 3 IQ 438 248 

v 36.359317 9 

10.975 1505 

1 75 03 29 

2315685 267 

36.373 067 

10.977 917 1 

1 75 29 76 

2 32O 940 224 

36.386810 8 

10.980 682 3 

1 75 56 25 

2 326 203 125 

36.4005494 

10.983 446 2 

1 75 82 76 

2 33 1 473 976 

36.414 282 9 

10.986 208 6 

1 76 09 29 

2 336 752 783 

36.428 on 2 

10.988 969 6 

1 7 6 35 84 

2 342 O39 552 

36.441 734 3 

10.991 7293 

1 7662 41 

2 347 334 289 

36.455 452 3 

10.994 487 6 

1 76 89 00 

2 352 637 OOO 

36.469 165 

10.9972445 

1 77 1561 

2357947691 

36.482 872 7 

11 

1 7742 24 

2 363 266 368 

36.4965752 

11.002 7541 

1 77 68 89 

2 368 593 037 

36.5102725 

11.005 5069 

1 77 95 56 

2 373 927 704 

36.5239647 

11.008 2583 

1 78 22 25 

2 379 270 375 

36.5376518 

11.011 008 2 

1 78 48 96 

2 384 621 056 

36.551333 8 

11.0137569 

1 78 75 69 

2 389979 753 

36.5650106 

11.016 504 1 

1 79 02 44 

2 395 346 472 

36.578 6823 

11.019 25 

1 79 29 21 

2 400 721 2x9 

36-592 348 9 

11.021 9945 

1 79 56 00 

2 406 104000 

36.6060104 

11.024 7377 

1 79 82 81 

2 411 494821 

36.6196668 

11.0274795 

i 80 09 64 

2 416 893 688 

36.633 318 1 

11.030 2199 













296 

Nimsfia. 

T o 43 

*344 

*345 

1346 

1347 

1348 

1349 

1350 

i35i 

1352 

1353 

1354 

1355 

1356 

1357 

1358 

1359 

1360 

1361 

1362 

1363 

i3 6 4 

1365 

1366 

1367 

1368 

1369 

1370 

i37i 

1372 

1373 

1374 

1375 

1376 

1377 

1378 

1379 

1380 

1381 

1382 

1383 

1384 

1385 

1386 

1387 

1388 

1389 

139° 

I 39 I 

1392 

1393 

1394 

1395 

1396 

1397 

1398 


SQUABKS, CUBES, AND BOOTS. 


So.uA.ftE. 

1803649 
I 80 63 36 
I 80 90 25 
I 81 17 16 
i 81 44 09 
1 81 71 04 
1 81 9801 
1 82 25 00 
1 82 52 01 
1 82 79 04 
1 83 06 09 
1 83 33 16 
1 83 60 2 ^ 
1838736 
1 84 14 49 
1 84 41 64 
1846881 
1 849600 
1 85 23 21 
185 5044 
1 85 7769 
1 86 04 96 
1 86 32 25 
1 86 59 56 
1 86 86 89 
1 87 1424 
1 87 41 61 
1 87 69 GO 
1 879641 
1 88 23 84 
1 88 51 29 
1 88 78 76 
1 89 06 25 

189 3376 
1 8961 29 
1 89 88 84 
1 90 1641 
1 90 44 00. 

190 71 61 
1909924 
1 91 26 89 

1 9 1 54 56 
1 91 82 25 
1 92 09 96 
1 92 37 69 
1 92 65 44 
1 92 93 21 
1 93 21 00 
1 93 48 81 
1937664 
1940449 
1 94 3 2 3 6 
1 94 60 25 
1 94 88 16 
195 1609 
195 4404 


Cube. 

2 422 3OO 607 
2 427 7x5 584 
2 433 138 625 
2 438 569 736 
2 444 008 923 
2 449 456 192 
2 454 911 549 

2 460 375 OOO 
2465846 55I 
2 471 326 208 
2 476 813 977 
2 482 309 864 

2487813875 

2493326016 
2 498 846 293 
2504374712 
2 509 911 279 

2 5I5 456oOO 

2 521 008 88l 
2 526 569 928 
2 53 2 I 39 147 
2 537 7 i 6 544 
2 543 3 ° 2 I2 5 
2 548 895 896 

2 554 497 863 

2 560 108 032 
2565 726409 

2 57 i 3530 oo 

2 576 987 811 
2 582 630 848 
2 588 282 117 
2 593 94 i 624 
2599609375 

2 605 285 376 
2 610 969 633 
2 616662 152 
2 622 362 939 
2 628 072 OOO 
2633789341 
2639514968 
2 645 248 887 
2 650 99I IO4 
2 656 741 625 
2 662 500 456 
2 668 267 603 
2 674 O43 072 
2 679 826 869 
2 685 619OOO 
2 691 41947I 
2 697 228 288 

2 703 045 457 
2 708 870 984 
2714704875 
2 720 547 136 
2 726397 773 
2 732 256 792 


Sou abe Root. 


36.6469644 
36.660 605 6 
36.674 241 6 
36.687 8726 
36.701 4986 
36.7151195 
36.728 7353 
36.742 3461 
36.755 9519 
36.769 55 2 6 
36.7831483 

36.796739 

36.8103246 
36.823 905 3 
36.8374809 
36.851 051 5 
36.864 617 2 
36.878 1778 

36.891 733 5 

36.905 284 2 
36.918 8299 

36.932 370 6 

36.9459064 

36 . 959437 2 
36.972 963 1 
36.986484 
37 

37.013 511 
37.027 017 2 
37.0405184 
37.0540146 
37.067 506 
37.0899924 
37,094 474 
37- io 7 95o6 
37.121 4224 
37- I 34 8893 
37.148 351 2 
37.161 8084 
37.175 2606 
37.1887079 
37.202 1505 
37.215 588 1 
3 7.2290209 
37.242 448 9 
37.255872. 
37.2692903 
37.282 703 7 
37.2961124 

37-309 5 i 6 2 

37.3229152 

37-3363094 

37-349698 8 

37.3630834 

37.3764632 

37.3898382 


Cube Root. 


II.032 959 
II.0356967 
II.O38433 
II.041 168 
11.043 901 7 
11.0466339 
11.049 3649 
11.0520945 
11.054 822 7 
11.0575497 
11.060275 2 
11.062 999 4 
11.065 722 2 
11.0684437 
11.071 1639 
11.073 882 8 
11.0766003 
11.0793165 
11.082 031 4 
11.084 744 9 
11.087 457 1 
11.090 1679 
11.0928775 
11.095 5857 
11.098 292 6 
11.100998 2 
11.1037025 
11.106405 4 
11.109 107 
11.hi 8073 
11.1145064 
11.117 204 1 
11.119 900 4 
11.122 5955 
11.125 2893 
11.127 981 7 
11.1306729 
H-I 33 3628 
11.136 0514 
11.138 7386 
11.141 4246 
11.144 1093 
11.146 792 6 
n.149474 7 
ii-I 52 I 55 5 
II - I 54 835 
1I - I 57 5 I 3 3 
11.160 1903 
11.162 865 9 
11.165 5403 
11.168 2134 
11.170885 2 
II - I 73 555 8 
11.176225 
11.178893 
11.181 5598 







Numbeb 

1399 

1400 

1401 

1402 

1403 

1404 

1405 

1406 

1407 

1408 

1409 

1410 

1411 

1412 

1413 

1414 

1415 

1416 

14x7 

1418 

1419 

1420 

1421 

1422 

1423 

1424 

1425 

1426 

1427 

1428 

1429 

143 ° 

I43 1 

1432 

1433 

1434 

1435 

1436 

1437 

143? 

1439 

1440 

1441 

1442 

1443 

1444 

1445 

1446 

1447 

1448 

1449 

1450 

I4SI 


SQUARES, CUBES, AND ROOTS. 


Square. 


I 95 72 OI 
I 96 OO OO 

I 96 28 OI 
1965604 
I 96 84 09 
I 97 12 l6 
I 97 40 25 
i 97 68 36 

1 97 9^49 
1982464 
1 98 52 81 
1 98 81 00 
1 9909 21 

1 99 37 44 
19965 69 
19993 96 
2002225 

2 00 50 56 
2 OO 78 89 

2 OI 07 24 

2OI3561 

2 OI 64 OO 
2 OI92 41 
2 02 20 84 
2 02 49 29 
2 02 77 76 
2 03 06 25 
2 03 34 76 
2036329 
2039184 
2 04 20 41 

2 04 49 OO 
2 04 77 6l 
2 05 06 24 
2053489 
2 05 63 56 
2 05 92 25 
2 06 20 96 
2 06 49 69 
2 06 78 44 
2 07 07 21 
2 07 36 OO 
2 07 64 8l 
2 07 93 64 
2 08 22 49 
2085136 
2 08 80 25 
20909 l6 
2 09 38 09 
2 09 67 04 
2099601 

2 IO 25 OO 
2 IO 54 01 
2 IO 83 04 
2 II 12 09 
2 II 41 l6 


Cube. 

2 738 124 199 

2 744 OOO OOO 

2 749 884 201 
2 755 776 808 
2 761 677 827 
2 767 587 264 

2 773 505 125 
2779 431416 
2 785 366 143 
2 79I 309312 
2 797 260 929 

2 803 221 OOO 
2 809 189 53I 
2 815 l66 528 
2 821 I5I997 
2 827 145 944 
2 833 148 375 
2 839 159 296 
2845 178713 
2 851 206632 
2857243 059 
2 863 288 OOO 
2 869 34I 461 
2 875 403 448 
2881473967 
2887553 024 
2 893 640 625 
2 899 736 776 
2 905 841 483 
29II 954 752 
2918076589 1 

2 924 207 OOO 

2 930 345 991 
2 936 493 568 
2 942 649 737 
2948814504 
2954987875 
2 961 169 856 
2 967 360 453 
2 973 559 672 
2 979 767 519 
2 985 984 000 
2 992 209 121 

2 998 442 888 

3 004 685 307 
3010936384 
3017 196 125 
3023464536 
3029 741 623 
3036027392 
3042 321 849 
3 048 625 000 
3 054 936 851 
3061 257408 
3067 586677 

3073 924664 


Square Root. 


37.403 208 4 
37 - 4 I 6 573 8 

37-429934 5 
37.4432904 
37.4566416 
37.469 988 

37-483 329 6 

37.4966665 
37.509 998 7 
37-523 3261 
37.5366487 
37.549 966 7 
37.563 279 9 
37.5765885 
37.5898922 
37.603 191 3 
37.616485 7 
37.629 775 4 
37.6430604 
37.656 340 7 
37.6696164 
37.682 8S74 
376961536 
37.7094153 
37.722 672 2 
37-735 924 5 
37-749 I 7 2 2 
37.7624152 

37-775 653 5 
37.788 8873 
37.802 1163 
37.8153408 
37.828 5606 
37.841 7759 
37-854 9864 
37.808 1924 

37 8813938 
37.8945906 
37.907 7828 
37.9209704 
37-934 153 5 
37-947 33i 9 
37.960 505 8 

37- 973 675 1 
37.986 8398 

38 

38.0131556 
38.026 306 7 
38.0394532 
38.052 595 2 
38.065 732 6 
38.078 865 5 
38.091 993 9 
38.105 1178 
38.1182371 

38- 13! 35 1 9 


297 

Cube Root. 


II.184 225 2 
II. 186 8894 
II.1895523 
II.192 2139 
II.1948743 
H-I97 533 4 
11.200 191 3 
11.262 847 9 
11.205 5032 
11.208 1573 
11.210 810 1 
11.213461 7 
11.216 112 
11.218 761 1 
11.221 408 9 
1 i.224005 4 
11.226 700 7 
11.2293448 
11.231 987 6 
1 r.234 6292 
11.237 269 6 
11.239908 7 
11.242 5465 
11.245 183 1 
11.2478185 
11.250 452 7 
11.253 0856 
11.255 7173 
11.258 3478 
11.260 977 
11.263605 
11.266 231 8 
11.268 857 3 
11.271 481 6 
11.274 io 4 7 
11.276 7266 
11.2793472 
11.281 9666 
11.2845849 
11.287 201 9 
11.289 817 7 
11.2924323 
11.2950457 
11.2976579 
11.300 268 8 
11.3028786 
11.305487 1 
11.3080945 
11.310 7006 
11.3133056 
ii- 3 I 59094 
11.3185119 
11.321113 2 
11.323 713 4 
11.3263124 
11.3289102 











2g8 

Number. 

1455 

1456 

1457 

1458 

1459 

1460 

1461 

1462 

1463 

1464 

1465 

1466 

1467 

1468 

1469 

1470 

1471 

1472 

1473 

1474 

147s 

1476 

1477 

1478 

1479 

1480 

I4SI 

1482 

1483 

1484 

1485 

i486 

1487 

1488 

1489 

1490 

1491 

1492 

1493 

1494 

1495 

1496 

1497 

1498 

1499 

1500 

1501 

1502 

1503 

1504 

1505 

1506 

1507 

1508 

1509 

1510 


SQUARES, CUBES, AND ROOTS. 


Square. 

2 11 70 25 
2 11 99 36 
2 12 28 49 
2 12 57 64 

2 12 86 8l 
2 13 l6 OO 

2 1345 21 
2 13 74 44 
2 140369 
2 14 32 96 
2 14 62 25 
2 I49I 56 
2 15 20 89 
2 15 50 24 
2 IS 7961 
2 1609OO 
2 163841 
2 1667 84 
2 l6 97 29 
2 17 26 76 
2 1756 25 
2 1785 76 
2 l8 15 29 
2 l8 44 84 
2 18 74 41 
2 1904OO 
2 1933 61 
2 196324 
2 1992 89 
2 20 22 56 
2 20 52 25 
2 20 8l 96 
2 21 II 69 
2 21 41 44 
2 21 71 21 
2 22 OI OO 
2 22 30 8l 
2 22 60 64 
2 22 90 49 
2 23 20 36 
2 23 SO 25 
2 23 80 l6 

2 24 IO 09 

2 24 40 04 

2 24 70 OI 
2 25 OO OO 
2 25 30 OI 

2 25 60 04 
2 25 90 09 

2 2620 l6 

2 26 50 2S 
2 26 80 36 

2 27 IO 49 

2 27 40 64 
2 27 70 8l 
2 28 OI OO 


Cube. 


3080271 375 
3 086 626 816 
3 092 990 993 
3°99 363 9 12 
3105 745 579 
3 112 136000 
3 118535 181 
3 I2 4943 128 
3 I 3 1 359 847 
3 i 37 785 344 
3 144219625 
3 150 662 696 
3 157 11 4 563 
3 l6 3 575 232 
3170044 709 
3 176523000 
3 183010 hi 
3 189 506048 
3 196010817 
3 202 524 424 
3 209 046 875 
3215578 176 
3222 118333 
3 228 667 352 
3 235 225 239 
3 241 792 000 
3248 367641 
3254952 168 
3 261 545 587 
3 268 147 904 

3 274 759 I2 5 
3281379256 
3 288 008 303 
3 294 646 272 
3 30 1 293 169 
3 307 949 000 

3314613771 

3 321 287 488 
3327970157 
3 334 661 784 
3 34 1 362 375 
3348071936 

3 354 79°473 
3361 517992 
3 368 254 499 
3375000 000 
338i 754 501 
3388518008 
3 395 290 527 
3 402 072 064 
3 408 862 625 
3 415 662 216 
3 422 470 843 
3429 288512 
3436 115 229 
3442 951 000 


Square Root. 


38.144 4622 
38.1575681 
3S.17O 6693 
38.183 766 2 
38.196 858 5 
38.2099463 
38.223 0297 
38.236 108 5 
38.249 182 9 
38.262 252 9 
38.2753184 
38.2883794 
38.301 436 
38.3x44881 
38 . 3 2 7 535 8 
38.340579 
38.353 6 I 7 8 
38.366 652 2 
38.379 682 I 
38.392 707 6 
38.405 728 7 
38.4187454 

3 S. 43 1 757 7 
38.444 765 6 
3S.457 769 1 
38.470 768 1 
38.483 762 7 
3S.496 753 
38.509739 
38.522 720 6 
38.535 697 7 
38.5486705 
38.5616389 
38.574603 
38.587562 7 
38.600 518 1 
38.6134691 
38.626415 8 
38-639 358 2 
38.652 296 2 
38.665 229 9 
38.678 1593 
38.691 084 3 
38.704 005 
38.716 921 4 
38.729 8335 
38.742 741 2 

38-7556447 

38.768 5439 
38.781 4389 
38.7943294 
38.8072158 
38.820097 8 
38.832975 7 
38.845 849 1 
38.8587184 


Cube Root; 


n- 33 1 5o6 7 
n-334 1022 
II.336 6964 
II.3392894 
II.341 8813 

II.344 47I9 

II.3470614 
II.349 6497 
11-352 2368 
II.354 822 7 
II.3574075 

11 -359 991 1 
11-3625735 
11.3651547 
11.367 734 7 

ii - 37 0 3 i 36 
11.372 8914 
11.375 4679 
11.3780433 
11.3806175 

11-383 I 9° 6 

11.385 7625 
ti- 3 88 333 2 
11.390902 8 
n -393 47 i 2 
11.396 0384 
11.398 6045 
11.401 169 5 
ii -403 733 2 
11.406 295 9 
11.408 8574 
11.4114177 
11.4139769 
11.4165349 

11.419 091 8 

11.420 647 6 
11.424 202 2 
11.426 755 6 
11.429 3079 
n-43 1 859 1 
11.4344092 
11.436958 1 
11.439 5059 
11.442 0525 
11.444 598 
11.447 1424 
11.449 685 7 
11.452 227 8 
11.454 7688 
11.4573087 
11.459 847 4 
11.462 385 
11.464 921 
11.467456 
11.469991 1 
11.4725242 


00 Or 









Number, 

I5II 

1512 

1513 

1514 

1515 

1316 

1517 

1518 

ISI9 

1520 

1521 

1522 

1523 

1524 

1525 

1526 

1527 

1528 

1529 

1530 

IS 3 I 

1532 

1533 

1534 

1535 

1536 

1537 

1533 

1539 

1540 

1541 

1542 

1543 

1544 

I 545 

1546 

1547 

1543 

1549 

1550 

i55i 

1552 

1553 

1554 

1555 

1556 

1557 

1558 

1559 

1560 

1561 

1562 

1563 

1564 

1565 

1566 


SQUARES, CUBES, AND ROOTS. 


299 


Square. 

2 28 31 21 
2 28 6l 44 
2 28 91 69 
2 2991 96 
2 29 52 25 
2 29 82 56 
230 12 89 
2304324 
2 30 73 61 

2 31 04 OO 

2 31 34 41 
23164 84 
2319529 
2 32 25 76 
232 5625 
2 32 86 76 
2331729 
2 33 47 84 
2 33 78 41 
2 34 09 00 
2343961 
2 34 70 24 
2 35 00 89 
2 35 3 T 56 
2 35 62 25 
2 35 92 96 
2 q 6 23 6g 
2 36 5444 
236 8521 
2 37 16 00 
2 37 46 81 
2 37 77 64 
2 38 08 49 
2383936 

2 38 70 25 
2 3901 16 
2 39 32 09 
2396304 
2399401 
2 40 25 00 
2405601 
2408704 
2 41 18 09 
241 49 16 
2 41 80 25 
2 42 11 36 
2 42 42 49 
2 42 73 64 
2430481 
2 43 36 00 
2436721 
2 43 98 44 
2 44 29 69 
2 44 60 96 
2449225 
i 2452356 


Cube. I Square Root. 


3 449 795 831 
3 456 649 728 
3463512697 
3 470 384 744 
3 477 265 875 
3484 156096 
3 49 1 °55 4i3 
3 497 963 832 
3504881359 
3 511 808000 
3 5i8 743 761 
3 525 688 648 
3 532 642 667 
3 539605824 
3546578125 
3 553 559 576 
3560 558183 
3 567 549 952 
3 574 558 889 
3581 577000 
3 588 604 291 
3 595 640 768 
3 602 686 437 
3 609 741 304 
3 616 805 375 
3 623 878 656 
3630961 153 
3 638 052 872 

3645 153819 

3 652 264 000 

3659383 421 

3 666 512 088 
3 673 650 007 
3680797 184 
3.687 953 625 
3695 119 336 
3 702 294 323 
3709478 592 
3716672 149 
3 723 875 000 
3731087 151 
3738308 608 
3 745 539 377 
3 752 779 464 
3 760 028 875 
3 767 287 616 
3 774 555 693 
3781833112 
3789119879 
3 796416000 
3803721 481 
38x1036328 
3 818 360 547 
3 825 641 144 

3833037 I2 5 

3 840 389 496 


38.8715834 
38.884 444 2 
38.8973006 
38.910 1529 
38.923 0009 
38.935 844 7 
38.948 684 1 
38.961 5194 
38.974 350 5 
38.9871774 
39 

39.012 8184 
39.025 632 6 
39.038 442 6 
39.051 2483 
39.0640499 
39.076 847 3 
39.089 640 6 
39.102 4296 
39.115 2144 

39- I2 7 995 x 
39.140 771 6 
39- x 53 543 9 

39.166312 
39.179076 
39.19 1 8359 
39.204 591 5 
39.2173431 
39.2300905 
39.242 833 7 
39.255 572 8 
39.268 307 8 
39.281 038 7 
39,293 7654 
39.306488 
39.3192065 
39.3319208 
39-344 631 1 
39-357 337 3 
39.3700394 
39.382 737 3 
39-395 43i 2 
39.408 121 
39.420 806 7 
39-433 488 3 
39.446 165 8 
39.458 8393 
39.471 508 7 
39.484174 
39.496 835 3 
39.509 492 5 
39.522 145 7 
39-534 794 8 
39-547 4399 
39.560 080 9 
39.5727179 


Cube Root. 


II.475 0562 
II -477 587 1 
11.480 1169 
11.4826455 
11.485 1731 
11.487 6995 
11.4902249 
11.492 749 1 
11.4952722 
11.497 794 2 
11.500 315 1 
11.5028348 
1 1.505 353 5 
11.507871 1 
11.5103876 
11.512903 
ii- 5 i 54 i 73 
n.5179305 

11.5204425 
11.522 9535 
11.5254634 
11.527 972 2 
11.530 4799 
11.532 9865 
11 -535 492 
11.5379965 
11.5404998 
11.543002 1 
11.545 503 3 
11.548 0034 
11.550 5025 
11-5530004 
n -555 497 3 
11 -557 993 1 
11.560 487 8 
11.562 981 5 
11.565474 
11.5679655 
11-570 455 9 
11.5729453 
n -575 433 6 
11.5779208 
11.5804069 
11.582 891 9 
ii-585 375 9 
11.587 8588 
11.5903407 
11.592 821 5 
11.595 3013 
n -597 7799 
11.600 257 6 
11.602 7342 
11.605 209 7 
11.607 684 1 
11.610 1575 
11.612 6299 











300 SQUARES, CUBES, AND ROOTS. 


Number. 

Square. 

Cube. 

Square Root. 

Cube Root. 

1567 

2 45 54 89 

3847 751 263 

39-585 350 8 

11.615 IOI 2 

1568 

2 45 86 24 

3 855 123432 

39 597 979 7 

11.617 57 1 5 

1569 

2 46 17 61 

3 862 503 009 

39.610 604 6 

11.620 040 7 

1570 

2 46 49 00 

3 869 893 000 

39.623 225 5 

11.622 508 8 

1571 

2 46 80 41 

3877292411 

39.635 842 4 

11.6249759 

1572 

2 47 if 84 

3 884 701 248 

39.648 455 2 

11.627 442 

1573 

2 47 43 29 

3 892 X19 517 

39.661 064 

11.629 907 

1574 

2 47 74 7 6 

3 899 547 224 

39.673 668 8 

11.632371 

1575 

2 48 06 25 

3 906 984 375 

39.686 269 6 

11.634 8339 

I 57 6 

2 4S 37 76 

3914430976 

39.698 866 5 

11.637 295 7 

1577 

2 48 69 29 

3921 887033 

39.711 4593 

11-639 756 6 

1578 

2 49 00 84 

3 929 352 552 

39.724048 1 

11.642 216 4 

1579 

2493241 

3 93 6 827 539 

39-7366329 

11.644675 1 

1580 

2 49 64 00 

3944312000 

39.7492138 

11.647 1329 

I 5 Sl 

2 49 95 61 

3951 805 941 

39.761 7907 

11.649 5895 

1582 

2 5027 24 

3 959 3°9 368 

39-774 3636 

11.652045 2 

1583 

2 50 58 89 

3 966 822 287 

39.786 9325 

11.6544998 

1584 

2 50 90 56 

3 974 344 7°4 

39.799 4975 

11-656 953 4 

1585 

2 51 22 25 

3981 876 625 

39.812058 5 

11.659 405 9 

1586 

251 53 96 

3989418056 

39.8246155 

11.661 857 4 

1587 

251 85 69 

3 996 969 003 

39.837 1686 

11.664 307 9 

1588 

252 17 44 

4 004 529 472 

39.849717 7 

11.666 757 4 

1589 

252 4921 

4 012 099 469 

39.862 262 8 

11.669 205 8 

I 59 ° 

2 52 81 00 

4 019 679 000 

39.874 804 

11.671 6532 

i 59 i 

2 53 12 81 

4 027 268 071 

39.887 341 3 

11.674 °99 6 

1592 

2 53 44 64 

4 034 866 688 

39.899 874 7 

11.676 5449 

I 593 

2 53 76 49 

4042 474 857 

39.912 404 1 

11.678 989 2 

1594 

2 54 08 36 

4 050 092 584 

39 924 929 5 

11.681 4325 

1595 

2 54 40 25 

4057 719875 

39 937 451 1 

11.683 874 8 

1596 

2 54 72 16 

4 065 356 736 

39.949 968 7 

11.686 316 1 

1597 

2 55 04 09 

4073 003 173 

39.962 482 4 

11.688 7563 

1598 

2 55 3 6 04 

4 080 659 192 

39.974 992 2 

11.691 195 5 

1599 

2 55 68 01 

4 088 324 799 

39.987 498 

11.693633 7 

1600 

2 56 00 00 

4 096 000 000 

40 

11.696 070 9 


Uses of preceding table may be greatly extended by aid of following 
Rides: 


To Compute Square or Cube of a liiglxer Number tliaix 
is contained iix Table. 

When Number is divisible by a Number without leaving a Remainder. 

Rule.— If number exceed by 2, 3, or any other number of times, any number 
contained in table, multiply square or cube of that number in table by square of 2, 
3, etc., and product will give result. 

Example. —Required square of 1700. 

1700 is 10 times 170, and square of 170 is 2 8900. 

Then, 2 89 00 X io 2 — 2 89 00 00. 

2.—What is cube of 2400? 

2400 is twice 1200, and cube of 1200 is 1728000000. 

Then 1 728 000 000 X 2 3 = 13 824 000 000. 

When Number is an Odd Number. 

Rule.— Take the two numbers nearest to each other, which, added together, 
make that sum; then from sum of squares or cubes of these two numbers, multi¬ 
plied by 2, subtract 1, and remainder will give result. 











SQUARES, CUBES, AND ROOTS. 


301 


Example. —What is square of 1745 ? 

Two nearest numbers are | f ^31 _ I745> 

Then per table, f o 73 ^ = 7^ 21 29 

’ ’ (072 ~ = 76 03 84 

1 5 2 25 *3 X 2 = 3 045 026 — 1 = 3 04 50 25. 

To Compute Square or Cube Root of a lriglier TSTu-rnber 
tlaan is contained, in Table. 

When Number is divisible by 4 or 8 ivithout leaving a Remainder. 

Rule. —Divide number by 4 or 8 respectively, as square or cube root is required; 
take root of quotient in table, multiply it by 2, and product will give root required. 

Example.—W hat are square and cube roots of 3200? 

3200 -r- 4 = 800, and 3200 -j- 8 = 400. 

Then, square root for 800, per table, 1328.284271 2, which, being x 2 = 56.5685 424 
root. 

Cube root for 400, per table, is 7.368 063, wbich, being X 2 = 14.736126 root. 

When the Root (which is taken as Number ) does not exceed 1600 . 

The Numbers in table are roots of squares or cubes, which are to be taken 
as numbers. 

Illustration. —Square root of 6400 is 80, and cube root of 512000 is 80. 

When a Number has Three or more Ciphers at its right hand. 

Rule. —Point off number into periods of two or three figures each, according as 
square or cube root is required, until remaining figures come within limits of table; 
then take root for these figures, and remove decimal point one figure for every pe¬ 
riod pointed off. 

Example. —What are square or cube roots of 1 500000? 

1 500000 = 150, remaining figure, square root of which=i2.247 45; hence 1224.745, 
square root. 

1500000 = 1500, remaining figures, cube root of which = n. 447 14 ; hence 
114.4714, cube root. 

To Ascertain Cube Root of any Number over 1600. 

Rule. —Find by table nearest cube to number given, and term it assumed cube; 
multiply it and given number respectively by 2; to product of assumed cube add 
given number, and to product of given number add assumed cube. 

Then, as sum of assumed cube is to sum of given number, so is root of assumed 
cube to root of given number. 

Example. —What is cube root of 224 809? 

By table, nearest cube is 216000, and its root is 60. 

216000 x 2-f 224809 = 656809, 

And 224 809 X 2 4-216000 = 665 618. 

Then 656 809 : 665 618 :1 60 : 60.8044-, root . 

To Ascertain Square or Cube Root of a ZNTnm'ber con¬ 
sisting of Integers and. Decimals. 

Rule. —Multiply difference between root of integer part and root of next higher 
integer by decimal, and add product to root of integer given; the sum will give root 
of number required. 

This is correct for Square root to three places of decimals, and for Cube root to seven. 

C c 



302 SQUARES, CUBES, AND ROOTS. 

Example. —What is square root of 53.75, and cube root of 843.75? 


V 54 — 7 ’ 34^4 

^844 =9.4503 

V 53 =7.2801 

ij/843 =9.4466 

.0683 

.0037 

•75 

•75 

.051 225 

.002 775 

V 53 — 7 - 2 %° 1 

-^843 =9.4466 

V 53-75 — 7 - 33 1 3 2 5 

i/843.7 5 = 9.449 375 


When the Square or Cube Root is required for Numbers not exceeding Roots 

given in Table. 

Numbers in table are squares and cubes of roots. 

Rule. — Find, by table, in column of numbers that number representing figures 
of integer and decimals for which root is required, and point it off decimally by' 
places of 2 or 3 figures as square or cube root is required; and opposite to it, in 
column of roots, take root and point off 1 or 2 additional places of decimals to those 
in root, as square or cube root is required, and result is root required. 

Example x. —What ai'e square roots of .15, 1.50, and 15.00? 

In table, 15 has for its root 3.87298; hence .387298 = square root for .15. 

150 has for its root 12.24 74 5 i hence 1.22 47 45 = square root for 1.50. 

1500 has for its root 38.7298; hence 3.87 29 8 = square root for 15. 

2.—What are cube roots of .15, 1.50, and 15.00? 

Add a cipher to each, to give the numbers three places of figures, as .150, 1.500, 
and 15.000. 

In table 150 has for its l’oot 5.3133; hence .531 33 = cube root of. 15. 

1500 has for its root 11.447; hence 1.1447 = cube root of 1.50. 

15 has for its root 2.4662; and 15.000, by addition of 3 places of figures, has 
24.662; hence 2.4662 = cube root of 15.00. 

To Ascertain Square or Cube Hoots of Decimals alone. 

Rule. —Point off number from decimal point into periods of two or three figures 
each, as square or cube root is required. Ascertain from table or by calculation 
root of number coi'responding to decimal given, the same being read off by remov¬ 
ing the decimal point one place to left for every period of 2 figures if square root is 
required, and one place for every period of 3 figures if cube root is required. 

Example. —What are square and cube roots of .810, .081, and .0081 ? 


.810, 

when pointed off = .81, 

and y/.8i =.9. 

.081, 

“ “ “ =.081, 

“ -\/.o8i =.2846. 

.0081, 

“ “ “ = .0081, 

“ y.oo8i=.o9. 

.810, 

when pointed off = .810, 

and ^/. 810 = 93217. 

w 

00 

0 

“ “ “ =.081, 

“ y .081 =.43267. 

.0081, 

“ t£ = .0081, 

“ y/. 0081 = .200 83. 


To Compute 4 tli Hoot of a jSTnxnDer. 
Rule. —Take square root of its square root. 

Example. —What is the -ty of 1600? 

1600 = 40, and y/40 — 6.32 45 55 3. 

To Compute 6th. Hoot of a INTnmDer. 
Rule.—T ake cube root of its square root. 

Example.—W hat is the -£/ of 441 ? 

vQ.41 = 21, and ^21 = 2,7 589 243. 









FOURTH AND FIFTH POWERS OF NUMBERS. 303 
4th. and. 5th Powers of Numbers. 


From 1 to 150 . 


Number. 

4th Power. 

5th Power. 

Number. 

4tb Power. 

5th Power. 

I 

I 

I 

64 

16777216 

1 073 741 824 

2 

16 

32 

65 

17850625 

1 160 290 625 

3 

81 

243 

66 

18974736 

1 2 S 2 332 576 

4 

256 

I 024 

67 

20 151 121 

I 350125107 

5 

625 

3125 

68 

21 381 376 

1453933568 

6 

1 296 

7 776 

69 

22 667 121 

1564031349 

7 

2 4OI 

16807 

70 

24 OIO OOO 

I 680700000 

8 

4096 

32 768 

71 

25 411 68l 

I 804 229 351 

9 

6 561 

59049 

72 

26873856 

1934917632 

IO 

IOOOO 

100000 

73 

28 398 241 

2073071 593 

II 

14641 

161051 

74 

29986576 

2219006624 t 

12 

20736 

248 832 

75 

31640625 

2 373 046 875 

13 

28 561 

371 293 

76 

33 362176 

2 535 525 376 

14 

38 416 

537824 

77 

35 153 041 

2706784157 

is 

50625 

759375 

78 

37615056 

2 887 174 368 

16 

65536 

1048 576 

79 

38950081 

3 °77 056 399 - 

17 

83 521 

1419857 

80 

40 960 OOO 

3 276 800 OOO 

18 

104976 

1 889 568 

81 

43046721 

3 486 784 401 

19 

130 321 

2476099 

82 

45212.176 

3 707 398 432 

26 

160000 

3 200 000 

83 

47 458 321 

3939O40643 

21 

194 481 

4 084 IOI 

84 

49 787136 

4 182 119424 

22 

234 256 

5153632 

85 

52 200 625 

4 437 053125 

23 

279 841 

6436 343 

86 

54 708016 

4704 270176 

24 

331776 

7 962 624 

87 

57 289 761 

4 984 209 207 

25 

390 625 

9765625 

88 

59969 536 

5277319168 

26 

456 976 

11 881 376 

89 

62 742 241 

5 584 059 449 

27 

531441 

14 348 907 

9° 

65 610000 

5 904 900 OOO 

28 

614656 

17 210368 

9i 

68574961 

6240321451 

29 

707 281 

20 511149 

92 

71639296 

6 590815 232 

3° 

810000 

24 306000 

93 

74 805 201 

6956883693 

31 

923 521 

28629 151 

94 

78074896 

7339040224 

32 

1048 57 6 

33 554432 

95 

81 450 625 

7 737 809 375 

33 

1185921 

39 r 35393 

96 

84034656 

8 153 726 976 

34 

1 336 336 

45 435 424 

97 

88 529 281 

8587 340257 

35 

1 500625 

52521875 

98 

92 236 816 

9039207968 

36 

1679616 

60 466176 

99 

96059601 

9 509900 499 

37 

1 874161 

69 343 957 

100 

IOO OOO OOO 

IO OOO OOO OOO 

38 

2 085136 

79235168 

IOI 

104 060401 

10510 IOO 501 

39 

2313441 

90 224199 

102 

108 243.216 

II 040808 032 

40 

‘2 560' OOO 

102400000 

103 - 

112 550 881 

11592740743 

4i 

2 825 761 

115 856 201 

104 

116985856 

12 166 529 024 

42 

3 in 696 

130691 232 

105 , 

121 550625 

12 762 815 625 

43 

3 418 801 

147 008 443 

106 

126 247 696 

13382255776 

44 

3748096 

164916224 

107 

131 079 601 

14025517307 

45 

4 100625 

184 528 125 

168 

136 048 896 

14693 280768 

46 

4 477 456 

205.962 976 

109 

141 158 161 

I.5386239549 

47 

4 879681 

229345007 

no 

146 410 OOO 

16 105 IOO OOO 

48 

5 308 416 

254803968 

III 

151 807 041 

16 850581551 

49 

5764801 

282475249 

112 

x 57 35 1 936 

17623416832 

5o 

6 250000 

312500000 

H3 

163 047 361 

18424351793 

51 

6 765 201 

345025251 

114 

168 896 016 

19 254145 824 

52 

7311616 

380204O32 

H5 

174900 625 

20 II3581 875 

53 

7 890481 

4 l8 195 493 

Il6 

181 063 936 

21 OO3 416 576 

54 

8 503 056 

459165024 

117 

187 388 721 

21924480357 

55 

9 150625 

503 284 375 

Il8 

193 877 776 

22 877 577 568 

56 

9834496 

550731.776 

ng 

200 533 9 21 

23 863 536 599 

57 

10556001. 

601 692 057 

120 

207 360 OOO 

24 883 200 OOO 

58 

11 316496 

656356768 

121 

214 358 881 

25937424601 

59 

12 117 361 

714924299 

122 

22 i 533 456 

27 027 b8l 632 

60 

12 960 OOO 

777 600 OOO 

123 

228 886 641 

28153056843 

61 

I3845S4I 

844 596 301 

124 

236421376 

29 316 250624 

62 

14776336 

916 132 832 

125 

244 140625 

• 30 517 578 125 

63 

15752961 

992 436 543 

126 

252047 376 

31 757 969 376 















304 


POWERS OF NUMBERS.-RECIPROCALS. 


Number. 

4 th Power. 

5 th Power. 

Number. 

4 th Power. 

5 th Power. 

127 

260 144 641 

33 °S8 369 407 

139 

373 3 01 641 

51 888 844 699 

128 

268435456 

34 359 738 368 

140 

384 160000 

53 782 400000 

129 

276 922 881 

35723 051649 

141 

395 254 * 6 i 

55 73 0 836 7 01 

130 

285 610000 

37 129 300000 

142 

406 586 896 

57 735 339 2 32 

131 

294 499 921 

38579 489651 

M3 

418 161 601 

59 797 108 943 

132 

3°3 595 776 

40074 642432 

144 

429 981 696 

61 917 364 224 

133 

312 900 721 

41615 795 893 

145 

442 050 625 

64 097 340625 

134 

322417936 

43 204 003 424 

146 

454371 856 

66 338 290 976 

135 

332 150625 

44 840 334 375 

147 

466 948 881, 

68 641 485 507 

136 

342 102 016 

46525 874176 

148 

479785 216 

71 008 211 968 

137 

352275361 

48 261724457 

149 

492 884 401 

73 439 775 749 

138 

362 673936 

50049 003168 

150 

506 250 000 

75 937 5 00000 


Compute “1th IPower of a, 3STnm'ber greater than is 
contained in Table. 

Rule. —Ascertain square of number by preceding table or by calculation, and 
square it; product is power required. 

Example.—W liat is 4th power of 1500? 

1500 2 = 2 250 000, and 2 250 ooo 2 = 5 062 500 000 000. 

To Compute £3th. Power of a Number greater than is 

contained, in Table. 

Rule. —Ascertain cube of number by preceding table or by calculation, and mul¬ 
tiply it by its square; product is power required. 

To Compute 4th. and 5th. Powers by another NEethod. 

Rule. —Reduce number by 2 until it is one contained within table. Take power 
which is required of that number, and multiply it by 16, 16 2 , or i 63 respectively 
for each division, by 2 for 4th power, and by 32, 32 2 , or 323 respectively for each 
division by 2 for 5th power. 

Example. —What are the 4th and 5th powers of 600? 

600 -j- 2 = 300, and 300 -T- 2 = 150. 

The 4th power of 150, per table,=3 5o6 25oooo, which x 16 2 , multiplier for a second 
division 256 = 129 600 000 000, 4 th power. 

Again, the 5th power of 150 = 75 937 500 000, which X 32 s , multiplier for a second 
division 1024 77 760000000000 — power. 

To Compute 6th Power of a 1ST umber. 

Rule. —Square its cube. 

Example. —What is the 6th power of 2? 

2 3 = 64. 

To Compute 4th or 5th Root of a IN'umber per Table. 

Rule. —Find in column of 4th and 5th powers number given, and number from 
which that power is derived will give root required. 

Example.— What is the 5th root of 3 200000? 

3 200000 in table is 5th power of 20; hence 20 is root required. 


RECIPROCALS. 

Reciprocal of a number is quotient arising from dividing 1 by number; thus re¬ 
ciprocal of 2 is 1 — 2 =. 5 

Product of a number and its reciprocal is always equal to 1; thus, 2 X .5 = 1. 
Reciprocal of a vulgar fraction is denominator divided by numerator; thus, - = .5. 












LOGARITHMS. 


305 


LOGARITHMS. 

Logarithms of ^mribers. 

Logarithms are a series of numbers adapted to facilitate the operation of 
numerical computation, 

Addition being substituted for Multiplication, Subtraction for Division, 
Multiplication for Involution, and Division for Evolution. 

The Logarithm of a number is the exponent of a power to which 10 
must be raised to give that number. 

It is not necessary, however, that the base should be 10, it may be any other num¬ 
ber; but Tables of Logarithms, in common use, are computed with 10 as the base. 

Thus, Number 100 Log. = 2, as io 2 base and exponent =s zoo. 

“ 10000 “ =4, “ 10 4 “ “ “ =10000. 

The Unit or Integral part of a Logarithm is termed the Index , and the Decimal 
part the Mantissa; the sum of the index and mantissa is the Logarithm. 

The Index of the Logarithm of any number , Integral or Mixed , w r hen the base is 10, 
is equal to the number of digits to the left of the decimal point less 1. From o to 
9, it is o; from 10 to 99, it is 1, and from 100 to 999, it is 2, etc. 

Thus, logarithm of 3304 = 3.51904, 3 being the index and .51904 the mantissa. 

The Index of the Logarithm of a Decimal Fraction is a negative number, and is 
equal to the number of places which the first significant figure of the decimal is re¬ 
moved from the place of units. 

Thus, index of logarithm .005 is 3 or —3, the first significant figure, 5, being re¬ 
moved three places from that of units. The bar or minus sign is placed over an 
index to indicate that this alone is negative, while the decimal part is positive. 

The Difference is the tabular difference between the two nearest logarithms. 

The Proportional Part is the difference bettveen the given and the nearest less 
tabular logarithm. 

The Arithmetical Complement of a number is the remainder after subtracting it 
from a number consisting of 1, with as many ciphers annexed as the number lias 
integers. When the index of a logarithm is less than 10, its complement is ascer¬ 
tained, by subtracting it from 10. 


Number. 
4743 • • • 
474 - 3 -• 
47 - 43 - 
4-743 


Illustrations. 


Logarithm. 

3.676053 

2.676053 

1.676053 
.676053 


N umber. 

•474 3 •• 
.04743 . 
.004 743 


Logarithm. 

^.676053 

2.676053 

3-676053 


Computation of INTegative Indices. 

To add two Negative Indices. Add them and put the sum negative. As 5 -f- 3 = 3 . 

To add a Positive and Negative Index. Subtract the less from the greater, and 
to remainder give the positive or negative sign, according as the positive or nega¬ 
tive index is the greater. As 6 -j- 2 = 4, and 6 + 2 = 4. 

Illustration.— Add 6.387 57 and 2.924 59. 6.387 57 

2 - 9 2 4 59 

5.31216 

Here the excess of 1 from 13 in the first decimal place, being positive, is carried 
to the positive 6, which makes 7, and 7 — 2 = 5. 

To Subtract a Negative Index. Change its sign to plus or positive, and then add 
it as in addition. As 3 from 2, = 3 -f- 2 = 5. And 5 from 2, = 5 + 2 = 3 ; also 
3 from 5 , = 3 + 5 = 2. 

Illustration. —Subtract 5.765 52 from 2.346 74. 2.34674 

5-765 5 2 

2.58122 

Here, excess of 1 in the first decimal place used with the .3 in subtracting the .8 
from the 1.3 is to be subtracted from the upper number 2, which makes it 3; then 
3 + 5 = 2 - Cc * 












LOGARITHMS 


306 

To Subtract a Positive Index. Change its sign to negative, and then add as in 
addition. As 2 — 2 = 2 + 2 = 4. 

To Multiply a Negative Index. Multiply the fractional parts by the ordinary rule, 
then multiply the negative index , which will give a negative product, and when an 
excess over 10 is to be carried, subtract the less index from the greater, and the re¬ 
mainder gives the positive or negative index , according as the positive or negative 

index is the greater. As 2 X 5 = 10, and 1 to be carried = 9. 

Illustration.—M ultiply 2.3681 by 2, and 3.7856 by 6. 

2.3681 3-7856 

2 6 

4.7362 14-7136 

Here 2X2 = 4, also 3X6 = 18, with a positive excess of 4 = 14. 

To Divide a Negative Index. If index is divisible by divisor, without a remain¬ 
der, put quotient with a negative sign. If negative exponent is not divisible by 
divisor, add such a negative number to it as will make it divisible, and prefix an 
equal positive integer to fractional part of logarithm; then divide increased nega¬ 
tive exponent and the other part of logarithm separately by ordinary rules, and for¬ 
mer quotient, taken negatively, will be index to fractional part of quotient. As 

6 = 3 = 2. 10 -r- 3 requires 2 to be added or 2 to be subtracted, to make it divisible 

without a remainder, then 10 + 2 = 12, 12 = 3 = 4, and 2 (the sum subtracted) = 
3 = .66, the quotient therefore is 4.66. 

Illustration i.— Divide 6.324282 by 3. 

6.324 282 -4-3 = 2.108 094. 

2.—Divide 14.326745 by 9. 

14.326 745 = 9 = 18 + 4.326 745 = 9 = 2.480 749+. 

Here 4 is added to 14, that the sum 18 may be divided by 9, and as 4 is added, 4 
must be prefixed to the fractional part of the logarithm, and thus the value of the 

logarithm is unchanged, for there is added 4, and 4 = o, or 4 is subtracted and 4 
added. 


To ^Ascertain Logarithm, of a Number by Table. 

When the Number is less than roi. 

Look into first page of table, and opposite to number is its logarithm with its 
index prefixed. 

Illustration. —Opposite 7 is .845098, its logarithm; hence 70=1.845098, .7 = 
1.845098, and .07 = 2.845098. 

When the Number is between 100 and 1000. 

Rule. —Find the given number in left-hand column of table headed No., and un¬ 
der 0 in next column is decimal part of its logarithm, to which is to be prefixed a 
whole number for an index , of 1 or 2, according as the number consists of 2 or 3 
figures. 

Example.— What is logarithm of 450, and what of .45 ? 

Log. 450 = 2.653213, and of .45= 1.653213. 

When the Number is between 1000 and 10 000. 

Rule. —Find the three left-hand figures of the number in the left-hand column 
of the table headed No., and under the 4th figure at top of table is the four last 
figures of the decimal part of logarithm, to which is to be prefixed the proper 
index. 

Example.— What is logarithm of 4505, and what of .045 05 ? 

Log. 4505 = 3.653 695, and of .045 05 = 2.653 695- 




LOGARITHMS, 


307 


When the Number consists of Five Figures. 

Rule.—F ind the logarithm of the number composed of the first four figures as 
preceding, then take the tabular difference from the right-hand column under D 
and multiply it by the fifth figure; reject the right-hand figure of the product and 
add the other figures, which are, and are termed, a proportional part to the logarithm 
found as above, observing that the right-hand figure of the proportional part is to 
be added to that of the logarithm, and the rest in order. 

Example. —Required logarithm of 83 407 ? 

Note.—W hen the number consists of less than 4 figures conceive a cipher an¬ 
nexed to make it four. 

Log. of 8340 (83 407) = 4.921166 

Tabular difference 52, which x 7 (5th figure) = 364 = 364 

4.921 202 4 logarithm. 

The difference of the numbers is nearly proportionate to the difference of their 
logarithms. 

Thus, difference between the numbers 8340 and 8341, the next in order, is 1, and 
the difference between their logarithms or tabular difference is 52. 

The log. of this 1 in the 4th place is therefore 52. The correction then, for the 7 
of the 5th place, which is .7 of 1 in the 4th place, is ascertained by the proportion 
1 : 52 ;; .7 : 36.4. 

The correction is obtained by multiplying the tabular difference by 7, rejecting 
the right hand figure of the product, if the log. is to be confined to six decimal 
places. 

When the Number consists of any Number over Four Figures. 

Rule. —Proceed as for four figures for the first four, multiplying the tabular dif¬ 
ference by the excess of figures over 4 and rejecting one right-hand figure of the 
product for a number of five figures, and two for one of six, and so on. 


Example i. —Required logarithm of 834079? 

Log. of 8340 (834079) = 5.921166 
Tabular difference 52, which x 79 = 4108 

5.92120708 logarithm. 

2.—Required logarithm of 8 340 794? 

Log. of 8340 (8 340 794) = 6.921166 

Tab. diff. 52, which X 794 (5th, 6th, and 7th figures) = 41 288 

6.921207 288 logarithm. 

Or, Log. of 8340 = .921166 

“ “ 7 (5th figure) X 52 tab. dif. = 364 

“ “ 9 (6th “ ) X 52 “ “ == 468 

“ £< 4 (7th “ ) X 52 l! “ = 208 

Log. with index for 7 figures.6.921 207 288 

To Ascertain. Logarithm of a, IVlixecl USTumber. 
Rule. —Take out logarithm of the number as if it were an integer or whole num¬ 
ber, to which prefix the index of the integral part of the number. 

Example. —What is logarithm of 834.0794? 

Mantissa of log. of 8 340794 = 9 212073; hence log. of 834.0794 = 2.921 207 3. 


To Ascertain FogarifKm of a Decimal Fraction.. 

Rule. —Take logarithm from table as if the figures were all integers, and prefix 
index as by previous rules. 

Example. —Logarithm of .1234 = 1.091 305. 

To Ascertain Logarithm of a Vulgar Fraction. 

Rule. —Reduce the fraction to a decimal, and proceed as by preceding rule. Or, 
subtract logarithm of denominator from that of numerator, and the difference will 
give logarithm required. 

Example. —Logarithm of^? 

^ = .1875. Log. .1875 =1.273001 logarithm. 

Or, Log. 3 = .477 121 
“ 16 = 1.20412 

1.273 001 logarithm. 








3°8 


LOGARITHMS. 


To .Ascertain th.e Number Corresponding to a Given 

Logarithm. 

When the given or exact Logarithm is in the Table. 

Operation. —Opposite to first two figures of logarithm, neglecting the index , in 
column o, look for the remaining figures of the log. in that column or in any of the 
nine at the right thereof; the first three figures of the number will be found at the 
left in column under No., and the fourth at top directly over the log. 

The number is to be made to correspond to index of logarithm, by pointing off 
decimals or prefixing ciphers. 

Illustration. —What is number corresponding to log. 3.963 977 ? 

Opposite to 963977, in page 329, is 920, and at top of column is 4; hence, num¬ 
ber = 9204. 

When the given or exact Logarithm is not in the Table. 

Operation. —Take the number for the next less logarithm from table, which will 
give first four figures of required number. 

To ascertain the other figures, subtract the logarithm in table from the given 
logarithm, add ciphers, and divide by the difference in column D opposite the 
logarithm. Annex quotient to the four figures already ascertained, and place deci¬ 
mal point. 

Illustration i. —What is number corresponding to log. 5.921 207 ? 

Given log. =r 5-921 207 

Next less in table 5.921166 8340 

D — 52) 4100 (78-f- _78 

834078 

460 
416 

Hence, number = 834 078. 44 

2.—What is number corresponding to log. 3.922853? 

Given log. = 3.922853 

Next less in table 3-922 829 

D = 52) 2400 (46 -f- 

' 208 

320 

Hence, number = 8372.46. 8 

Nlnltiplication. 

Rule. —Add together the logarithms of the numbers and the sum will give the 
logarithm of the product. 

Example i. —Multiply 345.7 by 2.581. 

L °g-345-7 =2.538699 

“ 2.581= .411788 

2.950487 log. of product. Number = 892.251. 

2.—Multiply .03902, 59.71, and .003 147. 

Log. .03902 =2.591287 
“ 59 - 7 i =£-776047 

“ .003147 = 3,497 897 

3.865 231 log. of product. Number = .007 332 15. 
Division. 

Rule.—F rom logarithm of dividend subtract that of divisor, and remainder will 
give logarithm of the quotient. 

Example.—D ivide 371.4 by 52.37. 

Log. 371.4 =2.569842 
“ 52.37 = 1.719083 

.850 759 log. of quotient. Number = 7.091 85. 


8372 


46 


837 246 








LOGARITHMS. 


309 


Ltnle of Three, or Proportion. 

Rule.— Add together the logarithms of the second and third terms, from their 
sum subtract logarithm of the first, and the remainder will give logarithm of the 
fourth term. 

Or, instead of subtracting logarithm of first term, add its Arithmetical Comple¬ 
ment, and subtract 10 from its index. 

Example i. —What is fourth proportional to 723.4, .025 19, and 3574? 

As 723.4 log. = _ 2.859379 

Is to .02519 “ —2.401228 
So is 3574 “ — 3 - 553 I 55 

i -954 383 

First term “ 2.859379 

1.095 004 log. of 4</i term. Number —. 124 453. 

By Arithmetical Complement. 

Illustration.— As 723.4 log. = 2.859 379 ) - A - r - com - = 7-140621 

Is to .02519 u = 2.401228 

So is 3574 “ = 3-553 I55 

1.095 004 log. of 4th term. 

Number =. 124 453. 

2. —If an engine of 67 IP can raise 57 600 cube feet of water in a given time, what 
IP is required to raise 8 575 000 cube feet in like time ? 

Log. 8 575 000 = 6.933 234 
“ 67 — 1.826 075 

8.759 309 
“ 57600 = 4.760422 

3.998 877 log. of 4th term. Number = 9974.4 cube feet. 

3. —If 14 men in 47 days excavate 5631 cube yards, what time will it require to 

excavate 47 280 at same rate of excavation ? 394.626 days. 

Involution. 

Rule. —Multiply logarithm of given number by exponent of the power to which 
it is to be raised, and the product will give the logarithm of the required power. 
Example. —What is cube of 30.71 ? 

Log. 30.71 = 1.487 28 
_ 3 

4.46184 log. of power. Number = 28 962.73. 
Evolution. 

Rule. —Divide logarithm of given number by exponent of the root which is to be 
extracted, and quotient will give logarithm of required root. 

Example i. —What is cube root of 1234? 

Log. 1234 = 3.091 315 

Divide by 3 = 1.030438 log. of root. Number = 10.72601. 

2.—What is 4th root of .007 654 ? 

Log. .007654 = 3.883888 

Divide by 4 (here 34-1 + 1)== 1.470 972 log. of root. Number = .295 78. 

To Ascertain Reciprocal of a UNTnmber. 

Rule. —Subtract decimal of logarithm of the number from .000000; add 1 to in¬ 
dex of logarithm and change its sign. The result is logarithm of the reciprocal. 

Example. —Required reciprocal of 230? 

.000000 
Log. 230 = 2.361 728 

3.638 272 = log. of .004 348 reciprocal. 










3 io 


LOGARITHMS. 


Simple Interest. 

Rule. —Add together logarithm of principal, rate per cent., and time in years, from 
the sum subtract 2, and the remainder will give logarithm of the interest. 

Example.— What is interest on $500, @ 6 per cent., for 3 years? 

Log. 500 = 2.698 97 
6= .778151 
3 — -477 121 
3.954 242 
2 

x.954 242 log. of interest. Number = 90 dollars. 

Compound Interest. 

Rule. —Compute amount of $ 1 or £ 1, etc., at the given rate of interest for one 
year for the first term, which is termed the ratio. 

Multiply logarithm of ratio by the time, add to product logarithm of the principal, 
and sum is logarithm of the amount. 


JLiOgaritlims of Ratios at given Rates Per Cent. 


Rate. 

Log. of Ratio. 

Rate. 

Log. of Ratio. 

Rate. 

Log. of Ratio. 

Rate. 

Log. of Ratio 

I 

.004321 4 

3-25 

.013 890 1 

5-5 

.023252 5 

7-75 

•032 4 W 3 

1.25 

•005 395 

3-5 

■0149403 

5-75 

.024 2804 

8 

.0334238 

i -5 

. 006 466 

3-75 

.015 988 X 

6 

.025 305 9 

8.25 

•034 427 9 

i -75 

•007 534 4 

4 

.0170333 

6.25 

.026 328 9 

8.5 

•035 429 7 

2 

. 008 600 2 

4-25 

.018 076 1 

6-5 

.027 3496 

8-75 

.0364293 

2.25 

.009 6633 

4-5 

.019 116 3 

6-75 

.028 763 9 

9 

.037 4265 

2.5 

.0107239 

4-75 ' 

.020 154 

7 

.0293838 

9-25 

.0384214 

2-75 

.011 781 8 

5 

.021 189 3 

7-25 

•030 397 3 

9-5 

.039414 1 

3 

.012 837 2 

5-25 

.022 222 1 

7-5 

.0314085 

9-75 

.040 4045 


Example.—W hat will $364, at 6 per cent, per annum, compounded yearly, amount 
to in 23 years ? 

Log. of ratio from above table .025 305 9 

•5420357 

“ “ 364 2.561 IOI 

3.103 1367 log. of amount. Number = 1268.05 doM. 

Miscellaneous Illnstrations. 

1. What is area and circumference of a circle of 21.72 feet in diameter? 

1.336 860 
2 

Log. of 21.72 2 =2.673720 
“ “ .7854 = 1.895091 

“ “ 2.568811 =370.54 feet area. 

Log. of 21.72 =2.33686 

“ “ 3-1416= -49715 

“ “ 1.83971=68.236 feet circum. 

2. Sides of a triangle are 564, 373, and 747 feet; what is its area ? 

Log. of sides 5 ^ 4 + 373 + 747 — 2.925 312 
2 

u 11 .5 side — a = 842 — 564 = 2.444045 
“ “ . 5 side — b = 842 — 373 = 2.671173 

“ “ .5 side —c =842 — 747 = 1.977724 

2)10.018 254 

Area = Number of 5.009127 = 1021.24. feet. 

3. —What is logarithm of 8 3-6 ? 

_ 8 X 36 36 . 

Log. —= — X log. 8 = 3.6 X .90309 = 3.251124. Number = 1782.89. 



























LOGARITHMS OF NUMBERS. 


31 I 


LogaritTmas of IN’viiii "bers. 
From 1 to 10 000. 


No. | 

Logarithm; | 

No. | 

Logarithm. 

No. 

Logarithm. | 

No. 

Logarithm. 

1 

.O 

26 

i-4i4 973 

51 

I.70757 

76 

I.880 814 

2 

.301 03 

27 

I -43 I 3 6 4 

52 

I.716 003 

77 

1.886 491 

3 

•477 121 

28 

1.447 158 

53 

I.724 276 

78 

1.892 095 

4 

.602 06 

29 

1.462 398 

54 

i-72»2 394 

79 

1.897 627 

5 

.698 97 

30 

1.477 121 

55 

1-740363 

80 

1.90309 

6 

•778 151 

31 

1.49 1 362 

56 

1.748 188 

81 

1.908 485 

7 

.845 098 

32 

1-505 15 

57 

1-755 875 

82 

i-9 x 3 814 

8 

.903 09 

33 

1.518 514 

58 

1.763 428 

83 

1.919 078 

9 

•954 243 

34 

I-53 1 479 

59 

1.770 852 

84 

1.924 279 

10 

1 

35 

1.544 068 

60 

1.778 151 

85 

1.929 419 

11 

1-041 393 

36 

i-556 303 

61 

1-785 33 

86 

1.934 498 

12 

1.079 J 8i 

37 

1.568 202 

62 

1.792 392 

87 

i-939 5 x 9 

13 

I * II 3 943 

38 

1-579 784 

63 

i-799 34i 

88 

I-944 483 

14 

1.146 128 

39 

1.591 065 

64 

1.806 18 

89 

1-949 39 

15 

1.176 091 

40 

1.602 06 

65 

1.812 913 

9° 

i-954 243 

16 

1.204 12 

41 

1.612 784 

66 

1.819544 

91 

1.959 041 

17 

1.230 449 

42 

1.623 249 

67 

1.826 075 

92 

1.963 788 

18 

i-255 273 

43 

1.633468 

68 

1.832 509 

93 

1.968 483 

19 

1.278 754 

44 

I-643 453 

69 

1.838 849 

94 

1.973 128 

20 

1.301 03 

45 

1-653213 

70 

1.845 098 

95 

1.977 724 

21 

1.322 219 

46 

1.662 758 

71 

1.851 258 

96 

1 982 271 

22 

1.342 423 

47 

1.672 09S 

72 

I-857 332 

97 

1.986 772 

23 

1.361 728 

48 

1.6S1 241 

73 

1-863 323 

98 

1.991 226 

24 

1.380 211 

49 

1.690 196 

74 

1.869 232 

99 

I-995 635 

25 

1-397 94 

50 

1.698 97 

75 

1.875 061 

100 

0 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

n 

100 

OCr- OOOO 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

346 i 

3891 

432 

IOI 

OO- 4321 

4751 

5!8i 

5609 6038 

6466 6894 

7321 

7748 

8174 

428 

102 

00- 86 

9026 9451 

9876 

— 

— 

— 

— 

— 

— 

425 

102 

01- — 

— 

— 

— 

03 

0724 

1147 

157 

1993 

2415 

424 

103 

01- 2837 

3259 

368 

41 

4521 

494 

536 

5779 6197 

6616 

420 

104 

01- 7033 

7451 

7868 

8284 

87 

9116 9532 

9947 

— 

— 

417 

104 

02- - 








0361 

0775 

416 

105 

02- 1189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

oc 

'Tf 

VO 

00 

412 

106 

02- 5306 

5715 

6125 

6533 6942 

735 

7757 

8164 

8571 

8978 

408 

107 

02- 9384 

9789 









405 

107 

03- — 

— 

0195 

06 

1004 

1408 

1812 

2216 

2619 

3021 

404 

108 

03- 3424 

3826 

4227 

4628 

5029 

543 

583 

623 

6629 

7028 

400 

109 

03- 7426 

7825 

8223 

862 

9017 

9414 

9811 

— 

— 

— 

398 

109 

04- — 

— 

— 

— 

— 

— 

— 

0207 

0602 

0998 

397 

110 

04- 1393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

454 

4932 

393 

hi 

04- 5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

883 

389 

112 

04- 9218 

9606 

9993 

— 

— 

— 

— 

— 

— 

— 

388 

112 

05- — 

— 

— 

038 

0766 

ii53 

1538 

1924 

2309 

2694 

386 

xx 3 

05- 3078 

3463 

3846 

423 

4613 

4996 

5378 

576 

6142 

6524 

383 

114 

05- 6905 

7286 

7666 

8046 

8426 

8805 

9 i8 5 

9563 

9942 

— 

383 

114 

06- - 









032 

379 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 






































312 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

115 

06- 0698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 4083 

376 

116 

06- 4458 

4832 

5206 

558 

5953 

6326 6699 

7071 

7443 

7815 

373 

117 

06- 8186 

8557 

8928 

9298 

9668 

— 

— 

— 

— 

— 

380 

117 

07- — 

— 

— 

— 

— 

0038 

0407 

0776 

ii 45 

1514 

370 

118 

07- 1882 

225 

2617 

2985 

3352 

37 i 8 

4085 

445 i 

4816 

5182 

366 

119 

07- 5547 

5912 

6276 

664 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

07- 9181 

9543 

9904 

— 

— 

— 

— 

— 

— 

— 

362 

120 

08- — 

— 

— 

0266 

0626 

0987 

1347 

1707 

2067 

2426 

360 

121 

08- 2785 

3 i 44 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122 

08- 636 

6716 

7071 

7426 

778 i 

8136 

849 

8845 

9198 

9552 

355 

123 

08- 9905 










355 

123 

09- — 

0258 

0611 

0963 

1315 

1667 

2018 

237 

2721 

3071 

353 

124 

09- 3422 

3772 

4122 

4471 

482 

5169 5518 

5866 6213 

6562 

349 

125 

09- 691 

7257 

7604 

7951 

8298 

co 

899 

9335 

9681 

— 

348 

125 

10- — 

— 

— 

— 

— 

— 

— 

— 

— 

0026 

346 

126 

10- 0371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3 ii 9 

3462 

343 

127 

10- 3804 

4146 4487 

4828 

5169 

551 

5851 

6191 

6531 

6S71 

34 i 

128 

10- 721 

7549 

7888 

8227 

8565 

8903 

9241 

9579 99 io 

— 

338 

128 

11- — 









0253 

337 

129 

11- 059 

0926 

1263 

1599 

1934 

227 

2605 

294 

3275 

3609 

335 

130 

11- 3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 6608 

694 

333 

131 

11- 7271 

7603 

7934 

8265 

8595 

8926 9256 9586 

9915 

— 

33 i 

131 

12- — 









0245 

330 

132 

12- 0574 

° 9°3 

1231 

156 

1888 

2216 

2544 

2871 

3198 

3525 

328 

133 

12- 3852 

4178 

4504 483 

5156 

548 i 

5806 

6131 

6456 

6781 

325 

134 

12- 7105 

7429 

7753 

8076 

8399 

8722 

9°45 

9368 

969 

— 

323 

134 

13 









0012 

323 

135 

13- 0334 

0655 

0977 

1298 

1619 

1939 

226 

258 

29 

3219 

321 

136 

13- 3539 

3858 

4177 

4496 4814 

5133 

545 i 

5769 

6086 

6403 

318 

137 

13- 6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 9249 

9564 

316 

138 

13- 9879 










3 i 5 

138 

14- — 

0194 

0508 

0822 

1136 

145 

1763 

2076 

2389 

2702 

3 i 4 

139 

14- 3015 

3327 

3639 

.3951 

4263 

4574 

4885 

5 i 9 6 

5507 

5818 

311 

140 

14- 6128 

6438 

6748 

7058 

73 6 7 

7676 

7985 

8294 

8603 

8911 

309 

141 

14- 9219 

9527 

9835 

— 

— 

— 

— 

— 

— 

— 

308 

141 

15- — 

— 

— 

0142 

0449 

0756 

1063 

137 

1676 

1982 

307 

142 

15- 2288 

2594 

29 

3205 

35 i 

3815 

412 

4424 

4728 

5032 

305 

143 

15- 5336 

564 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

144 

15- 8362 

8664 

8965 

9266 

9567 

9868 

— 

— 


— 

302 

144 

16- — 

— 

— 

— 

— 

— 

0168 

0469 

0769 

1068 

301 

145 

16- 1368 

1667 

1967 

2266 

2564 

2863 

3161 

346 

3758 

4055 

299 

146 

16- 4353 

465 

4947 

5244 

554 i 

5838 

6134 

643 

6726 

7022 

297 

147 

16- 7317 

7613 

7908 

8203 

8497 

8792 

9086 

938 

9674 

9968 

295 

14S 

17- 0262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

149 

17- 3186 

3478 

3769 

406 

435 i 

4641 

4932 

5222 

55 i 2 

5802 

291 

150 

17- 6091 

6381 

667 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

151 

17- 8977 

9264 

9552 

9839 

— 

— 

.— 

— 

— 

— 

287 

151 

18- — 

— 

— 

— 

0126 

0413 

0699 

0986 

1272 

1558 

287 

152 

18- 1844 

2129 

2415 

27 

2985 

327 

3555 

3839 

4123 

4407 

285 

153 

18- 4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

154 

18- 7521 

7803 

8084 

8366 

8647 

8928 

9209 

949 

9771 


281 

154 

19- — » 









0051 

281 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 
















LOGARITHMS OF NUMBERS. 3 I 3 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

155 

19- 0332 

0612 

0892 

1171 

I 45 i 

173 

20X 

2289 

2567 

2846 

279 

156 

x 9- 3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

157 

19- 59 

6176 

6453 6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

158 

19- 8657 

8932 

9206 9481 

9755 

— 

— 

— 

— 

— 

275 

158 

20- —• 

— 

— 

— 

— 

0029 

° 3°3 

0577 

085 

1124 

274 

159 

20- 1397 

167 

1943 

2216 

00 

00 

rt- 

O 

2761 

3033 

3305 

3577 

3848 

272 

160 

20- 412 

439 1 

4663 

4934 

5204 

5475 

5746 6016 

6286 

6556 

271 

161 

20- 6826 

7096 

7365 

7634 

7904 

8 i 73 

8441 

871 

8979 9247 

269 

162 

20- 9515 

9783 









268 

162 

21- —• 

— 

0051 

03 1 9 0586 

0853 

1121 

1388 

1654 

1921 

267 

163 

21- 2188 

2454 

272 

2986 

3252 

35 i 8 

3783 

4049 

43 H 

4579 

266 

164 

21- 4844 

5109 

5373 

5638 

5902 

6166 

643 

6694 6957 

7221 

264 

165 

21- 7484 

7747 

801 

8273 

853 6 

8798 

906 

9323 

9585 

9846 

262 

166 

22- 0108 

037 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

167 

22- 2716 

2976 

3236 

3496 3755 

4015 

4274 

4533 

4792 

5051 

259 

168 

22- 5309 

556 S 

5826 

6084 6342 

66 

6858 

7 JI 5 

7372 

763 

258 

169 

22- 7887 

8144 

84 

8657 

8913 

917 

9426 9682 9938 

— 

257 

169 

23- — 









0193 

256 

170 

23- °449 

0704 096 

1215 

147 

1724 

1979 

2234 

2488 

2742 

255 

171 

23- 2996 325 

3504 

3757 

4011 

4264 4517 

477 

5023 

5276 

253 

172 

23- 5528 

578 i 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

173 

23- 8046 

8297 

8548 

8799 9049 

9299 

955 

98 

— 

— 

251 

173 

24- 








005 

03 

250 

174 

24- 0549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

279 

249 

175 

24- 3°3S 

3286 3534 3782 

403 

4277 

4525 

4772 

5019 

5266 

248 

176 

24- 5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

24- 7973 

8219 

8464 

8709 

8954 

9x98 9443 9687 

9932 

— 

246 

177 

25- - 









0176 

245 

178 

25- 042 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

261 

243 

179 

25- 2853 

3 ° 9 6 

3338 

358 

3822 

4064 

4306 

4548 

479 

5031 

242 

180 

25- 5273 

55 i 4 

5755 

599 6 

6237 

6477 

6718 

6958 

7198 

7439 

241 

181 

25- 7 6 79 

79 i 8 

8x58 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

26- 0071 

051 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

183 

26- 2451 

2688 

2925 

3162 

3399 

36 36 

3873 

4109 

4346 

4582 

237 

184 

26- 4818 

5054 

529 

5525 

576 i 

5996 

6232 

6467 

6702 

6937 

235 

185 

26- 7172 

7406 

7641 

7875 

811 

8344 

8578 

8812 

9046 

9279 

234 

186 

26- 9513 

9746 

998 

— 

— 

— 

— 

— 

— 

— 

234 

186 

27- — 

— 

— 

0213 

0446 

0679 

0912 

1144 

1377 

1609 

233 

187 

27- 1842 

2074 

2306 

2538 

277 

3001 

3233 

3464 

3696 

3927 

232 

188 

27- 4158 

4389 

462 

485 

5081 

53 ii 

5542 

5772 

6002 

6232 

230 

189 

27^ 6462 

6692 

6921 

7151 

738 

7609 

7838 

8067 

8296 

8525 

229 

190 

27- 8754 

8982 

9211 

9439 

9667 

9895 

— 

— 

— 

— 

228 

190 

28- — 

— 

— 

— 

— 

— 

0123 

0351 

0578 

0806 

228 

191 

28- 1033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

28- 3301 

3527 

3753 

3979 

4205 

443 t 

4656 

4882 

5107 

5332 

226 

193 

28- 5557 

5782 

6007 

6232 

6456 

6681 

6905 

7 X 3 

7354 

7578 

225 

194 

28- 7802 

8026 

8249 

8473 

8696 

892 

9 I 43 

9366 

9589 

9812 

223 

195 

29 ~ °°35 

0257 

048 

0702 

0925 

1147 

1369 

i 59 i 

1813 

2034 

222 

196 

2 Q- 2 - 2^6 

2478 

2699 

292 

3 J 4 1 

3363 

3584 

3804 

4025 

4246 

221 

197 

29- 4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

29- 6665 

6884 

7 io 4 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

29- 8853 

9071 

9289 

9507 

9725 

9943 

— 

— 

— 

— 

2l8 

199 

30 - — 


— 

— 

— 

— 

0161 

0378 

0595 

0813 

2l8 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 


D D 















314 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

200 

30- 103 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

298 

217 

201 

30- 3196 

3412 

3628 3844 4059 

4275 

4491 

4706 4921 

5136 

216 

202 

30 - 53 Si 

5566 

578 i 

5996 

6211 

6425 6639 6854 

7068 

7282 

215 

203 

30- 7496 

771 

7924 

8137 

8351 

8564 8778 

8991 

9204 

9417 

213 

204 

3 °~ 9 6 3 

9843 









213 

204 

31- “ 

— 

0056 

0268 

0481 

0693 

0906 

IIl8 

133 

1542 

212 

205 

31- 1754 

1966 

2177 

2389 

26 

2812 

3023 

3234 

3445 

3656 

211 

206 

31- 3867 

4078 

4289 4499 

47 i 

492 

513 

534 

555 i 

576 

210 

207 

3 i- 597 

618 

639 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

208 

3 I_ 8063 

8272 

8481 

8689 

8898 

9x06 

9314 

9522 

973 

9938 

208 

209 

32- 0146 

0354 

0562 

0769 

0977 

1184 

I 39 1 

1598 

1805 

2012 

207 

210 

32- 2219 

2426 

2633 2839 3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

32- 4282 

4488 

4694 4899 

5 io 5 

53 i 

55 i 6 

572 i 

5926 

6131 

205 

212 

32- 6336 

6541 

6745 

695 

7 i 55 

7359 

7563 

7767 

7972 

8176 

204 

213 

32- 838 

8583 

S787 

8991 

9194 

9398 

9601 

9805 

— 

— 

204 

213 

33 " 








0008 

0211 

203 

214 

33 - 0414 

0617 

0 

00 

HI 

VO 

1022 

1225 

1427 

163 

1832 

2034 

2236 

202 

215 

33 " 2438 

264 

2842 

3044 

3246 

3447 

3649 385 

4051 

4253 

202 

216 

33 - 4454 4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

626 

201 

217 

33- 646 

666 

686 

706 

726 

7459 

7659 

7858 

8058 

8257 

200 

218 

33 - 8456 

S656 8855 

9054 

9 2 53 

945 i 

965 

9849 

— 

— 

200 

218 

34 - 








0047 

0246 

I99 

219 

34- 0444 0642 

0841 

1039 

1237 

1435 

1632 

183 

2028 

2225 

198 

220 

34- 2423 

262 

2817 

3 OI 4 

3212 

3409 

3606 3802 

3999 

4196 

197 

221 

34 " 4392 

4589 4785 

498 i 

5178 

5374 

557 

5766 

5962 

6157 

196 

222 

34- 6353 

6549 

6744 6939 

7135 

733 

7525 

772 

79 L 5 

8ll 

195 

223 

34- 8305 

85 

8694 

888> 

9083 

9278 

9472 

9666 

986 

— 

I94 

223 

35 





— 




OO54 

I94 

224 

35- 0248 

0442 

0636 0829 

1023 

1216 

141 

1603 

1796 

1989 

J 93 

225 

35- 2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

226 

35- 4108 

4301 

4493 

4685 

4876 

5068 

526 

5452 

5643 

5834 

192 

227 

35- 6026 

6217 

6408 

6599 6 79 

6981 

7172 

73 6 3 

7554 

7744 

191 

228 

35 - 7935 

8125 

S316 

8506 

8696 

8886 

9076 9266 9456 

9646 

190 

229 

35 - 9835 










189 

229 

36- — 

0025 

0215 

0404 

0593 

0783 

0972 

1161 

135 

1539 

189 

230 

36- 1728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

188 

231 

36- 3612 

3 S 

3988 

4176 

4363 

455 i 

4739 

4926 

5113 

53 oi 

188 

232 

36- 5488 

5675 

5862 

6049 

6236 

6423 

661 

6796 

6983 

7169 

187 

233 

3 6 ' 7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9°3 

186 

234 

36- 9216 

9401 

9587 

9772 

9958 

— 

— 

— 

— 


186 

235 

37 - — 

— 

— 

— 

— 

oi 43 

0328 

0513 

0698 

0883 

185 

235 

37- 1068 

1253 

I 437 

1622 

1806 

1991 

2175 

236 

2544 

2728 

184 

236 

37- 29x2 

3096 

328 

3464 

3 6 47 

3831 

4015 

4198 

4382 

4565 

184 

237 

37 - 4748 

4932 

5 ii 5 

5298 

548 i 

5664 

5846 

6029 

6212 

6394 

183 

238 

37 - 6 577 

6759 

6942 

7124 

7306 

7488 

767 

7852 

8034 

8216 

182 

239 

37- 8398 

858 

8761 

8943 

9 I2 4 

9306 

9487 

9668 

9849 

— 

182 

239 

38 - 









003 

181 

240 

38- 02x1 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

241 

38- 2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

38- 3815 

3995 

4174 

4353 

4533 

4712 

4891 

507 

5249 

5428 

179 

243 

38- 5606 

5785 

5964 6142 6321 

6499 6677 

6856 

7034 

72x2 

178 

244 

38- 739 

7568 

7746 

7923 

8101 

8279 8456 

8634 

8811 

8989 

178 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

















LOGARITHMS OF NUMBERS. 3 I 5 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

245 

38- 9166 

9343 

952 

9698 

9875 

— 

— 

— 

— 

— 

177 

245 

39 - — 

— 

— 

— 

— 

0051 

0228 

0405 

0582 

0759 

1 77 

246 

39 " 0935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

39- 2697 

2873 

3048 

3224 

34 

3575 

3751 

3926 

4x01 

4277 

176 

248 

39 - 4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

585 

6025 

175 

249 

39 " 6i 99 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

39 - 794 

8114 

8287 

8461 

8634 

8808 

8981 

9 T 54 

9328 

9501 

173 

251 

39 - 9674 

9847 









*73 

251 

40- — 

— 

002 

0192 

0365 

0538 

0711 

0883 

1056 

1228 

x 73 

252 

40- 1401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

40- 3121 

3292 

3464 3635 

3807 

3978 

4149 

432 

4492 

4663 

171 

254 

40- 4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 637 

171 

255 

40- 654 

671 

6881 

7051 

7221 

7391 

756 i 

773 i 

7901 

807 

170 

256 

40- 824 

841 

8579 

8749 

8918 

9087 

9257 

9426 9595 9764 

169 

257 

40- 9933 










169 

257 

41- — 

0102 

0271 

044 

0609 

0777 

0946 

1114 

1283 

1451 

169 

258 

41- 162 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

259 

4 i- 33 

3467 

3635 

3803 

397 

4 X 37 

4305 

4472 

4639 

4806 

167 

260 

4 i- 4973 

5 i 4 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

41- 6641 

6807 

6973 

7 i 39 

7306 

7472 

7638 

7804 

797 

8135 

166 

262 

41- 8301 

8467 

8633 

8798 

8964 

9 I2 9 

9295 

946 

9625 

979 1 

165 

263 

41- 9956 










165 

263 

42- — 

0121 

0286 

0451 

0616 

0781 

0945 

in 

1275 

1439 

165 

264 

42- 1604 

1768 

1933 

2097 

2261 

2426 

259 

2754 

2918 

3082 

164 

265 

42- 3246 

34 i 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

47 x 8 

164 

266 

42- 48S2 

5045 

5208 

537 i 

5534 

5697 

586 

6023 

6186 

6349 

163 

267 

42- 6511 

6674 

6836 

6999 

7161 

7324 

74S6 

7648 

7811 

7973 

162 

268 

42- 8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

959 i 

162 

269 

42- 9752 

9914 









162 

269 

43 - — 

— 

0075 

0236 

0398 

0559 

072 

0881 

1042 

1203 

161 

270 

43 - 1364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

43- 2969 

3 i 3 

329 

345 

361 

377 

393 

409 

4249 

4409 

160 

272 

43 - 4569 

4729 

4888 

5048 

5207 

53 6 7 

5526 

5685 

5844 

6004 

159 

273 

43- 61'63 

6322 

6481 

664 

6799 

6957 

7116 

7275 

7433 

7592 

159 

274 

43 - 7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

275 

43 - 9333 

9491 

9648 

9806 

9964 

— 

— 

— 

— 

— 

158 

275 

44 - — 

— 

— 

— 

— 

0122 

0279 

0437 

0594 

0752 

158 

276 

44- 0909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

157 

277 

44- 248 

2637 

2793 

295 

3 iq 6 

3263 

34 i 9 

35 76 

3732 

3889 

x 57 

278 

44 " 4045 

4201 

4357 

45 i 3 

4669 

4825 

4981 

5137 

5293 

5449 

156 

279 

44- 5604 

576 

59 x 5 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

280 

44 - 7158 

73 1 3 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

155 

281 

44- 8706 

8861 

90 1 5 

9 I 7 

9324 

9478 

9 6 33 

9787 

9941 

— 

154 

281 

45 









0095 

154 

282 

45- 0249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

16 33 

154 

283 

45- 1786 

194 

2093 

2247 

24 

2553 

2706 

2859 3° 12 

3165 

153 

284 

45 - 33i8 

347 i 

3624 

3777 

393 

4082 

4235 

4387 

454 

4692 

153 

285 

45 - 4845 

4997 

5 i 5 

5302 

5454 

5606 

5758 

59 1 

6062 

6214 

152 

286 

45- 6366 6518 

667 

6821 

6973 

7125 

7276 

7428 

7579 

773 i 

152 

287 

45- 7882 

8033 

8184 

8336 

8487 

8638 

8789 

894 

9091 

9242 

151 

288 

45 - 939 2 

9543 

9694 

9845 

9995 

— 

— 

— 

— 

— 

151 

288 

46- — 

— 

— 

— 

— 

0x46 0296 0447 

0597 

0748 

151 

289 

46- 0898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 




















3 1 *: 

No. 

0 

LOGARITHMS 0 ] 

1234 

F NUMBERS. 

567 

8 

9 

D 

290 

46- 2398 

2548 

2697 

2847 

2997 

3 i 46 3296 3445 

3594 

3744 

150 

291 

46- 3893 

4042 

4191 

434 

449 

4639 4788 4936 5085 

5234 

149 

292 

46- 5383 

5532 

568 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

293 

46- 6868 

7016 

7 i6 4 

7312 

746 

7608 

7756 

7904 

8052 

82 

148 

294 

46- 8347 

8495 

8643 

879 

8938 

9085 

9233 

938 

9527 

9 6 75 

148 

295 

46- 9822 

9969 









147 

295 

47 - — 

—' 

0116 

0263 

041 

0557 

0704 

0851 

0998 

1145 

147 

296 

47- 1292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

261 

146 

297 

47- 2756 

2903 

3049 

3 i 95 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

47- 4216 

4362 

4508 

4653 

4799 

4944 

509 

5235 

538 i 

5526 

146 

299 

47 - 5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

47- 7121 

7266 

7411 

7555 

77 

7844 

7989 

8 i 33 

8278 

8422 

145 

3 ° x 

47- 8566 

8711 

8855 

8999 

9 I 43 

9287 

9431 

9575 

9719 

9863 

144 

302 

48- 0007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

144 

303 

48- 1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

304 

48- 2874 

3016 

3 i 59 

3302 

3445 

3587 

373 

3872 

4015 

4157 

143 

305 

48- 43 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

306 

48- 5721 

5863 

6005 

6147 

6289 

643 

6572 

6714 

6855 

6997 

142 

307 

48- 7*38 

728 

7421 

7563 

7704 

7845 

7986 

S127 

8269 

841 

141 

308 

48- 8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

309 

48- 9958 










140 

309 

49 - — 

0099 

0239 

038 

052 

0661 

0801 

0941 

1081 

1222 

140 

310 

49- i3 62 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

3 11 

49- 276 

29 

304 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

3 12 

49 - 4 T 55 

4294 

4433 

4572 

4711 

485 

4989 

5128 

5267 

5406 

139 

3 i 3 

49 - 5544 

5683 

5822 

59 6 

6099 

6238 

6376 

6515 

6653 

6791 

139 

3 i 4 

49 - 693 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

315 

49- 8311 

8448 

8586 

CO 

8862 

CO 

0 

0 

vO 

9*37 

9275 

9412 

955 

138 

316 

49 - 9687 

9S24 

9962 

— 

— 

— 





137 

3 l6 

50- — 

— 

— 

0099 

0236 

0374 

0511 

0648 

0785 

0922 

137 

3 X 7 

50- 1059 

1196 

I 333 

147 

1607 

1744 

188 

2017 

2154 

2291 

137 

318 

50- 2427 

2564 

27 

2837 

2973 

3 io 9 

3246 

3382 

35 i 8 

3 6 55 

136 

3 X 9 

50 - 379 1 

3927 

4063 

4199 

4335 

447 1 

4607 

4743 

4878 

5014 

136 

320 

50 - 515 

5286 

542 i 

5557 

5693 

5828 

5964 

6099 

6234 

637 

136 

321 

50 - 6505 

664 

6776 

6911 

7046 

7181 

73 i 6 

745 i 

7586 

7721 

135 

322 

50- 7856 

7991 

8126 

826 

8395 

853 

8664 

8799 

8934 

9068 

x 35 

323 

50- 9203 

9337 

947 i 

9606 

974 

9874 

— 

— 

— 

— 

I 34 

323 

5 i- — 

— 

— 

— 

— 

— 

0009 

0143 

0277 

0411 

!34 

324 

5 i- 0545 

0679 

0813 

0947 

1081 

1215 

1349 

1482 

1616 

I 75 

134 

325 

51- 1883 

2017 

2151 

2284 

2418 

255 i 

2684 

2818 

2951 

3084 

133 

326 

5 i- 3218 

335 i 

3484 

3617 

375 

3883 

4016 

4149 

4282 

44 I 5 

133 

327 

5 i- 4548 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

574 i 

133 

328 

5 i- 5874 

6006 

6139 

6271 

6403 

6 535 

6668 

68 

6932 

7064 

132 

329 

51- 7196 

7328 

746 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

5 i- 8514 

8646 

8777 

8909 

904 

9171 

9303 

9434 

9566 

9697 

I 3 I 

33 i 

51- 9828 

9959 

— 

— 

— 

— 

— 

— 

— 


131 

33 i 

52- — 

—■ 

009 

0221 

0353 

0484 

0615 

0745 

0S76 

1007 

131 

332 

52- 1138 

1269 

14 

153 

1661 

1792 

1922 

2053 

2183 

2314 

131 

333 

52- 2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

334 

52- 3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

478 s 

4915 

130 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 














LOGARITHMS OF NUMBERS. 3 I / 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

335 

52 - 5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

621 

129 

336 

52- 6339 6469 6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

337 

52 - 763 

7759 

7888 

8016 

8 i 45 

8274 

8402 

8531 

866 

8788 

129 

338 

52- 8917 

9045 

9 I 74 

9302 

943 

9559 9 6 87 

9815 

9943 

— 

128 

338 

53 " 





'- 




0072 

128 

339 

53“ 02 

0328 

0456 0584 

0712 

084 

0968 

1096 

1223 

I 35 I 

128 

340 

53 - 1479 

1607 

1734 

1862 

199 

2117 

2245 

2372 

25 

2627 

128 

34 i 

53 - 2754 

2882 

3°°9 

3136 

3264 

339 i 

35 x 8 

3645 

3772 

3899 

127 

342 

53- 4026 4153 

428 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

53 - 5294 

542 i 

5547 

5674 

58 

5927 

6053 

618 

6306 

6432 

126 

344 

53 “ 6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

345 

53 - 7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

53- 9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

— 

— 

126 

346 

54 - 








0079 

0204 

125 

347 

54- 0329 

0455 

058 

0705 

083 

0955 

108 

1205 

133 

1454 

125 

348 

54 - 1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

270.1 

125 

349 

54“ 2825 

295 

3074 

3199 

3323 

3447 

357 i 

3696 

382 

3944 

124 

350 

54- 4068 

4192 

43 x 6 

444 

4564 

4688 

4812 

4936 

506 

5183 

124 

35 i 

54 - 5307 

543 i 

5555 

5678 

5802 

5925 

6049 6172 

6296 

6419 

124 

352 

54 - 6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

54 - 7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

354 

54 - 9°°3 

9126 9249 

937 i 

9494 

9616 9739 9861 

9984 

— 

123 

354 

55 " 









0106 

123 

355 

55- 0228 

0351 

0473 

0595 

0717 

084 

0962 

1084 

1206 

1328 

122 

356 

55 - 145 

1572 

1694 

1816 

1938 

206 

2181 

2303 

2425 

2547 

122 

357 

55- 2668 

279 

2911 

3033 

3155 

3276 3398 

3519 

364 

3762 

121 

358 

55 - 3883 

4004 

4126 

4247 

4368 

4489 

461 

473 i 

4852 

4973 

121 

359 

55 “ 5094 

52x5 

5336 

5457 

5578 

5699 

582 

594 

6061 

6182 

121 

360 

55" 6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

361 

55 - 7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

55- 8709 

8829 

8948 

9068 9188 

9308 

9428 

9548 9667 

9787 

120 

363 

55 - 9907 










120 

363 

56- — 

0026 

0146 

0265 

0385 

0504 

0624 

0743 

0863 

0982 

119 

364 

56- IIOI 

1221 

134 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

56- 2293 

2412 

253 1 

265 

2769 

2887 

3006 

3125 

3244 

33 6 2 j 

119 

366 

56- 3481 

3 6 

37 x 8 

3837 

3955 

4074 

4192 

43 X 1 

4429 

4548 

119 

367 

56- 4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

573 

118 

368 

56- 5848 

5966 

6084 

6202 

632 

6437 

6555 

6673 

6791 

6909 

11S 

369 

56- 7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

56- 8202 

83 I 9 

8436 

8554 

8671 

8788 

8905 

9023 

9 r 4 

9257 

117 

37 i 

56 - 9374 

949 1 

9608 

9725 

9842 

9959 

— 

— 

— 

— 

117 

37 1 

57 - — 

— 

— 

— 

— 

— 

0076 

0193 

0309 

0426 

117 

372 

57 - 0543 

066 

0776 

0893 

IOI 

1126 

1243 

1359 

1476 

1592 

117 

373 

57 - 1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

57- 2872 

2988 

3104 

322 

3336 

3452 

3568 

3684 

3 § 

39 I 5 

116 

375 

57 " 4031 

4147 

4263 

4379 

4494 

461 

4726 

4841 

4957 

5072 

116 

376 

57- 5188 

5303 

5419 

5534 

565 

5765 

588 

5996 

61XI 

6226 

115 

377 

57 " 6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

378 

57 " 7492 

7607 

7722 

7836 

795 i 

8066 

8181 

8295 

841 

8525 

xi 5 

379 

57" 8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 


D D* 













3 I 8 LOGARITHMS OF NUMBERS. 


No. 

380 

380 

381 

382 

383 

384 

385 

386 

387 

388 

389 

389 

390 

391 

39 2 

393 

394 

395 

39 6 

397 

398 

398 

399 

100 

401 

402 

403 

404 

405 

40 6 

407 

407 

408 

409 

410 

411 

412 

413 

414 

415 

416 

416 

417 

418 

419 

420 

421 

422 

423 

424 

No. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

57- 9784 

9898 









114 

58- - 

— 

0012 

0126 

0241 

0355 

0469 

0583 

0697 

0811 

114 

58 - 0925 

1039 

ii53 

1267 

1381 

1495 

1608 

1722 

1836 

195 

114 

58 - 2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3 0 85 

1 14 

58- 3199 

3312 

3426 3539 3652 

3765 

3879 

3992 

4105 

4218 

**3 

58- 4331 

4444 

4557 

467 

4783 

4896 

5009 

5122 

5235 

5348 

1 13 

58 - 5461 

5574 

5686 

5799 

5912 

6024 

6137 

625 

6362 

6475 

i*3 

58 - 6587 

67 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

58- 77ii 

7823 

7935 

8047 

816 

8272 

8384 

8496 

8608 

872 

112 

58 - 8832 

8944 9056 9167 

9279 

9391 

9503 

9615 

9726 9838 

. 1 X 2 

58- 995 










112 

59- — 

006 X 

oi73 

0284 0396 

0507 

0619 

073 

0842 

0953 

112 

59 - 1065 

1176 

1287 

1399 

151 

1621 

1732 

1843 

1955 

2066 

III 

59" 2177 

2288 

2399 

251 

2621 

2732 

2843 

2954 

3064 

3175 

III 

59 - 3286 

3397 

3508 

3618 

3729 

384 

395 

4061 

4171 

4282 

III 

59- 4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

no 

59- 549 6 

5606 

57i7 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

IIO 

59- 6 597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

IIO 

59- 7695 

7805 

79 1 4 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

IIO 

59- 8791 

89 

9009 

9XX9 

9228 

9337 

9446 9556 9 66 5 

9774 

IO 9 

59- 9883 

9992 









IO 9 

60 - — 

— 

0101 

021 

0319 

0428 

0537 

0646 

0755 

0864 

IO 9 

60 - 0973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

i95i 

IO 9 

60 - 206 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

IOS 

60 - 3144 

3253 

33 61 

3469 

3577 

3686 

3794 

3902 

40 X 

4118 

IOS 

60 - 4226 

4334 

4442 

455 

4658 

4766 

4874 

4982 

5089 

5197 

IOS 

60 - 5305 

5413 

5521 

5628 

5736 

5844 

595i 

6059 

6166 

6274 

IOS 

60 - 6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

IO 7 

60 - 745 s 

7562 

7669 

7777 

7884 

7991 

809 S 

8205 

8312 

8419 

IO 7 

60 - 8526 

86 33 

874 

8847 

8954 

9061 

9167 

9274 

938i 

9488 

IO 7 

60 - 9594 

9701 

9808 

9914 

— 

— 

— 

— 

— 

— 

IO 7 

61 - — 

— 

— 

— 

0021 

0128 

0234 

0341 

0447 

0554 

IO 7 

61 - 066 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

61 - 1723 

1829 

1936 

2042 

2148 

2254 

236 

2466 

2572 

2678 

106 

61 - 2784 

289 

2996 

3 102 

3207 

3313 

3419 

3525 

363 

3736 

106 

61 - 3842 

3947 

4053 

4i59 

4264 

437 

4475 

45Si 

4686 

4792 

106 

61 - 4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

574 

5845 

*05 

61 - 595 

6055 

616 

6265 

637 

6476 

6581 

6686 

679 

6 S 95 

105 

61 - 7 

7 io 5 

72 X 

73i5 

742 

7525 

7629 

7734 

7839 

7943 

105 

61 - 8048 

8i53 

8257 

8362 

8466 

8571 

8676 

878 

8884 

8q8q 

*05 

61 - 9093 

9 i 98 

9302 

9406 

95ii 

9615 

9719 

9824 

9928 

— 

105 

62 - — 









0032 

IO 4 

62 - 0136 

024 

0344 

0448 

0552 

0656 076 

0864 0968 

1072 

IO 4 

62 - 1176 

128 

1384 

1488 

1592 

1695 

I 799 

1903 

2007 

211 

IO 4 

62 - 2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3H6 

IO 4 

62 - 3249 

3353 

3456 3559 3663 

3766 3 S 69 

3973 

4076 

4179 

IO 3 

62 - 4282 

4385 

4488 

459* 

4695 

4798 

4901 

5004 

5107 

521 

*03 

62 - 5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

io 3 

62 - 634 

6443 

6546 6648 6751 

6853 6956 

7058 

7161 

7263 

*03 

62 - 7366 

7468 

7571 

7673 

7775 

7878 

798 

8082 

8185 

8287 

102 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 


















LOGARITHMS OF NUMBERS. 3 1 9 


No. 


0 

1 

2 

3 

4 

5 

6 

-T 

r 

8 

9 

D 

425 

62- 

8389 

8491 

8593 

8695 

8797 

89 

9002 

9 io 4 

9206 

9308 

102 

426 

62- 

941 

9512 

9613 

9715 

9817 

9919 

— 

— 

— 

— 

102 

426 

63- 

— 

— 

— 

— 

— 

— 

0021 

0123 

0224 

0326 

102 

427 

63- 

0428 

053 

0631 

0733 

0835 

0936 

1038 

ii 39 

1241 

1342 

102 

428 

63- 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

IOI 

429 

63- 

2 457 

2 559 

266 

2761 

2862 

2963 

3 ° 6 4 

3165 

3266 

3367 

IOI 

430 

63- 

3468 

3569 

367 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

IOI 

43 1 

63- 

4477 

4578 

4679 

4779 

488 

4981 

5081 

5182 

5283 

5383 

IOI 

43 2 

63- 

5484 

5584 

5685 

5785 

58S6 

5986 

6087 

6187 

6287 

6388 

IOO 

433 

63- 

6488 

6588 

6688 

6789 

6S89 

6989 

7089 

7189 

729 

739 

IOO 

434 

63- 

749 

759 

769 

779 

789 

799 

809 

819 

829 

8389 

IOO 

435 

63- 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

00 

N 

ON 

9387 

IOO 

436 

63- 

9486 

9586 

9686 

9785 

9885 

9984 

— 

— 

— 

— 

IOO 

436 

64- 

— 

— 

— 

— 

— 

— 

0084 

0183 

0283 

0382 

99 

437 

64- 

0481 

0581 

068 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

438 

64- 

1474 

1573 

1672 

1771 

1871 

197 

2069 

2x68 

2267 

2366 

99 

439 

64- 

2465 

2563 

2662 

2761 

286 

2959 

3 0 58 

3 X 56 

3255 

3354 

99 

440 

64- 

3453 

355 i 

3 6 5 

3749 

3847 

3946 

4044 

4143 

4242 

434 

99 

441 

64- 

4439 

4537 

4636 

4734 

4832 

493 i 

5029 

5127 

5226 

5324 

98 

44 2 

64- 

54 22 

552 i 

5619 

5717 

5815 

59 i 3 

6011 

611 

6208 

6306 

98 

443 

64- 

6404 

6502 

66 

6698 

6796 

6894 

6992 

70S9 

7187 

7285 

98 

444 

64- 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

445 

64- 

836 

8458 

8555 

8653 

875 

8848 

8945 

9043 

914 

9237 

97 

446 

64- 

9335 

9432 

953 

9627 

9724 

9821 

99 x 9 

— 

— 

— 

97 

446 

65- 

— 

— 

— 

— 

— 

— 

— 

0016 

0113 

021 

97 

447 

65 - 

030S 

0405 

0502 

0599 

0696 

0793 

089 

0987 

1084 

1181 

97 

448 

65- 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

215 

97 

449 

65- 

2246 

2343 

244 

2536 

2633 

273 

2826 

2923 

3019 

3h6 

97 

450 

65- 

3 2 i 3 

3309 

3405 

3502 

3598 

3695 

379 i 

3888 

3984 

408 

96 

45 i 

65- 

4 i 77 

4273 

4369 

4465 

4562 

4658 

4754 

485 

4946 

5042 

96 

45 2 

65 - 

5138 

5235 

533 i 

5427 

5523 

5619 

57 x 5 

581 

5906 

6002 

96 

453 

65- 

6098 

6194 

629 

6386 

6482 

6577 

6673 

6769 

6864 

696 

96 

454 

65- 

7056 

7 X 5 2 

7247 

7343 

7438 

7534 

7629 

7725 

782 

7916 

96 

455 

65 - 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

887 

95 

456 

65- 

8965 

906 

9 I 55 

925 

9346 

9441 

9536 

9631 

9726 

9821 

95 

457 

65 - 

9916 




0296 






95 

457 

66- 

— 

0011 

0106 

0201 

0391 

0486 

0581 

0676 

0771 

95 

458 

66- 

0865 

096 

1055 

115 

1245 

1339 

1434 

1529 

1623 

1718 

95 

459 

66- 

1813 

1907 

2002 

2096 

2191 

2286 

238 

2475 

2569 

2663 

95 

460 

66- 

2758 

2852 

2947 

3041 

3 i 35 

323 

33 24 

34 x 8 

3512 

3607 

94 

461 

66- 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

436 

4454 

4548 

94 

462 

66- 

4642 

4736 

483 

4924 

5 OI 8 

5112 

5206 

5299 

5393 

5487 

94 

463 

66- 

558 i 

5675 

5769 

5862 

595 6 

605 

6143 

6237 

6331 

6424 

94 

464 

66- 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7 X 73 

7266 

736 

94 

465 

66- 

7453 

7546 

764 

7733 

7826 

792 

8013 

8106 

8199 

8293 

93 

466 

66- 

83S6 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

467 

66- 

9317 

941 

9503 

9596 

9689 

9782 

9875 

9967 

— 

— 

93 

467 

67- 









006 

0153 

93 

468 

67- 

0246 

0339 

0431 

0524 

0617 

071 

0802 

0895 

0988 

108 

93 

469 

67- 

ii 73 

1265 

1358 

i 45 i 

1543 

1636 

1728 

1821 

I 9 I 3 

2005 

93 

No. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 



















LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 D 

470 

67- 2098 

219 

2283 

2375 

2467 

256 

2652 

2744 

2836 

2929 

1 

92 

47 i 

67- 3021 

3 ii 3 

3205 

3297 

339 

3482 

3574 

3666 3758 

385 

92 

47 2 

67- 3942 

4034 

4126 

4218 

43 i 

4402 

4494 

4586 

4677 

4769 

92 

473 

67- 4861 

4953 

5045 

5137 

5228 

532 

5412 

5503 

5595 

5687 

92 

474 

67- 5778 

587 

5962 

6053 

6145 

6236 

6328 

6419 6511 

6602 

92 

475 

67- 6694 

6785 

6876 

6968 

7059 

7 i 5 i 

7242 

7333 

7424 

7516 

9 1 

476 

67- 7607 

7698 

7789 

7881 

7972 

8063 

8 i 54 

8245 

8336 

8427 

9 i 

477 

67- 8518 

8609 

87 

8791 

8882 

8973 

9064 

9 I 55 

9246 

9337 

9 i 

478 

67- 9428 

9519 

961 

97 

9791 

9882 

9973 

— 

— 

— 

9 i 

478 

68- — 

— 

— 

— 

— 

— 

— 

0063 0154 

0245 

9 1 

479 

68- 0336 

0426 

0517 

0607 

0698 

0789 

0879 

097 

106 

1151 

9 1 

480 

68- 1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

90 

481 

68- 2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

482 

68- 3047 

3 i 37 

3227 

33 i 7 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

68- 3947 

4037 

4127 

4217 

4307 

439 6 

4486 

4576 4666 

4756 

90 

484 

68- 4845 

4935 

5025 

5 ii 4 

5204 

5294 

5383 

5473 

5563 

5652 

90 

485 

68- 5742 

5831 

592 i 

601 

61 

6189 

6279 

6368 

6458 

6547 

89 

486 

68- 6636 

6726 

6S15 

6904 

6994 

7083 

7172 

7261 

735 i 

744 

89 

487 

68- 7529 

7618 

7707 

7796 

7886 

7975 

8064 

8 i 53 

8242 

8331 

§9 

488 

68- 842 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9 I 3 I 

922 

89 

489 

68- 9309 

9398 

9486 

9575 

9664 

9753 

9841 

993 

— 

— 

89 

489 

69- 








0019 

0107 

89 

490 

69- 0196 

0285 

0373 

0462 

055 

0639 

0728 

0816 

0905 

0993 

89 

49 1 

69- 1081 

117 

1258 

1347 

1435 

1524 

1612 

17 

1789 

1877 

88 

492 

69- 1965 

2053 

2142 

223 

2318 

2406 

2494 

2583 

2671 

2759 

88 

493 

69- 2847 

2935 

3023 

3111 

3 i 99 

3287 

3375 

3463 

355 i 

3639 

88 

494 

69- 3727 

3815 

39°3 

399 i 

4078 

4166 

4254 

4342 

443 

4517 

88 

495 

69- 4605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

496 

69- 5482 

5569 

5657 

5744 

5832 

59 i 9 

6007 

6094 

6182 

6269 

87 

497 

69- 6356 

6444 

6531 

6618 

6706 

6793 

688 

6968 

7055 

7142 

87 

498 

69- 7229 

7317 

7404 

749 1 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

69- 8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

69- 897 

9057 

9 T 44 

9231 

9317 

9404 

949 1 

9578 9664 

975 i 

87 

501 

69- 9838 

9924 









87 

501 

70- — 

— 

0011 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

87 

502 

70- 0704 

079 

0877 

0963 

105 

1136 

1222 

1309 

1395 

1482 

86 

503 

70- 1568 

1654 

1741 

1827 

1913 

I 999 

2086 

2172 

2258 

2344 

86 

504 

70- 2431 

2517 

2603 

2689 

2775 

2861 

2947 

30 33 

3119 

3205 

86 

505 

70- 3291 

3377 

34 63 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

506 

70- 4151 

4236 

4322 

4408 

4494 

4579 

4665 

475 i 

4837 

4922 

86 

507 

70- 5008 

5094 

5179 

5265 

535 

5436 

5522 

5607 

5693 

5778 

86 

508 

70- 5864 

5949 

6035 

612 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

70- 6718 

6803 

6888 

6974 

7059 

7144 

7229 

73 i 5 

74 

7485 

85 

510 

7 °- 757 

7655 

774 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

85 

5 H 

70- 8421 

8506 

859 1 

8676 

8761 

8846 

8931 

9 OI 5 

9 1 

9185 

85 

512 

70- 927 

9355 

944 

9524 

9609 

9694 

9779 

9863 

9948 


85 

512 

71- — 

— 

— 

— 

— 

— 

— 

— 

— 

0033 

85 

5 i 3 

71- 0117 

0202 

0287 

0371 

0456 

054 

0625 

071 

0794 

0879 

85 

514 

71- 0963 

1048 

1132 

1217 

1301 

1385 

147 

1554 

1639 

1723 

84 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 
















LOGARITHMS OF ^NUMBERS. 


321 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

515 

71- 1807 

1892 

1976 

206 

2144 

2229 

2313 

2397 

2481 

2566 

84 

5 i 6 

71- 265 

2734 

2818 

2902 

2986 

307 

3*54 3238 

3323 

3407 

84 

5 i 7 

7 i- 3491 

3575 

3659 

3742 

3826 

391 

3994 

4078 

4162 

4246 

84 

5 i 8 

7 i- 433 

4414 

4497 

458 r 

4665 

4749 

4833 

49 l6 

5 

5084 

84 

5 i 9 

71- 5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

592 

84 

520 

71- 6003 

6087 617 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

83 

521 

71- 6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

83 

522 

71- 7671 

7754 

7837 

792 

8003 

8086 

8169 

8253 

8336 

8419 

83 

523 

71- 8502 

8585 

8668 

8751 

8834 

8917 

9 

9083 

9 i6 5 

9248 

83 

524 

7 i- 933 1 

9414 

9497 

958 

9663 

9745 

9828 

9911 

9994 

— 

83 

524 

72 - 









0077 

83 

525 

72- 0159 

0242 

0325 

° 4°7 

049 

0573 

0655 

0738 

0821 

0903 

83 

526 

72- 0986 

1068 

1151 

1233 

i 3 i 6 

1398 

1481 

1563 

1646 

1728 

82 

527 

72- 1811 

1893 

1975 

2058 

214 

2222 

2305 

2387 

2469 

2552 

82 

528 

72- 2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

S29 

72- 3456 

3538 

362 

3702 

3784 

3866 

3948 

403 

4112 

4194 

82 

530 

72- 4276 4358 

444 

4522 

4604 

4685 

4767 

4849 

493 i 

5013 

82 

53 1 

72- 5095 

5176 

5258 

534 

5422 

5503 

5585 

5667 

5748 

583 

82 

532 

72- 59 12 

5993 

6075 

6156 

6238 

632 

6401 

6483 

6564 

6646 

82 

533 

72- 6727 

6809 

689 

6972 

7053 

7 r 34 

7216 

7297 

7379 

746 

81 

534 

72- 7541 

7623 

7704 

7785 

7866 

7948 

8029 

811 

8191 

8273 

81 

535 

72- 8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

72- 9165 

9246 

9327 

9408 

9489 

957 

9 6 5 i 

9732 

9813 

9893 

81 

537 

72- 9974 










81 

537 

73 " — 

0055 

0136 

0217 

0298 

0378 

0459 

054 

0621 

0702 

81 

538 

73- 0782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

539 

73 " 1589 

1669 

175 

183 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

73 " 2394 

2474 

2555 

2635 

2 7 i 5 

2796 

2876 

2956 

3037 

3 ii 7 

80 

54 1 

73 - 3 i 97 

3278 

3358 

3438 

35 i 8 

3598 

3679 

3759 

3839 

39 I 9 

80 

542 

73 - 3999 

4079 

416 

424 

432 

44 

448 

456 

464 

472 

80 

543 

73 - 48 

488 

496 

504 

512 

52 

5279 

5359 

5439 

5519 

80 

544 

73 - 5599 5679 

5759 

5838 

59 i 8 

5998 

6078 

6157 

6237 

6317 

80 

545 

73 - 6 397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7 ii 3 

80 

546 

73 - 7*93 

7272 

7352 

743 i 

75 ii 

759 

767 

7749 

7829 

7908 

79 

547 

73 - 7987 

8067 

8146 

8225 

83 0 5 

8384 

8463 

8543 

8622 

8701 

79 

548 

73- 8781 

886 

8939 

9018 

9°97 

9177 

9256 

9335 

9414 

9493 

79 

549 

73 - 9572 

9651 

973 i 

981 

9889 

1 99 68 

— 

— 

— 

— 

79 

549 

74 " — 

— 

— 


— 

1 - 

O 

O 

0126 

0205 

X) 

(N 

0 

79 

550 

74- 0363 

0442 

0521 

06 

0678 

0757 

0836 

0915 

0994 

1073 

79 

55 i 

74 - 1152 

123 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

186 

79 

552 

74 - 1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

553 

74" 2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

554 

74 " 35 i 

3588 

3667 

3745 

3823 

39 ° 2 

398 

4058 

4136 

4215 

78 

555 

74 " 4293 

437 i 

4449 

4528 

4606 

4684 

4762 

484 

4919 

4997 

78 

55 6 

74 - 5075 

5153 

523 1 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

74 - 5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

74- 6634 

6712 

679 

6868 

6945 

7023 

7101 

7 J 79 

7256 

7334 

78 

559 

74 - 74 12 

7489 

7567 

7645 

7722 

78 

7878 

7955 

8033 

811 

78 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 




















322 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

560 

74- 8188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

77 

561 

74- 8963 

904 

9x18 

9*95 

9272 

935 

9427 

9504 

9582 

9 6 59 

77 

562 

74 - 9736 

9814 

9891 

9968 

— 

— 

— 

— 

— 

— 

77 

562 

75 - — 

— 

— 

— 

0045 

0123 

02 

0277 

0354 

0431 

77 

563 

75- 0508 

0586 

0663 

074 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

75- 1279 

1356 

1433 

151 

1587 

1664 

1741 

1818 

1895 

T 97 2 

77 

565 

75- 2048 

2125 

2202 

2279 

2356 

24 33 

2509 

2586 

2663 

274 

77 

566 

75- 2816 

2893 

297 

3047 

3123 

32 

3277 

3353 

343 

35 o 6 

77 

567 

75 " 3583 

366 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

568 

75 - 4348 

4425 

4501 

4578 

4654 

473 

4807 

4883 

496 

5036 

76 

569 

75- 5112 

5189 

5265 

534 i 

5417 

5494 

557 

5646 

5722 

5799 

76 

570 

75 - 5875 

5951 

6027 

6103 

618 

6256 

6332 

6408 

6484 656 

76 

57 i 

75- 6636 

6712 

6788 

6864 

694 

7016 

7092 

7168 

7244 

732 

76 

572 

75 " 7396 

7472 

7548 

7624 

77 

7775 

7851 

7927 

8003 

8079 

76 

573 

75 - 8155 

823 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

574 

75- 8912 

8988 

9063 

9 i 39 

9214 

929 

9366 

9441 

9517 

9592 

76 

575 

75- 9668 

9743 

9819 

9894 

997 

— 

— 

— 

— 

— 

76 

575 

76- — 

— 

— 

— 

— 

0045 

0121 

0196 

0272 

0347 

75 

576 

76- 0422 

0498 

0573 

0649 

0724 

0799 

0875 

095 

1025 

IIOI 

75 

577 

76- 1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

75 

578 

76- 1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

579 

76- 2679 

2754 

2829 

2904 

2978 

3053 

3 I2 8 

3203 

3278 

3353 

75 

580 

76- 3428 

3503 

3578 

3653 

3727 

3802 

3877 

3952 

4027 

4101 

75 

581 

76- 4176 

4251 

4326 

44 

4475 

455 

4624 

4699 

4774 

4848 

75 

582 

76- 4923 

4998 

5072 

5147 

5221 

5296 

537 

5445 

552 

5594 

75 

583 

76- 5669 

5743 

5818 

5892 

5966 

6041 

6x15 

619 

6264 6438 

74 

584 

76- 6413 

6487 

6562 

6636 

671 

6785 

6859 

6 933 

7007 

7082 

74 

585 

76- 7156 

723 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

586 

76- 7898 

7972 

8046 

812 

8194 

S268 

8342 

8416 

849 

8564 

74 

587 

76- 8638 

8712 

8786 

886 

8934 

9008 

9082 

9156 

923 

9303 

74 

588 

76- 9377 

945 i 

9525 

9599 

9673 

9746 

982 

9894 

9968 

— 

74 

588 

77 









0042 

74 

589 

77 - 0115 

0189 

0263 

0336 

041 

0484 

0557 

0631 

0705 

0778 

74 

590 

77" 0852 

0926 

0999 

1073 

1146 

122 

1293 

1367 

144 

1514 

74 

59 i 

77 " 1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

592 

77- 2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

593 

77 " 3°55 

3128 

3201 

3274 

3348 

342 i 

3494 

3567 

364 

3713 

73 

594 

77 - 3786 

386 

3933 

4006 

4079 

4152 

4225 

4298 

437 i 

4444 

73 

595 

11 - 45 1 7 

459 

4663 

473 6 

4809 

4882 

4955 

5028 

5 i 

5 i 73 

73 

59 6 

11 - 5246 

53 i 9 

5392 

5465 

5538 

561 

5683 

5756 

5829 

5902 

73 

597 

77 - 5974 

6047 

612 

6193 

6265 

6338 

6411 

6483 

6556 6629 

73 

598 

77- 6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

599 

77 - 7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

77 - 8151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

873 

8802 

72 

601 

77- 8874 

8947 

9019 

9091 

9163 

9236 

9308 

938 

9452 

9524 

72 

602 

77 - 9596 

9669 

9741 

9813 

9885 

9957 

— 

— 



72 

602 

78- — 

— 

— 

— 

— 

— 

0029 

0101 

01 73 

0245 

72 

603 

78- 0317 

0389 

0461 

0533 

0605 

0677 

°749 

0821 

0 893 

0965 

72 

604 

78- 1037 

1109 

1181 

1253 

!324 

1396 

1468 

i 54 

1612 

1684 

72 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 


















LOGARITHMS OF NUMBERS. 


No. | 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

605 

78 - 1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

606 

78- 2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 3 0 46 

3^7 

72 

607 

78- 3189 

326 

3332 

3403 

3475 

3546 3618 3689 3761 

3832 

7 i 

608 

78- 3904 

3975 

4046 4118 

4189 

4261 

4332 

4403 

4475 

4546 

7 i 

609 

78- 4617 

4689 476 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

7 i 

610 

78- 533 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

597 

71 

611 

78- 6041 

6112 

6x83 

6254 

6325 

6396 6467 6548 66oq 

668 

7 i 

612 

78- 6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

73 I 9 

739 

7 i 

613 

78- 746 

753 i 

7602 

7673 

7744 

7815 

7885 

7956 8027 

8098 

7 i 

614 

78- 8168 

8239 

831 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

7 i 

615 

78- 8875 

8946 9016 9087 9157 

9228 

9299 

9369 

944 

95 i 

71 

616 

78- 9581 

9 6 5 i 

9722 

9792 

9863 

9933 

— 

— 

— 

— 

70 

616 

79 " — 

— 

— 

— 

-— 

— 

0004 

0074 

0144 

0215 

70 

617 

79- 0285 

0336 0426 0496 0367 

0637 

0707 

0778 

0848 

0918 

70 

618 

79- 0988 

1059 

1129 

IT 99 

1269 

134 

141 

148 

155 

162 

70 

6x9 

79- 1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

79 - 2 392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

79 - 3092 

3 j 62 

3231 

3301 

337 i 

344 i 

35 H 

358 i 

3 6 5 i 

3721 

70 

622 

79 - 379 

386 

393 

4 

407 

4139 

420Q 

4279 

4349 

44x8 

70 

623 

79 - 4488 

4558 

4627 4697 

4767 

4836 4906 4976 5045 

5 H 5 

70 

624 

79- 5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

574 i 

5811 

70 

625 

79- 588 

5949 

6019 6088 

6158 

6227 6207 6466 6446 

6505 

69 

626 

79 - 6574 

6644 6713 

6782 

6852 

6921 

699 

706 

7129 

7198 

69 

627 

79- 7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

789 

69 

628 

79 - 796 

8029 

8098 

8167 

8236 

8305 

8374 8443 

8513 

8582 

69 

629 

79- 8651 

872 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

79 " 934 i 

9409 9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

69 

63 1 

80- 0029 0098 

0167 

0236 0305 

0373 

0442 

0511 

058 

0648 

69 

632 

80- 07x7 

0786 0854 0923 

0992 

1061 

1129 

1198 

1266 

1335 

69 

633 

80- 1404 

1472 

1541 

1609 

1678 

1747 

18x5 

1884 

1952 

2021 

69 

634 

80- 2089 

2158 

2226 

2295 

2363 

2432 

25 

2568 

2637 

2705 

69 

635 

80- 2774 

2842 

291 

2979 

3047 

3116 3184 

3252 

332 i 

33 89 

68 

636 

80- 3457 

3525 

3594 3662 

373 

3798 

3867 

3935 

4003 

4071 

68 

637 

80- 4139 4208 

4276 

4344 

4412 

448 

4548 

4616 4685 

4753 

68 

638 

80- 4821 

4889 

4957 

5025 

5093 

5161. 5229 

5297 

5305 

5433 

68 

639 

80- 5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 6044 

6112 

68 

640 

So- 618 

6248 

6316 6384 6451 

6519 6587 6655 

6723 

679 

68 

641 

80- 6858 

6026 6qq4 

7061 

7129 

7197 

7264 

7332 

74 

7467 

68 

642 

80- 7535 

7603 

767 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

643 

80- 8211 

82 70 8446 8414 

8481 

8549 

8616 

8684 8751 

8S18 

67 

644 

80- 8886 

8953 

9021 

9088 

9^6 

9223 

929 

9358 

9425 

9492 

67 

645 

80- 956 

<3627 q6q4 9762 9829 

g8g6 9964 

— 

— 

— 

67 

645 

81- — 

— 

— 

— 

— 

— 

— 

0031 

0098 

0165 

67 

646 

81- 0233 

03 

0367 

0434 

0501 

0569 0636 

0703 

077 

0837 

67 

647 

81- 0904 

0971 

1039 

1106 

ii 73 

124 

1307 

1374 

1441 

1508 

67 

648 

81- 1575 

1642 

1709 

1776 

1843 

191 

1977 

2044 

2111 

2178 

67 

649 

81- 2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

278 

2847 

67 

650 

81- 2913 

298 

3047 

3114 

3181 

3247 

3314 

338 i 

3448 

35 1 4 

67 

651 

81- 3581 

3648 

37 H 

378 i 

3848 

39 i 4 

398 i 

4048 

4114 

4181 

67 

652 

81- 4248 

4314 

438 i 

4447 

45 i 4 

458 i 

4647 

4714 

47S 

4847 

67 

653 

81- 4913 

498 

5046 

5 ii 3 

5 i 79 

5246 

53 i 2 

5378 

5445 

55 ii 

66 

654 

81- 5578 

5644 

57 ii 

5777 

5843 

59 i 

5976 6042 

6109 

6 i 75 

66 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

-8 

9 

D 














324 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

655 

81- 6241 

6308 

6374 

644 

6506 

6573 

6639 

6705 

6771 

6838 

66 

656 

81- 6904 

697 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

81- 7565 

7631 

76q8 

7764 

783 

7896 

7962 

8028 

8094 

816 

66 

658 

81- 8226 

8292 

8358 

8424 

849 

8556 

8622 

8688 

8754 

882 

66 

659 

81- 8885 

8951 

9 OI 7 

9 o8 3 

9 J 49 

9215 

9281 

9346 

9412 

9478 

66 

GGO 

81- 9544 961 

9676 

9741 

9807 

9873 

9939 

— 

— 

— 

66 

660 

82- — 

— 

— 

— 

— 

— 

— 

0004 

007 

0136 

66 

661 

82- 0201 

0267 

0333 

0399 

0464 

053 

0595 

0661 

0727 

0792 

66 

662 

82- 0858 

0924 

0989 

1055 

112 

1186 

1251 

I 3 I 7 

1382 

1448 

66 

663 

82- 1514 

1579 

1645 

171 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

82- 2168 

2233 

2299 

2364 

243 

2495 

256 

2626 

2691 

2756 

65 

665 

82- 2822 

2887 

2952 

3018 

30S3 

3 H 8 

3213 

3279 

3344 

3409 

65 

666 

82- 3474 

3539 

3 6 o 5 

367 

3735 

38 

3865 

393 

399 6 

4061 

65 

667 

82- 4126 

4191 

4256 

4321 

4386 

4451 

45 i 6 

4581 

4646 

4711 

65 

668 

82- 4776 4841 

4906 

497 i 

5036 

5101 

5166 

5231 

5296 

53 61 

65 

669 

82- 5426 

549 1 

5556 

5621 

5686 

5751 

5815 

588 

5945 

60 X 

65 

670 

82- 6075 

614 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65 

671 

82- 6723 

6787 

6852 

6917 

6981 

7046 

7111 

7 T 75 

724 

73°5 

65 

672 

82- 7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

795 i 

6 5 

67 3 

82- 8015 

808 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

82- 866 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

675 

82- 9304 

9368 

9432 

9497 

956 x 

9625 

969 

9754 

9818 

9882 

64 

676 

82- 9947 










64 

676 

83- - 

oori 

0075 

0139 

0204 

0268 

0332 

0396 

046 

0525 

64 

677 

83- 0589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

678 

83- 123 

1294 

1358 

1422 

i486 

155 

1614 

1678 

1742 

1806 

64 

679 

83- 187 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

83- 2509 

2573 

2637 

27 

2764 

2828 

2892 

2956 

302 

3083 

64 

681 

83- 3 T 47 

3211 

3275 

3338 

3402 

3466 

353 

3593 

3657 

3721 

64 

682 

83- 3784 

3848 

39 12 

3975 

4039 

4103 

4166 

423 

4294 

4357 

64 

683 

83- 442i 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

83- 5056 

512 

5183 

5247 

53 i 

5373 

5437 

55 

5564 

5627 

6 3 

6 S 5 

83- 5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

686 

83- 6 324 6387 

6451 

6514 

6577 

6641 

6704 

6767 

683 

6894 

63 

687 

83- 6957 

702 

7083 

7146 

721 

7273 

733 6 

7399 

7462 

7525 

63 

688 

83- 7588 

7652 

77 I 5 

7778 

7841 

7904 

7967 

803 

8093 

8156 

6 3 

689 

83- 8219 

8282 

8345 

8408 

8471 

8534 

8597 

866 

8723 

8786 

63 

690 

83- 8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

63 

691 

83- 9478 

9541 

9604 

9667 

9729 

9792 

9855 

99 l8 

9981 

— 

63 

691 

84- — 

— 

— 

— 

— 

— 

— 

— 

— 

0043 

63 

692 

84 0106 

0169 

0232 

0294 

0357 

042 

0482 

0545 

0608 

0671 

63 

693 

84- 0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

!234 

1297 

63 

694 

84- 1359 

1422 

1485 

1547 

161 

1672 

1735 

1797 

186 

1922 

63 

695 

84- 1985 

2047 

211 

2172 

2235 

2297 

236 

2422 

2484 

2547 

62 

696 

84- 2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3 T 7 

62 

697 

84- 3233 

3295 

3357 

342 

3482 

3544 

3606 

366Q 

3731 

3793 

62 

698 

84- 3855 

39 l8 

398 

4042 

4104 

4.166 

4229 

4291 

4353 

44 i 5 

62 

699 

84- 4477 

4539 

460 X 

4664 

4726 

4788 

485 

4912 

4974 

5 ° 3 6 

62 

700 

84- 5098 

516 

5222 

5284 

5346 

5408 

547 

5532 

5594 

5656 

62 

701 

84- 5718 

578 

5842 

59°4 

5966 

6028 

609 

6151 

6213 

6275 

62 

7 P2 

84- 633?-. 6399 

6461 

6523 

6585 

6646 

6708 

677 

6832 

6894 

62 

703 

84- 6955 

7 OI 7 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

75 ii 

62 

704 

84- 7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8127 

62 

No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D 


















No. 

705 

706 

707 

708 

709 

710 

711 

712 

7 I 3 

714 

715 

716 

717 

718 

719 

720 

721 

722 

723 

724 

724 

725 

726 

727 

728 

729 

730 

73 1 

732 

733 

734 

735 

736 

737 

738 

739 

740 

74i 

741 

742 

743 

744 

745 

746 

747 

748 

749 

750 

75i 

752 

753 

754 

No. 


D 

62 

61 

61 

61 

61 

61 

61 

61 

61 

61 

61 

61 

61 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

60 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

59 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

58 

D 


LOGARITHMS OF NUMBERS. 


0 

\ 

2 

3 

4 

5 

6 

7 

8 

9 

84- 8189 

8251 

8312 

8374 

8435 

8497 

8559 

862 

8682 

8743 

84- 8805 

8866 

8928 

8989 

9051 

9112 

9*74 

9235 

9297 

9358 

84- 9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

85- 0033 

0095 

0156 

0217 

0279 

034 

0401 

0462 

0524 

0585 

85- 0646 

0707 

0769 

083 

0891 

0952 

1014 

1075 

1136 

1197 

85- 1258 

132 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

85- 187 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

85- 248 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

85- 309 

3i5 

3211 

3272 

3333 

3394 

3455 

35i6 

3577 

3637 

85- 3698 

3759 

382 

3881 

394i 

4002 

4063 

4124 

4185 

4245 

85- 430 6 

4367 

4428 

4488 

4549 

461 

467 

4731 

4792 

4852 

85- 49 I 3 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

85- 5519 

558 

564 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

85- 6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

85- 6729 

6789 

685 

691 

697 

7031 

7091 

7152 

7212 

7272 

85- 7332 

7393 

7453 

75L3 

7574 

7634 

7694 

7755 

7815 

7875 

85- 7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

85- 8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

85- 9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

85- 9739 

9799 

9859 

9918 

9978 

— 

— 

— 

—•. 

— 

86- — 

— 

— 

— 

— 

0038 

0098 

0158 

0218 

0278 

86- 0338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

86- 0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

86- 1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

86- 2131 

2191 

2251 

231 

237 

243 

2489 

2549 

2608 

2668 

86- 2728 

2787 

2847 

2906 

2966 

3025 

3085 

3 T 44 

3204 

3263 

86- 3323 

3382 

3442 

3501 

356 i 

362 

368 

3739 

3799 

3S58 

86- 3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

86- 45ix 

457 

463 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

86- 5104 

5163 

5222 

5282 

534i 

54 

5459 

55i9 

5578 

5637 

86- 5696 

5755 

5814 

5874 

5933 

5992 

6051 

611 

6169 

6228 

86- 6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

676 

6819 

86- 6878 

6937 

6996 

7055 

7114 

7*73 

7232 

7291 

735 

7409 

86- 7467 

7526 

7585 

7644 

7703 

7762 

7821 

788 

7939 

7998 

86- 8056 

8115 

8174 

8233 

8292 

835 

8409 

8468 

8527 

8586 

86- 8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9 I 73 

86- 9232 

929 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

976 

86- 9818 

9877 

9935 

9994 

— 

— 

— 

— 

— 

— 

87- - 

— 

— 

— 

0053 

OIII 

017 

0228 

0287 

0345 

87- 0404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

093 

87- oq8q 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

87- 1573 

1631 

169 

1748 

1806 

1865 

1923 

1981 

204 

2098 

87- 2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

87- 2739 

2797 

2855 

2913 

2972 

303 

3088 

3 i 46 

3204 

3262 

87- 332i 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

87- 3902 

396 

4018 

4076 

4134 

4192 

425 

4308 

4366 

4424 

87- 4482 

454 

4598 

4656 

4714 

4772 

483 

4888 

4945 

5003 

87- 5061 

5ii9 

5i77 

5235 

5293 

535i 

5409 

5466 

5524 

5582 

87- 564 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

616 

87- 6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

668 

6737 

87- 6795 

6853 

691 

6968 

7026 

7083 

7141 

7199 

7256 

73H 

87- 737i 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


E E 










326 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

755 

87- 7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

756 

87- 8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

757 

87- 9096 

9 X 53 

9211 

9268 

9325 

9383 

944 

9497 

9555 

9612 

758 

87- 9669 

9726 

9784 

9841 

9898 

9956 

— 

— 

— 

— 

758 

88- — 

— 

— 

— 

— 

— 

0013 

007 

0127 

0185 

759 

88- 0242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

88- 0814 

0871 

0928 

0985 

1042 

io 99 

1156 

1213 

1271 

1328 

761 

88- 1385 

1442 

1499 

1556 

1613 

167 

1727 

1784 

1841 

1898 

762 

88- 1955 

2012 

2069 

2126 

2183 

224 

2297 

2354 

2411 

2468 

763 

88- 2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

298 

3037 

764 

88- 3093 

3 T 5 

3207 

3264 

3321 

3377 

3434 

349 x 

3548 

3605 

765 

88- 3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4 ii 5 

4172 

766 

88- 4229 

4285 

4342 

4399 

4455 

45 i 2 

4569 

4625 

4682 

4739 

767 

88- 4795 

4852 

4909 

4965 

5022 

5078 

5 i 35 

5192 

5248 

5305 

768 

88- 5361 

54 i 8 

5474 

553 1 

5587 

5644 

57 

5757 

5813 

587 

769 

88- 5926 

5983 

6039 

6096 

6x52 

6209 

6265 

6321 

6 378 

6434 

770 

88- 6491 

6547 

6604 

666 

6716 

6773 

6829 

6885 

6942 

6998 

771 

88- 7054 

7111 

7167 

7223 

728 

7336 

7392 

7449 

7505 

756 i 

772 

88- 7617 

7674 

773 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

773 

88- 8179 

8236 

8292 

8348 

8404 

846 

8516 

8573 

8629 

8685 

774 

88- 8741 

8797 

8853 

8909 

8965 

9021 

9°77 

9 X 34 

919 

9246 

775 

88- 9302 

9358 

9414 

947 

9526 

9582 

9638 

9694 

975 

9806 

776 

88- 9862 

99x8 

9974 

— 

— 

— 

— 

— 

— 

— 

776 

89- - 

— 

— 

003 

0086 

0141 

0197 

0253 

0309 

03 6 5 

777 

89- 0421 

0477 

0533 

0589 

0645 

07 

0756 

0812 

0868 

0924 

778 

89- 098 

1035 

1091 

1147 

1203 

1259 

1314 

137 

1426 

1482 

779 

89- 1537 

1593 

1649 

1705 

176 

1816 

1872 

1928 

1983 

2039 

780 

89- 2095 

215 

2206 

2262 

2317 

2373 

2429 

2484 

254 

2595 

781 

89- 2651 

2707 

2762 

2818 

2873 

2929 

2985 

304 

3096 

3151 

782 

89- 3207 

3262 

33 x 8 

3373 

3429 

3484 

354 

3595 

3651 

3706 

783 

89- 3762 

3817 

3873 

3928 

3984 

4039 

4094 

4 i 5 

4205 

4261 

784 

89- 4316 

437 i 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

785 

89- 487 

4925 

498 

5036 

5091 

5 X 46 

5201 

5257 

53!2 

5367 

786 

89- 5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

592 

787 

89- 5975 

603 

6085 

614 

6 i 95 

6251 

6306 

6361 

6416 

6471 

788 

89- 6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

789 

89- 7077 

7132 

7 i8 7 

7242 

7297 

7352 

7407 

7462 

75 x 7 

7572 

790 

89- 7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

79 1 

89- 8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

867 

792 

89- 8725 

878 

8835 

889 

8944 

8999 

9054 

9109 

9164 

9218 

793 

89- 9 2 73 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

794 

89- 9821 

9875 

993 

9985 

— 

— 

— 

— 

— 

— 

794 

90- — 

— 

— 

— 

°°39 

0094 

0149 

0203 

0258 

0312 

795 

90- 0367 

0422. 

0476 

0531 

0586 

064 

0695 

0749 

0804 

0859 

796 

90- 0913 

0968 

1022 

io 77 

1131 

1186 

124 

1295 

x 349 

T 4 ° 4 

797 

9 °- J 458 

1513 

*567 

1622 

1676 

x 73 i 

1785 

184 

1894 

1948 

798 

90- 2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

9 °- 2547 

2601 

2655 

271 

2764 

2818 

2873 

2927 

2981 

3036 

800 

90- 309 

3144 

3199 

3253 

3307 

336 i 

34 i 6 

347 

3524 

3578 

801 

9 °“ 3 6 33 

3687 

374 i 

3795 

3849 

3904 

3958 

4012 

4066 

412 

802 

9 °- 4174 

4229 

4283 

4337 

439 1 

4445 

4499 

4553 

4607 

4661 

803 

90- 4716 

477 

4824 

4878 

4932 

4986 

504 

5094 

5148 

5202 

804 

90- 5256 

53 i 

5364 

54 i 8 

5472 

5526 

558 

5634 

5688 

5742 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


D 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

56 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

55 

54 

54 

54 

54 

54 

54 

54 

54 

D 








LOGARITHMS OF NUMBERS. 32/ 


No. 

0 

' 

2 

3 

4 

5 

6 

7 

8 

9 

D 

805 

90- 5796 

585 

5904 

5958 

6012 

6066 

6119 6174 

6227 

6281 

54 

806 

90- 6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

682 

54 

807 

90- 6874 

6927 

6981 

7035 

7089 

7143 

7196 

725 

7304 

7358 

54 

808 

90- 7411 

7465 

75i9 

7573 

7626 

768 

7734 

7787 

7841 

7895 

54 

809 

90- 7949 

8002 

8056 

811 

8163 

8217 

827 

8324 

8378 

8431 

54 

810 

90- 8485 

8539 8592 

8646 8699 

8753 

8807 

886 

8914 

8967 

54 

811 

90- 9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

812 

90- 9556 9609 9663 9716 977 

9823 

9877 

993 

9984 


54 

812 

91- — 









0037 

53 

813 

91- 0091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

53 

814 

91- 0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

815 

91- 1158 

1211 

1264 

1317 

1371 

1424 

1477 

153 

1584 

1637 

53 

816 

91- 169 

1743 

1797 

185 

1903 

1956 

2009 

2063 

2116 

2169 

53 

817 

91- 2222 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

27 

53 

818 

9 I_ 2753 

2806 

2859 

2 9 T 3 

2966 

3019 

3072 

3125 

3178 

3231 

53 

819 

91- 3284 

3337 

339 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

820 

91- 3814 

3867 

392 

3973 

4026 

4079 

4132 

4184 

4237 

429 

53 

821 

9i- 4343 

4396 

4449 

4502 

4555 

4608 

466 

47i3 

4766 

4819 

53 

822 

91- 4872 

4925 

4977 

503 

5083 

5136 

5189 

5241 

5294 

5347 

53 

823 

9 1 " 54 

5453 

5505 

5558 

5611 

5664 

57i6 

5769 

5822 

5875 

53 

824 

91- 5927 

598 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

825 

91- 6454 

6507 

6559 

6612 

6664 

6717 

677 

6822 

6875 

6927 

53 

826 

91- 698 

7033 

7085 

7138 

719 

7243 

7295 

7348 

74 

7453 

53 

827 

91- 7506 

7558 

7611 

7663 

77x6 

7768 

782 

7873 

7925 

7978 

52 

828 

91- 803 

8083 

8i35 

8188 

824 

8293 

8345 

8397 

845 

8502 

52 

829 

91- 8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

91- 9078 

9 J 3 

9 i8 3 

9235 

9287 

934 

9392 

9444 

9496 

9549 

52 

831 

91- 9601 

9653 

9706 

9758 

981 

9862 

9914 

9967 

— 

— 

52 

831 

92- 








0019 

0071 

52 

832 

92- 0123 

0176 

0228 

028 

0332 

0384 

0436 

0489 

0541 

0593 

52 

833 

92- 0645 

0697 

0749 

0801 

0853 

0906 

0958 

IOI 

1062 

1114 

52 

834 

92- 1166 

1218 

127 

1322 

1374 

1426 

1478 

153 

1582 

1634 

52 

835 

92- 1686 

1738 

179 

1842 

1894 

1946 

1998 

205 

2102 

2154 

52 

836 

92- 2206 

2258 

231 

2362 

2414 

2466 

2518 

257 

2622 

2674 

52 

837 

92- 2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3i4 

3192 

52 

838 

92- 3244 

3296 

3348 

3399 

345i 

3503 

3555 

3607 

3658 

37i 

52 

839 

92- 3762 

3814 

3865 

39*7 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

92- 4279 

433i 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

52 

841 

92- 4796 

4848 

4899 

495i 

5003 

5054 

5106 

5i57 

5209 

5261 

52 

842 

92- 5312 

5364 

54 r 5 

5467 

55i8 

557 

5621 

5673 

5725 

5776 

52 

843 

Q2- ^828 

5879 

593i 

5982 

6034 

6085 

6137 

6188 

624 

6291 

5i 

844 

92- 6342 

6394 

6445 

6497 

6548 

66 

6651 

6702 

6754 

6805 

5i 

845 

92- 6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

5i 

846 

9 2 " 737 

7422 

7473 

7524 

7576 

7627 

7678 

773 

7781 

7832 

5i 

847 

02 - 7884 

7935 

7986 

8037 

8088 

814 

8191 

8242 

8293 

8345 

5i 

848 

92- 8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

5i 

849 

92- 8908 

8959 

901 

9061 

9112 

9163 

9215 

9266 

93i7 

9368 

5i 

850 

92- 9419 

947 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

5i 

851 

92- 993 

9981 









5i 

851 

93- — 


0032 

0083 

0134 

0185 

0236 0287 

0338 0389 

5i 

852 

93" °44 

°49 1 

0542 

0592 

0643 

0694 

0745 

0796 0847 

0898 

5i 

853 

93- 0949 

1 

1051 

1102 

ii53 

1203 

1254 

1305 

1350 

1407 

5i 

854 

93 - i '458 

1509 

156 

161 

1661 

1712 

i7 6 3 

1814 

1865 

I 9 I 5 

5i 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 













LOGARITHMS OF NUMBERS. 


328 


No. 

0 

1 

1 

3 

4 

5 

6 

7 

8 

9 

D 

855 

93 - * 9 66 

2017 

2068 

2118 

2169 

222 

2271 

2322 

2372 

2423 

5 i 

856 

93- 2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

293 

5 i 

857 

93- 2981 

3031 

3082 

3 T 33 

3183 

3234 

3285 

3335 

3386 3437 

5 i 

858 

93- 3487 

3538 

3589 

3639 

369 

374 

3791 

3841 

3892 

3943 

5 i 

859 

93 - 3993 

4044 

4094 

4145 

4195 

4246 4296 4347 

4397 

4448 

5 i 

860 

93 - 4498 

4549 

4599 

4 6 5 

47 

475 i 

4801 

4852 

4902 

4953 

50 

861 

93 “ S003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 5406 

5457 

50 

862 

93 - 5507 

5558 

5608 

5658 

5709 

5759 

5809 

586 

59 i 

596 

50 

863 

93- 601X 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 6463 

50 

864 

93" 6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 6966 

50 

865 

93- 7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

866 

93 - 75 i 8 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

867 

93- 8019 

8069 

8119 

8169 

8219 

8269 8319 837 

842 

847 

50 

868 

93- 852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

50 

869 

93 - 9° 2 

907 

912 

917 

922 

927 

932 

9369 9419 9469 

50 

870 

93 - 9519 

9569 

9619 

9669 

9719 

9769 9819 9869 9918 9968 

50 

871 

94- 0018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

50 

872 

94- 0516 

0566 

0616 

0666 

0716 

0765 

0815 0865 

0915 

0964 

50 

873 

94- 1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

874 

94 - 1511 

1561 

1611 

166 

171 

176 

1809 

1859 

I 9°9 

1958 

50 

875 

94- 2008 

2058 

2x07 

2157 

2207 

2256 2306 

2355 

2405 

2455 

50 

876 

94- 2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

295 

50 

877 

94 - 3 

3049 

3099 

3148 

3198 

3247 

3297 

3346 3396 

3445 

49 

878 

94 - 3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

389 

3939 

49 

879 

94 - 3989 

4038 

4088 

4137 

4186 

4236 4285 

4335 

4384 

4433 

49 

8S0 

94 " 4483 

4532 

458 i 

4631 

468 

4729 

4779 4828 

4877 

4927 

49 

881 

94 - 4976 

5025 

5074 

5124 

5173 ' 

5222 

5272 

532 i 

537 

5419 

49 

882 

94 - 5469 

55 i 8 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

59 12 

49 

883 

94 - 596 i 

601 

6059 

6108 

6157 

6207 

6256 6305 

6354 6403 

49 

884 

94- 6452 

6501 

6551 

66 

6649 

6698 

6747 6796 6845 

6894 

49 

885 

94- 6943 

6992 

7041 

709 

7x4 

7189 

7238 

7287 

733 6 

7385 

49 

886 

94 - 7434 

7483 

7532 

758 i 

763 

7679 

7728 

7777 

7826 

7875 

49 

887 

94 - 7924 

7973 

8022 

807 

8119 

8x68 

8217 

8266 

8315 

8364 

49 

888 

94- 8413 

8462 

8511 

856 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

94- 8902 

8951 

8999 

9048 

9097 

9x46 

9 i 95 

9244 

9292 

934 i 

49 

890 

94 - 939 

9439 

9488 

9536 

9585 

9634 9683 

973 i 

978 

Q82Q 

49 

891 

94 - 9878 

9926 

9975 

— 

— 

— 

— 

— 


— 

49 

891 

95 - — 

— 

— 

0024 

0073 

0121 

017 

0219 0267 

0316 

49 

892 

95 " 0365 

0414 

0462 

0511 

056 

0608 

0657 

0706 0734 0804 

49 

893 

95 - 0851 

09 

0949 

0997 

1046 

1095 

1143 

1192 

124 

1289 

49 

894 

95 - 1338 

1386 

1435 

1483 

1532 

158 

1629 

1677 

1726 

1775 

49 

895 

95- 1823 

1872 

192 

1969 

2017 

2066 

2114 

2163 

2211 

226 

48 

896 

95- 2308 

2356 

2405 

2453 

2502 

255 

2599 

2647 

2696 

2744 

48 

897 

95- 2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

318 

3228 

48 

898 

95 - 3276 

3325 

3373 

3421 

347 

35 i 8 

3566 

36 i 5 

3663 

3711 

48 

899 

95 - 376 

3808 

3856 

3905 

3953 

4001 

4049 4098 4146 

4194 

48 

900 

95 - 4243 

4291 

4339 

4387 

4435 

4484 

4532 

458 

4628 

4677 

48 

901 

95 " 4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

511 

5158 

48 

902 

95- 5207 

5255 

5303 

535 i 

5399 

5447 

5495 

5543 

5592 

564 

48 

903 

95- 5688 

5736 

5784 

5832 

588 

5928 

5976 6024 6072 

612 

48 

904 

95- 6168 

6216 

6265 

6313 

6361 

6409 6457 

6 505 

6553 

6601 

48 

No. 

0 

' 

2 

3 

4 

5 

6 

7 

8 

9 

D 











LOGARITHMS OP NUMBERS. 


No. 1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

905 

95- 6649 6697 6745 

6793 

684 

6888 

6936 

6984 

7032 

708 

48' 

906 

95 - 7 I2 § 

7176 

7224 

7272 

732 

7368 

7416 

7464 

7512 

7559 

48 

907 

95 - 7607 

7655 

7703 

775 T 

7799 

7847 

7894 

7942 

799 

8038 

48 

908 

95- 8086 

8 i 34 

8x81 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 

95 " 8564 

8612 

8659 8707 

8755 

8803 

885 

8898 

8946 8994 

48 

910 

95 - 9041 

9089 9137 

9 i8 5 

9232 

928 

9328 

9375 

9423 

9471 

48 

9 11 

95 - 95 l8 95 66 9 6i 4 9 661 

9709 

9757 

9804 

9852 

99 

9947 

48 

912 

95 " 9995 










48 

912 

96- — 

0042 

009 

0138 

0185 

0233 

028 

0328 

0376 

0423 

48 

9*3 

96- 0471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

914 

96- 0946 0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

915 

96- 1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

916 

96- 1895 

1943 

199 

2038 

2085 

2132 

218 

2227 

2275 

2322 

47 

9*7 

96- 2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

96- 2843 

289 

2937 

2985 

3032 

3079 

3 I2 6 

3 i 74 

3221 

3268 

47 

9 T 9 

96- 33 l6 33 6 3 

34 i 

3457 

3504 

3552 

3599 

3646 

3693 

374 i 

47 

920 

96- 3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

96- 426 

4307 

4354 

4401 

4448 

4495 

4542 

459 

4637 

4684 

47 

922 

96- 4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5 i 55 

47 

9 2 3 

96- 5202 

5249 

5296 

5343 

539 

5437 

5484 

553 i 

5578 

5625 

47 

924 

96- 5672 

5719 

5766 

5813 

586 

59°7 

5954 

6001 

6048 

6095 

47 

925 

96- 6142 

6189 6236 

6283 

6329 

6376 

6423 

647 

6517 

6564 

47 

926 

96- 6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

927 

96- 708 

7127 

7 T 73 

722 

7267 

73 1 4 

736 i 

7408 

7454 

75 oi 

47 

928 

96- 7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

47 

929 

96- 8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

839 

8436 

47 

930 

96- 8483 

853 

8576 

8623 

867 

8716 

8763 

881 

8856 

8903 

47 

93 i 

96- 895 

8996 

9043 

909 

9136 

9 i8 3 

9229 

9276 

9323 

9369 

47 

932 

96- 9416 9463 9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

933 

96- 9882 

9928 

9975 

— 

— 

— 

— 

— 

— 

— 

47 

933 

97 - — 

— 

— 

0021 

0068 

0114 

0161 

0207 

0254 

03 

47 

934 

97 - 0347 

0393 

044 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

97- 08x2 

0858 0904 

0951 

0997 

1044 

109 

ii 37 

1183 

1229 

46 

93 6 

97- 1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

937 

97 - 174 

17S6 

1832 

1879 

1925 

1971 

2018 

2064 

211 

2157 

46 

93 8 

Q 7 - 2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

939 

97- 2666 

2712 

2758 

2804 

2851 

2897 

2943 

29S9 

3035 

3082 

46 

940 

97“ 3128 3174 

322 

3266 

3313 

3359 

3405 

345 i 

3497 

3543 

46 

941 

91 - 359 

3636 3682 

3728 

3774 

382 

3866 

39 i 3 

3959 

4005 

46 

942 

97 - 4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

442 

4466 

46 

943 

97 - 45 i 2 

4558 

4604 

465 

4696 

4742 

4788 

4834 

488 

4926 

46 

944 

97 - 4972 

5 OI 8 

5064 

5 ii 

5156 

5202 

5248 

5294 

534 

5386 

46 

945 

97 - 5432 

5478 

5524 

557 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

97 - 5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

947 

97- 635 

64Q6 6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

948 

97- 6808 

6854 

69 

6946 

6992 

7037 

7083 

7129 

7 i 75 

722 

46 

949 

97- 7266 

7312 

7358 

7403 

7449 

7495 

754 i 

7586 

7632 

7678 

46 

950 

97 - 7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8 i 35 

46 

95 i 

97- 8181 

8226 

8272 

8317 

8363 

8409 

8454 

85 

8546 

859 1 

46 

952 

97- 8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

97 " 9093 

9138 

9 lS 4 

923 

9275 

9321 

9366 

94x2 

9457 

9503 

46 

954 

97 - 9548 9594 9639 

9685 

973 

9776 

9821 

9867 

9912 

9958 

46 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 


E E* 















330 LOGARITHMS OF NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

955 

98- 000;* 

0049 

0094 

014 

0185 

0231 

0276 

0322 

0367 

0412 

956 

98- 0458 

0503 

0549 

0594 

064 

0685 

073 

0776 

0821 

0867 

957 

98- 0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

132 

958 

98- 1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

959 

98- 1819 

1864 

1909 

1954 

2 

2045 

209 

2135 

2181 

2226 

960 

98- 2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

98- 2723 

2769 

2814 

2859 

2904 

2949 

2994 

304 

3085 

3 J 3 

962 

98- 3175 

322 

3265 

331 

3356 

3401 

3446 

3491 

3536 

358 i 

963 

98- 3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

964 

98- 4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

98- 4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

98- 4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

967 

98- 5426 

547 i 

55 i 6 

556 i 

5606 

5651 

5696 

5741 

5786 

583 

968 

98- 5875 

592 

5965 

601 

6055 

61 

6144 

6189 

6234 

6279 

969 

98- 6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

98- 6772 

6817 

6861 

6906 

6951 

6996 

704 

7085 

7 i 3 

7 i 75 

971 

98- 7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

98- 7666 

7711 

7756 

78 

7845 

789 

7934 

7979 

8024 

8068 

973 

98- 8113 

8 i 57 

8202 

8247 

8291 

8336 

8381 

8425 

847 

8514 

974 

98- 8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

896 

975 

98- 9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 

98- 945 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

985 

977 

98- 9895 

9939 

9983 

— 

— 

— 

— 

— 

— 

— 

977 

99 - ~ 

— 

— 

0028 

0072 

0117 

0161 

0206 

025 

0294 

978 

99 - 0339 

0383 

0428 

0472 

0516 

0561 

0605 

065 

0694 

0738 

979 

99 - 0783 

0827 

0871 

0916 

096 

1004 

1049 

1093 

ii 37 

1182 

9 S 0 

99- 1226 

127 

1315 

1359 

1403 

1448 

1492 

1536 

158 

1625 

981 

99- 1669 

1713 

1758 

1802 

1846 

189 

1935 

1979 

2023 

2067 

982 

99- 2111 

2156 

22 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

99 - 2 554 

2598 

2642 

2686 

273 

2774 

2819 

2863 

2907 

2951 

984 

99- 2995 

30.39 

3083 

3127 

3172 

3216 

326 

3304 

3348 

3392 

985 

99 - 343 6 

348 

3524 

3568 

3613 

3 6 57 

3701 

3745 

3789 

3833 

986 

99 - 3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

99 - 4317 

4361 

4405 

4449 

4493 

4537 

458 i 

4625 

4669 

4713 

988 

99 - 4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

989 

99 - 5 i 9 6 

524 

5284 

5328 

5372 

54 i 6 

546 

5504 

5547 

559 i 

990 

99 - 5635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

603 

991 

99- 6074 

6117 

6161 

6205 

6249 

6293 

6337 

638 

6424 

6468 

992 

99" 6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

99- 6949 

6993 

7037 

708 

7124 

7168 

7212 

7255 

7299 

7343 

994 

99 " 7386 

743 

7474 

7517 

756 i 

7605 

7648 

7692 

7736 

7779 

995 

99- 7823 

7867 

79 1 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

99" 8259 

83 0 3 

8347 

839 

8434 

8477 

8521 

8564 

8608 

8652 

997 

99- 8695 

8739 

8782 

8826 

8869 

89 j 3 

8956 

9 

9043 

9087 

998 

99 - 9 I 3 1 

9 X 74 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

999 

99 - 9565 

9609 

9652 

9696 

9739 

9783 

9826 

987 

9913 

9957 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


D 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

45 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

43 

D 










HYPERBOLIC LOGARITHMS OF NUMBERS. 


331 


Hyperbolic Logarithms of Numbers. 

From 1.01 to 30. 

In following table, the numbers range from 1.01 to 30, advancing by .01, 
up to the whole number 10; and thence by larger intervals up to 30. The 
hyperbolic logarithms of numbers, or Neperian logarithms, as they are some¬ 
times termed, are computed by multiplying the common logarithms of num¬ 
bers by the constant multiplier, 2.302 585. 

The hyperbolic logarithms of numbers intermediate between those which 
are given in the table may be readily obtained by interpolating proportional 
differences. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

I.OI 

.OO99 

1.41 

•3436 

I.8l 

•5933 

2.21 

•793 

2.6l 

•9594 

1.02 

.0198 

I.42 

•3507 

1.82 

.5988 

2.22 

•7975 

2.62 

.9632 

1.03 

.0296 

i -43 

•3577 

I.83 

.6043 

2.23 

.802 

2.63 

.967 

X.04 

.0392 

1.44 

.3646 

I.84 

.6098 

2.24 

.8065 

2.64 

.9708 

1.05 

.0488 

i -45 

.3716 

1-85 

.6152 

2.25 

.8109 

2.65 

.9746 

I.06 

• 058.3 

1.46 

•3784 

1.86 

.6206 

2.26 

.8154 

2.66 

•9783 

I.07 

.0677 

1.47 

•3853 

1.87 

.6259 

2.27 

.8198 

2.67 

.9821 

1.08 

.077 

1.48 

•392 

1.88 

•6313 

2.28 

.8242 

2.68 

.9858 

I.09 

.0862 

1.49 

•3988 

1.89 

.6366 

2.29 

.8286 

2.69 

•9895 

1.1 

•0953 

i -5 

•4055 

1.9 

.6419 

2-3 

.8329 

2.7 

•9933 

I.IX 

.IO44 

I - 5 I 

.4121 

1.91 

.6471 

2.3I 

•8372 

2.71 

.9969 

1.12 

•1133 

1.52 

.4187 

1.92 

•6523 

2.32 

.8416 

2.72 

1.0006 

1.13 

.1222 

i -53 

•4253 

i -93 

•6575 

2-33 

•8458 

2-73 

1.0043 

1.14 

•131 

i -54 

.4318 

1.94 

.6627 

2-34 

.8502 

2.74 

1.008 

1 .15 

.1398 

i -55 

•4383 

i -95 

.6678 

2-35 

•8544 

2-75 

1.0116 

1.16 

.1484 

1.56 

•4447 

1.96 

.6729 

2.36 

•8587 

2.76 

1-0152 

1.17 

•157 

i -57 

•4511 

1.97 

.678 

2-37 

.8629 

2.77 

1.0188 

1.18 

.1655 

1.58 

•4574 

1.98 

.6831 

2.38 

.8671 

2.78 

1.0225 

1.19 

.174 

i -59 

•4637 

I< 99 

.6881 

2-39 

•8713 

2.79 

1.026 

1.2 

.1823 

1.6 

•47 

2 

.6931 

2.4 

•8755 

2.8 

1.0296 

1.21 

.1906 

1.61 

.4762 

2.01 

.6981 

2.4I 

.8796 

2.81 

1.0332 

1.22 

.1988 

1.62 

.4824 

2.02 

•7 031 

2.42 

.8838 

2.82 

1.0367 

I.23 

.207 

1.63 

.4886 

2.03 

.708' 

2-43 

.8879 

2.83 

1.0403 

I.24 

.2151 

1.64 

•4947 

2.04 

.7129 

2-44 

.892 

2.84 

1.0438 

1.25 

.2231 

1.65 

.5008 

2.05 

• 7 i 78 

2-45 

.8961 

2.85 

1-0473 

1.26 

.23II 

1.66 

.5068 

2.06 

.7227 

2.46 

.9002 

2.86 

1.0508 

I.27 

•239 

1.67 

.5128 

2.07 

•7275 

2.47 

.9042 

2.87 

I-Q 543 

1.28 

.2469 

1.68 

.5188 

2.08 

•7324 

2.48 

.9083 

2.88 

1.0578 

1.29 

.2546 

1.69 

•5247 

2.09 

•7372 

2.49 

.9x23 

2.89 

1.0613 

1.3 

.2624 

1.7 

•5306 

2.1 

.7419 

2-5 

• 9 i6 3 

2.9 

1.0647 

i- 3 i 

.27 

i.li 

•5365 

2.11 

.7467 

2.51 

.9203 

2.91 

1.0682 

1.32 

.2776 

1.72 

•5423 

2.12 

•7514 

2.52 

•9243 

2.92 

1.0716 

i -33 

.2852 

i -73 

.5481 

2.13 

• 756 i 

2-53 

.9282 

2-93 

1-075 

i -34 

.2927 

1.74 

•5539 

2.14 

.7608 

2-54 

.9322 

2.94 

1.0784 

i -35 

• 3 °° 1 

i -75 

•5596 

2.15 

•7655 

2-55 

.9361 

2-95 

1.0818 

1.36 

•307S 

1.76 

•5653 

2.16 

.7701 

2.56 

•94 

2.96 

1.0852 

i -37 

• 3 i 48 

1.77 

• 57 i 

2.17 

•7747 

2.57 

•9439 

2.97 

1.0886 

1.38 

.3221 

1.78 

.5766 

2.18 

•7793 

2.58 

•9478 

2.98 

1.0919 

i .39 

•3293 

1.79 

.5822 

2 19 

•7839 

2-59 

• 95 i 7 

2.99 

1-0953 

1.4 

•3365 

1.8 

1 -5878 

2.2 

.7885 

2.6 

•9555 

3 

1.0986 











































332 


HYPEKBOLIC LOGARITHMS OF NUMBERS. 


No. 

Log. 1 

No. | 

Log. 

No. 

Log. 

No. 

Log. 

| No. 

Log. 

3- QI 

1.1019 

3-5i 

1.2556 

4.OI 

1.3888 

4-5i 

I.5063 

5.01 

I.6114 

3.02 

I - I °53 

3-52 

I.258S 

4.02 

I-39I3 

4-52 

1.5085 

5.02 

I.6134 

3-o 3 

1.1086 

3-53 

I.2613 

4-03 

1-3938 

4-53 

I.5107 

5-03 

I.6154 

3-°4 

1.1119 

3-54 

I.264I 

4.O4 

I.3962 

4-54 

I-5I29 

5-04 

I.6174 

3-°5 

1.1151 

3-55 

I.2669 

4-05 

I-39S7 

4-55 

I.5I5I 

5-05 

I.6194 

3.06 

1.1184 

3-56 

I.2698 

4.06 

1.4012 

4-56 

I-5I73 

5.06 

1.6214 

3-07 

1.1217 

3-57 

I.2726 

4.07 

1.4036 

4-57 

I-5I95 

5-07 

I.6233 

3.°8 

1.1249 

3-58 

1-2754 

4.08 

1.4061 

4-58 

I-52I7 

5.08 

1.6253 

3-09 

1.1282 

3-59 

1.2782 

4.09 

1.4085 

4-59 

1-5239 

5-09 

1.6273 

3- 1 

1.1314 

3- 6 

1.2809 

4.1 

1.411 

4.6 

1.5261 

5-i 

I.6292 

3-n 

1.1346 

3.61 

1.2837 

4.11 

I-4I34 

4.61 

1.5282 

5-n 

1.6312 

3.12 

1.1378 

3.62 

1.2865 

4.12 

I -4 I 59 

4.62 

1-5304 

5.12 

I.6332 

3-!3 

1.141 

3-63 

1.2892 

4-i3 

1.4183 

4-63 

I-53 2 6 

5-i3 

I-635I 

3- J 4 

1.1442 

3- 6 4 

1.292 

4.14 

1.4207 

4.64 

1-5347 

5-i4 

I.637I 

3-i5 

1.1474 

3- 6 5 

1.2947 

4-i5 

I -4 2 3 I 

4-65 

I-5369 

5-i5 

1.639 

3- 16 

x.1506 

3.66 

1-2975 

4.16 

I-4255 

4.66 

1-539 

5-i6 

1.6409 

3-i7 

I - I 537 

3-67 

1.3002 

4.17 

1.4279 

4.67 

1.5412 

5-i7 

1.6429 

3.18 

1.1569 

3.68 

1.3029 

4.18 

1-4303 

4.68 

1-5433 

5-i8 

1.6448 

3-i9 

1.16 

3- 6 9 

1-3056 

4.19 

1-4327 

4.69 

1-5454 

5-i9 

1.6467 

3-2 

1.1632 

3-7 

1-3083 

4.2 

1-4351 

4-7 

I-5476 

5-2 

1.6487 

3.21 

1.1663 

3-7i 

x-3 11 

4.21 

1-4375 

4.71 

1-5497 

5.21 

1.6506 

3.22 

1.1694 

3-72 

I -3 I 37 

4.22 

1.4398 

4.72 

1-5518 

5.22 

1-6525 

3-23 

1.1725 

3-73 

1.3164 

4-23 

1.4422 

4-73 

1-5539 

5-23 

1.6514 

3- 2 4 

1.1756 

3-74 

1.319 1 

4.24 

1.4446 

4-74 

i-556 

5-24 

1-6563 

3-25 

1.1787 

3-75 

1.3218 

4-25 

1.4469 

4-75 

i-558i 

5-25 

I.6582 

3.26 

1.1817 

3-76 

1.3244 

4.26 

1-4493 

4.76 

1.5602 

5.26 

1.6601 

3-27 

1.1848 

3-77 

1.327 1 

4.27 

1.4516 

4-77 

1-5623 

5-27 

1.662 

3.28 

1.1878 

3-78 

1.3297 

4.28 

1-454 

4.78 

1.5644 

5.28 

1.6639 

3-29 

1.1909 

3-79 

I -33 2 4 

4.29 

1-4563 

4-79 

1-5665 

5-29 

1.6658 

33 

I-I939 

3-8 

i-335 

4-3 

1.4586 

4.8 

1.5686 

5-3 

I.6677 

3-3i 

1.1969 

3.81 

I-3376 

4-3i 

1.4609 

4.81 

I -57°7 

5-3i 

1.6696 

3-32 

I - I 999 

3.82 

1-3403 

4-3 2 

1-4633 

4.82 

1.5728 

5-3 2 

1.6715 

3-33 

1 .203 

3-83 

1-3429 

4-33 

1.4656 

4-83 

i-5748 

5-33 

I-6734 

3-34 

1.206 

3-84 

1-3455 

4-34 

!-4679 

4.84 

1-5769 

5-34 

1.6752 

3-35 

1.209 

3-85 

i-348i 

4-35 

1.4702 

485 

i-579 

5-35 

1.6771 

3 36 

1.2119 

3.86 

1-3507 

4-36 

I-4725 

4.86 

1.581 

5-36 

1.679 

3-37 

1.2149 

3-87 

!-3533 

4-37 

1.4748 

4.87 

1-5831 

5-37 

I.6808 

3 38 

1.2179 

3-88 

1-3558 

4-38 

1.477 

488 

1-5851 

5-38 

1.6827 

3-39 

1.2208 

3-89 

!-3584 

4-39 

1-4793 

4.89 

1.5872 

5-39 

1.6845 

3-4 

1.2238 

3-9 

1.361 

4.4 

1.4816 

4-9 

1.5892 

5-4 

1.6864 

3 4^ 

1.2267 

3-9i 

1-3635 

4.41 

1.4839 

4 9 1 

I -59 I 3 

5-4i 

1.6882 

3-42 

1.2296 

3-92 

1.3661 

4.42 

1.4861 

4.92 

1-5933 

5-42 

1.6901 

3-43 

1.2326 

3-93 

1.3686 

4-43 

1.4884 

4-93 

1-5953 

5-43 

1.6919 

3-44 

1-2355 

3-94 

1.37x2 

4.44 

1.4907 

4.94 

1-5974 

5-44 

1.6938 

3-45 

1.2384 

3-95 

1-3737 

4-45 

1.4929 

4-95 

I -5994 

5-45 

1.6956 

3-46 

1.24x3 

3-96 

1.3762 

4.46 

1-4951 

4.96 

1.6014 

5 46 

1.6974 

3-47 

1.2442 

3-97 

1.3788 

4-47 

1.4974 

4-97 

1.6034 

5-47 

i - 6 993 

3-48 

1.247 

3-98 

1-3813 

4.4S 

1.4996 

4.98 

1.6054 

5 48 

1.7011 

3-49 

1.2499 

3-99 

1-3838 

4.49 

1.5019 

| 4-99 

1.6074 

5-49 

1.702Q 

3-5 

1.2528 

4 

1.3863 

1 4-5 

1.504 1 

l5 

1.6094 

5-5 

1.7047 




























HYPERBOLIC LOGARITHMS OF NUMBERS. 333 


No. 

Log. 

No. 

Log. 11 

No. | 

Log. 

No. | 

Log. 

No. 

Log. 

5 - 5 i 

1.7066 

6.01 

1-7934 I 

6.51 

1-8733 

7.OI 

*•9473 

7 - 5 * 

2.0162 

5-52 

1.7084 

6.02 

*• 795 * 

6.52 

I.8749 

7.02 

1.9488 

7-52 

2.OI76 

5-53 

1.7102 

6.03 

1.7967 : 

6-53 

1.8764 

7-03 

1.9502 

7-53 

2.0189 

5-54 

1.712 

6.04 

1.7984 

6-54 

I.8779 

7.04 

1.9516 

7-54 

2.0202 

5-55 

I.7138 

6.05 

1.8001 

6-55 

1-8795 

7-05 

*•953 

7-55 

2.0215 

5-56 

I.7156 

6.06 

1.8017 

6.56 

I.881 

7.06 

*•9544 

7-56 

2.0229 

5-57 

I.7174 

6.07 

1.8034 

6-57 

I.8825 

7.07 

1-9559 

7-57 

2.0242 

5-58 

I.7192 

6.08 

1.805 

6.58 

I.884 

7.08 

1-9573 

7-58 

2.0255 

5-59 

1.72I 

6.09 

1.8066 

6-59 

I.8856 

7.09 

I -9587 

7-59 

2.0268 

5-6 

I.7228 

6.1 

1.8083 

6.6 

I.8871 

7 -* 

1.9601 

7.6 

2.0281 

5.61 

I.7246 

6.11 

1.8099 1 

6.61 

1.8886 

7.11 

1.9615 

7.61 

2.O295 

5.62 

I.7263 

6.12 

1.81x6 

6.62 

1.8901 

7.12 

1.9629 

7.62 

2.0308 

5-63 

I.7281 

6.13 

1.8132 

6.63 

1.8916 

7-*3 

1.9643 

7-63 

2.0321 

5- 6 4 

I.7299 

6.14 

1.8148 

6.64 

1.8931 

7-*4 

I -9657 

7.64 

2-0334 

5-65 

*• 73*7 

6.15 

1.8165 

6.65 

1.8946 

7-*5 

1.9671 

7-65 

2.0347 

5.66 

1-7334 

6.16 

1.8181 

6.66 

1.8961 

7.16 

1.9685 

7.66 

2.036 

5-67 

* -7352 

6.17 

1.8197 

6.67 

1.8976 

7.17 

1.9699 

7.67 

2-0373 

5.68 

1-737 

6.18 

1.8213 

6.68 

1.8991 

7.18 

*• 97*3 

7.68 

2.0386 

5-69 

I -7387 

6.19 

1.8229 

6.69 

1.9006 

7.19 

1.9727 

7.69 

2.0399 

5-7 

I -7405 

6.2 

1.8245 

6.7 

1.9021 

7.2 

*• 974 * 

7-7 

2.0412 

5 - 7 i 

1.7422 

6.21 

1.8262 

6.71 

1.9036 

7.21 

*•9755 

7.71 

2.0425 

5-72 

1.744 

6.22 

1.8278 

6.72 

1-9051 

7.22 

1.9769 

7.72 

2.0438 

5-73 

1-7457 

6.23 

1.8294 

6-73 

1.9066 

7-23 

1.9782 

7-73 

2.0451 

5-74 

1-7475 

6.24 

1.831 

6.74 

1.9081 

7.24 

1.9796 

7-74 

2.0464 

5-75 

1.7492 

6.25 

1.8326 

6-75 

1.9095 

7-25 

1.981 

7-75 

2.0477 

5-76 

I -7509 

6.26 

1.8342 

6.76 

1.911 

7.36 

1.9824 

7.76 

2.049 

5-77 

I -7527 

6.27 

1.8358 

6.77 

1.9125 

7.27 

1.9838 

7-77 

2.0503 

5-78 

1-7544 

6.28 

I -8374 

6.78 

i- 9 i 4 

7.28 

1.9851 

7.78 

2.0516 

5-79 

1.7561 

6.29 

1.839 

6.79 

1 - 9*55 

7.29 

1.9865 

7-79 

2.0528 

5-8 

1-7579 

6-3 

1.8405 

6.8 

1.9169 

7-3 

1.9879 

7.8 

2.0541 

5 . 8 i 

I -7596 

6.31 

1.8421 

6.81 

1.9184 

7 - 3 * 

1.9892 

7.81 

2.0554 

5.82 

1.7613 

6.32 

1-8437 

6.82 

*• 9*99 

7-32 

1.9906 

7.82 

2.0567 

5-83 

1-763 

6-33 

1.8453 

6.83 

1.9213. 

7-33 

1.992 

7-83 

2.058 

5-84 

1.7647 

6-34 

1.8469 

6.84 

1.9228 

7-34 

*•9933 

7.84 

2.0592 

5-85 

1.7664 

6-35 

1.8485 

6.85 

1.9242 

7-35 

1.9947 

7-85 

2.0605 

5.86 

1.7681 

6.36 

1.85 

6.86 

i- 9 2 57 

7-36 

1.9961 

7.86 

2.0618 

5-87 

1.7699 

6-37 

1.8516 

6.87 

1.9272 

7-37 

1.9974 

7.87 

2.0631 

5-88 

1.7716 

6.38 

1-8532 

6.88 

1.9286 

7-38 

1.9988 

7.88 

2.0643 

5-89 

1-7733 

6-39 

1-8547 

6.89 

1.9301 

7-39 

2.0001 

7.89 

2.0656 

5-9 

1-775 

6.4 

1.8563 

6.9 

*• 93*5 

7-4 

2.0015 

7-9 

2.0669 

5 - 9 * 

1.7766 

6.41 

1-8579 

6.91 

*•933 

7.41 

2.0028 

7.91 

2.0681 

5-92 

I -7783 

6.42 

1.8594 

6.92 

*•9344 

7.42 

2.0042 

7.92 

2.0694 

5-93 

1.78 

6-43 

1.861 

6-93 

*•9359 

7-43 

2.0055 

7-93 

2.0707 

5-94 

1.7817 

6.44 

1.8625 

6.94 

*•9373 

7-44 

2.0069 

7-94 

2.0719 

5-95 

1.7834 

6-45 

1.8641 

6-95 

*•9387 

7-45 

2.0082 

7-95 

2.0732 

5 - 9 6 

1.7851 

6.46 

1.8656 

6.96 

1.9402 

7.46 

2.0096 

7.96 

2.0744 

5-97 

1.7867 

j 6.47 

1.8672 

6.97 

1.9416 

7-47 

2.0109 

7-97 

2.0757 

5 - 9 8 

1.7884 

6.48 

1.8687 

6.98 

*•943 

7.48 

2.0122 

7.98 

2.0769 

5-99 

1.7901 

6.49 

1.8703 

6-99 

*•9445 

7-49 

2.0136 

7-99 

2.0782 

6 

1.7918 

6.5 

1.8718 

7 

*•9459 

7-5 

2.0149 

8 

2.0794 























































334 


HYPERBOLIC LOGARITHMS OF NUMBERS. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

| No. 

Log. 

No. 

Log. 

8.01 

2.0807 

8.41 

2.1294 

8.8l 

2.1759 

9.21 

2.2203 

9.61 

2.2628 

8.02 

2.0819 

8.42 

2.1306 

8.82 

2.I77 

9.22 

2.2214 

9.62 

2.2638 

8.03 

2.0832 

8-43 

2.1318 

8.83 

2.1782 

9- 2 3 

2.2225 

9- 6 3 

2.2649 

8.04 

2.0844 

8.44 

2-133 

8.84 

2.1793 

9.24 

2.2235 

9.64 

2.2659 

8.05 

2.0857 

8-45 

2.1342 

8.85 

2.1804 

9-25 

2.2246 

9- 6 5 

2.267 

8.06 

2.0869 

8.46 

2.1353 

8.86 

2.1815 

9.26 

2.2257 

9.66 

2.268 

8.07 

2.0882 

8.47 

2.1365 

8.87 

2.1827 

9.27 

2.2268 

9.67 

2.269 

8.08 

2.0894 

8.48 

2.1377 

8.88 

2.1838 

9.28 

2.2279 

9.68 

2.270I 

8.09 

2.0906 

8.49 

2.1389 

8.89 

2.1849 

9.29 

2.2289 

9.69 

2.27H 

8.1 

2.0919 

8-5 

2.1401 

8.9 

2.l86l 

9-3 

2.23 

9-7 

2.272I 

8.11 

2.0931 

8.51 

2.1412 

8.91 

2.1872 

9 - 3 i 

2.23II 

9.71 

2.2732 

8.12 

2.0943 

8.52 

2.1424 

8.92 

2.1883 

9-32 

2.2322 

9.72 

2.2742 

8.13 

2.0956 

8-53 

2.1436 

8-93 

2.1894 

9-33 

2.2332 

9-73 

2.2752 

8.14 

2.0968 

8-54 

2.1448 

8.94 

2.1905 

9-34 

2.2343 

9-74 

2.2762 

8.15 

2.098 

8-55 

2.1459 

8-95 

2.1917 

9-35 

2.2354 

9-75 

2.2773 

8.16 

2.0992 

8.56 

2.1471 

8.96 

2.1928 

9-36 

2.2364 

9.76 

2.2783 

8.17 

2.1005 

8-57 

2.1483 

8.97 

2.1939 

9-37 

2.2375 

9-77 

2.2793 

8.18 

2.IOI7 

8.58 

2.1494 

8.98 

2.I95 

9-38 

2.2386 

9.78 

2.2803 

8.19 

2.1029 

8-59 

2.1506 

8.99 

2.1961 

9-39 

2.2396 

9-79 

2.2814 

8.2 

2.IO4I 

8.6 

2.1518 

9 

2.I972 

9.4 

2.2407 

9.8 

2.2824 

8.21 

2.IO54 

8.61 

2.1529 

9.01 

2.1983 

9.41 

2.2418 

9.81 

2.2834 

8.22 

2.1066 

8.62 

2.1541 

9.02 

2.I994 

9.42 

2.2428 

9.82 

2.2844 

8.23 

2.IO78 

8.63 

2.1552 

9-03 

2.2006 

9-43 

2.2439 

9-83 

2.2854 

8.24 

2.IO9 

8.64 

2.1564 

9.04 

2.2017 

9.44 

2.245 

9.84 

2.2865 

8.25 

2.1102 

8.65 

2.1576 

9-°5 

2.2028 

9-45 

2.246 

9-85 

2.2875 

8.26 

2.III4 

8.66 

2.1587 

9.06 

2.2039 

9.46 

2.2471 

9.86 

2.2885 

8.27 

2.1126 

8.67 

2.1599 

9.07 

2.205 

9-47 

2.2481 

9.87 

2.2895 

8.28 

2.II38 

8.68 

2.161 

9.08 

2.2o6l 

9.48 

2.2492 

9.88 

2.2905 

8.29 

2.II5 

8.69 

2.1622 

9.09 

2.2072 

9.49 

2.2502 

9.89 

2.2915 

8-3 

2.1163 

8.7 

2.1633 

9.1 

2.2083 

9-5 

2.2513 

9.9 

2.2925 

8.31 

2.II75 

8.71 

2.1645 

9.11 

2.2094 

9-51 

2.2523 

9.91 

2.2935 

8.32 

2.1187 

8.72 

2.1656 

9.12 

2.2105 

9-52 

2.2534 

9.92 

2.2946 

8-33 

2.II99 

8-73 

2.1668 

9 -L 3 

2.2Il6 

9-53 

2.2544 

9-93 

2.2956 

8-34 

2.1211 

8.74 

2.1679 

9.14 

2.2127 

9-54 

2.2555 

9.94 

2.2966 

8-35 

2.1223 

8-75 

2.1691 

9- J 5 

2.2138 

9-55 

2.2565 

9-95 

2.2976 

8.36 

2.1235 

8.76 

2.1702 

9.16 

2.2148 

9-56 

2.2576 

9.96 

2.2986 

8-37 

2.1247 

8.77 

2.1713 

9.17 

2.2159 

9-57 

2.2586 

9-97 

2.2996 

8.38 

2.1258 

8.78 

2.1725 

9.18 

2.217 

9-58 

2.2597 

9.98 

2.3006 

8-39 

2.127 

8.79 

2.1736 

9.19 

2.2l8l 

9-59 

2.2607 

9.99 

2.3016 

8.4 

2.1282 

8.8 

2.1748 

9.2 

2.2192 

9.6 

2.2618 

10 

2.3026 

10.25 

2.3279 

12.25 

2.5052 

14.25 

2.6567 

I 7-5 

2.8621 

23 

3- I 355 

10.5 

2.3513 

12.5' 

2.5262 

14-5 

2.674 

18 

2.8904 

24 

3.1781 

10.75 

2-3749 

12.75 

2-5455 

14-75 

2.6913 

18.5 

2 - 9 I 73 

25 

3.2189 

11 

2.3979 

13 

2.5649 

i 5 

2.7081 

19 

2.9444 

26 

3-2581 

11.25 

2.4201 

13-25 

2.584 

15-5 

2.7408 

19-5 

2.9703 

27 

3-2958 

n-5 

2-443 

13-5 

2.6027 

16 

2.7726 

2 

2-9957 

28 

3-3322 

n-75 

2.4636 

13-75 

2.6211 

16.5 

2.8034 

21 

3-0445 

29 

33673 

12 

I 2.4849 

14 

2.6391 

17 

2.8332 

22 

3 - 09 11 

3 

3.4012 



































MENSURATION OF AREAS, LINES, AND SURFACES. 335 


MENSURATION OF AREAS, LINES, AND SURFACES. 
Parallelograms. 

Definition.— Quadrilaterals, having their opposite sides parallel. 

To Compute Area of a Scqnare, Rectangle, Ftlio 111P>as, or 
Rhomboid.—Figs. 1 , £2, 3, and 4r. 


Rule.— Multiply length by breadth or height. 

Or, l X b == area, l representing length , and b breadth. 
Fig. 1. Fig. 3. 



Fig. 4. 


c 


Example. —Sides ab,b c, Fig. 1, are 5 feet 6 ins.; what is area? 

5-5 X 5.5 = 3°. 25 square feet. 

Note i. —Opposite angles of a Rhombus and a Rhomboid are equal. 

2. —In any parallelogram the four angles equal 360°. 

3. —Side of a square multiplied by 1.52 is equal to side of an equilateral triangle 
of equal area. 

GrrLOIXLOTX. 


Definition.— Space included between the lines forming two similar parallel¬ 
ograms, of which smaller is inscribed within larger, so that one angle in each is 
common to both, as shown by dotted lines, Fig. 1. 


To Compute Area of a Gnomon.—Fig. 1 . 


Rule.— Ascertain areas of the two parallelograms, and subtract less from 
greater. 

Or, a — a' — area, a and a' representing areas. 

Example.— Sides of a gnomon are 10 by 10 and 6 by 6 ins.; what is its area? 
10X10 = 100, and 6 X 6 = 36. Then 100 — 36 = 64 square ins. 

Triangles. 

Definition. —Plain superficies having three sides and angles. 


To Compute Area of a Triangle.—Rigs. 5 , 6, and 7 

Rule. —Multiply base by height, and divide product by 2. 

Or — Or, — = area, b representing base, and h height. 

2 2 

Note i.— Hypothenuse of a right angle is side opposite to right angle. 

2. —Perpendicular height of a triangle = twice its area divided by its base. 

3. —Perpendicular height of an equilateral triangle = a side X -866. 

4. —side of an equilateral triangle X -658 255 = side of a square of equal area, 

Or -4-1.3468 = diameter of a circle of equal area. 


Fig. 5 - 


Fig. 6. 


Fig. 7. 



Example. — Base ah, Fig. 
5, is 4 feet, and height c b, 6; 
what is area? 

4 x 6 = 24, and 24-7-2 = 12 
square feet. 



















336 MENSURATION OF AREAS, LINES, AND SURFACES. 


To Compute Area of a Triangle L>y Length, of its Sides.— 

Figs. 6 and ' 7 . 

Rule.—F rom half sum of the three sides subtract each side separately; 
then multiply half sum and the three remainders continually together, and 
take square root of product. 

Or, V(s — a) X (s — 6) X (s — c) S = area, a, b, c representing sides, and S half sum 
of the three sides. 

Example.— Sides of a triangle, Figs. 6 and 7, are 30, 40, and 50 feet; what is area? 

30 4 ~ 4 ° ~f~ 50 _ or |j a jf gum 0 f gideg. g 0 — 40 — 201 remainders. 

2 2 60 — 50 = 10) 

Whence, 30 x 20 X 10 X 60 = 360 000, and y/ 360 000 = 600 square feet. 

When all Sides are Equal. Rule.—S quare length of a side, and multi¬ 
ply product by .433. 

Or, S 2 X • 433 = area, S representing length of a side. 

To Compute Length, of One Side of a Light-A.ngled 

Triangle. 


When Length of the other Tivo Sides are given. 

To Ascertain Hypothenuse. — Fig. £5. 

Rule.—A dd together squares of the two legs, and take square root of 


sum. 


Or, Va b 2 -f- b c 2 = hypothenuse. Or, Vb 2 -f- K 2 . 

Example. —Base, a b, Fig. 5, is 30 ins., and height, be, 40; what is length of hy¬ 
pothenuse ? 

30 2 -f- 40 2 = 2500, and y /2500 = 50 ins. 


To Ascertain other Leg. 


When Hypothenuse and One of the Legs are given. —Fig. 5. Rule. —Sub¬ 
tract square of given leg from square of hypothenuse, and take square root 
of remainder. 


Or. {£“*; Or, y.*-{«•£»£ 

Example. —Base of a triangle, a b, Fig. 5, is 30 feet, and hypothenuse, a c, 50; 
what is height of it? 

50 2 — 30 2 = 1600, and f 1600 = 40 feet. 


To Compnte Length, of a Side. 

When Hypothenuse of a Right-angled Triangle of Equal Sides alone is 
given. —Fig. 8. Rule. —Divide hypothenuse by 1.414213. 

hyp. 


Fig. 8. 


Or, 



1.414213 


: length of a side. 


Example. — Hypothenuse a c of a right-angled triangle, Fig. 8, is 
300 feet; what is length of its sides ? 

300 -T-1.414 213 = 212.1321 feet. 


To Compute Perpendicular or Height of a Triangle. 

When Base and Area alone are given. —Fig. 9. Rule. —Divide twice 
area by its base. Or, 2a b = h. 

Example.— Area of a triangle, Fig. 9, is 10 feet, and length of its base, a b, 5; 
what is its perpendicular, c d? 

10X2 = 20, and 20 -7- 5 = 4 feet. 











MENSURATION OF AREAS, LINES, AND SURFACES. 337 


To Compute Perpendicular or ITeiglit of a, Triangle. 

When Base and Two Sides are given. Rule. —As base is to sum of the 
sides, so is difference of sides to difference of divisions of base. Half this 
difference being added to or subtracted from half base will give the two di¬ 
visions thereof. Hence, as the sides and their opposite division of base con¬ 
stitute a right-angled triangle, the perpendicular thereof is readily ascertained 
by preceding rules. 

b c-\- caxb c ^ ca 


Or. 



b a 

a c- -f- ab 2 - 
2 ab 


:b da. 


be 2 


ad; whence Vac 2 — ad 2 = dc. 


Example.— Three sides of a triangle, a be, Fig. 9, are 9.928, 
8, and 5 feet; what is length of perpendicular on longest side ? 


As 9.928 : 8 — 1 — 5 ;; 8 fo 5 
the base. 


3.928 = difference of divisions of 


Then 3.928 -f- 2 — 1.964, which, added to 


9.928 


4.964 -f- 


1.964 = 6 928 = length of longest division of base. 


Hence, there is a right-angled triangle with its base 6.928, and its hypothenuse 8; 
consequently, its remaining side or perpendicular is +(8 2 —6.92s 2 } — ^ feet. 

When any Two of the Dimensions of a Triangle and One of the corresponding 
Dimensions of a similar Figure are given, and it is required to ascertain 
the other corresponding Dimensions of the last Figure. 

Fig. 10. Fig. 11. 

Let ab c, a' b' c', be two similar triangles, Figs. 10 
and 11. 

Then a b : be ;: a' b' : b' c', or a' b': V c ':: ab : b c. 

Note. — Same proportion holds with respect to the 
similar lineal parts of any other similar figures, whether 
plane or solid. 

Example. _Shadow of a vertical stake 4 feet in length was 5 feet; at same time, 

shadow of a tree, both on level ground, was 83 feet; what was height of tree? 

5 a 1 b’ : 4 b' c' :: 83 a b : 66.4 feet. 



To Compnte Acreage. 

Divide area into convenient triangles, and multiply base of each triangle 
in links by half perpendicular in links ; cut off 5 figures at the right, remain¬ 
ing figures will give acres; multiply the 5 figures so cut off by 4, and again 
cut off 5, and remainder will give roods; multiply the 5 by 40, and again 
cut off 5 for perches. 

Trapezium. 

Definition.—A Quadrilateral having unequal sides of which no two are parallel. 

To Compute Area, of a Trapezium.-Fig. 12. 


Rule. —Multiply diagonal by sum of the two perpendiculars falling upon 
it from the opposite angles, and divide product by 2. 



Or, 


db X a + c 
-!— = area. 


2 


Example. —Diagonal d b, Fig. 12, is 125 feet, and perpen¬ 
diculars a and c 50 and 37; what is area? 

125 X 50 + 37 = 10 875, and 10 875 - 4 - 2 = 5437.5 square feet. 


















338 MENSURATION OF AREAS, LINES, AND SURFACES. 


When the Two opposite Angles are Supplements to each other , that is , when 
a Trapezium can be inscribed in a Circle , the Sum of its opposite Angles 
being equal to Two Right Angles, or i8o°. Rule. —From half sum of the 
four sides, subtract each side severally; then multiply the four remainders 
continually together, and take square root of product. 

Example. —In a trapezium the sides are 15, 13, 14, and 12 feet; its opposite an¬ 
gles being supplements to each other, required its area. 

15 +13 +14 + 12 = 54, and — = 27. 

27 27 27 27 2 

15 13 u ^ 

12 x 14 X 13 X 15 = 32760, and -^32760 = 180.997 square feet. 


Trapezoid. 

Definition.— A Quadrilateral with only one pair of opposite sides parallel. 

To Compute Area of a Trapezoid. —l-Cig. 13. 

Rule.—M ultiply sum of the parallel sides by perpendicular distance be¬ 
tween them, and divide product. 


Or, 


ab-\-dcXah 


Or 


s —|— s r X h 


; area , s and s ’ representing sides . 


Fig. 13. a 


c h 


Jj Example. —Parallel sides ab,cd , Fig. 13, are 100 and 132 

feet, and distance between them 62.5 feet; what is area? 
100 -)- 132 x 62.5 = 14 500, and 14 500 4 - 2 = 7250 square 

a M- 

Polygons. 

Definition. —Plane figures having three or more sides, and are either regular or 
irregular, according as their sides or angles are equal or unequal, and they are named 
from the number of their sides and angles. 

Regular Polygons. 

To Compute Area of a Hegwxlar Polygon.—Fig. 14. 
Rule.—M ultiply length of a side by perpendicular distance to centre; 
multiply product by number of sides, and divide it by 2. 

0r abxcexn 

- = area, n representing number of sides. 

2 

Fig. 14. g, Example. —What is area of a pentagon, side a b, Fig. 14, being 

' G 5 feet, and distance ce 4.25 feet? 

h 5 X 4-25 X 5 (n) = 106.25 — product of length of a side, dis- 
r tance to centre, and number of sides. 

Then, 106.254-2 = 53.125 square feet. 

To Compute Ifadins of a Circle tliat contains a Given 

Polygon. 

When Length of a Perpendicular from Centre alone is given. Rule. — 
Multiply distance from centre to a side of the polygon, by unit in column A 
of following Table. 

Example.— What is radius of a circle that contains a hexagon, distance to centre 
being 4.33 inches? 

4.33 X 1.156 = 5 ins. 

To Compute Length of a Side of a Polygon Chat is con¬ 
tained in a Given Circle. 



When Radius of Circle is given. Rule.—M ultiply radius of circle, by 
unit in column B of following Table. 

Example.— What is length of side of a pentagon contained in a circle 8.5 feet in 
diameter? 

8.5 4 - 2 = 4.25 radius, and 4.25 X 1-1756 = 5 feet. 











MENSURATION OF AREAS, LINES, AND SURFACES. 339 


To Compute Radius of a Circumscribing Circle. 

When Length of a Side is given. Rule. —Multiply length of a side of the 
polygon, by unit in column 0 of following Table. 

Example.—W hat is radius of a circle that will contain a hexagon, a side being 5 
inches? 

5 Xl = 5 MW- 

To Compute Radius of a Circle that can. be Inscribed 
in a Given Polygon. 


When Length of a Side is given. Rule. —Multiply length of a side of 
polygon, by unit in column D of following Table. 

Example.— What is radius of the circle that is bounded by a hexagon, its sides 
being 5 inches? 

5 X >866 = 4.33 ins. 


To Compute Area of a Regular Polygon. 


When Length of a Side only is given. Rule. —Multiply square of side, 
by multiplier opposite to term of polygon in following Table: 


No. of 
Sides. 

Polygon. 

Area. 

A. 

Radius of 
Circumscribed 
Circle. 

B. 

Length of a 
Side. 

C. 

Radius of 
Circumscrib¬ 
ing Circle. 

D. 

Radius of 
Inscribed 
Circle. 

3 

Trigon 

•433 01 

2 

1 732 

•5773 

.2887 

4 

Tetragon 

I 

1-414 

I.4I42 

.7071 

•5 

5 

Pentagon 

1.72048 

1.238 

1.1756 

.8506 

.6882 

6 

Hexagon 

2.59808 

1.156 

I 

I 

.866 

7 

Heptagon 

3-633 9 1 

I. II 

.8677 

1.1524 

1-0383 

8 

Octagon 

4.82843 

1.083 

•7653 

1.3066 

1.2071 

9 

Nonagon 

6.181 82 

1.064 

.684 

14619 

1-3737 

IO 

Decagon 

7.694 21 

1-051 

.618 

1.618 

1.5388 

II 

Undecagon 

9-36564 

I.O42 

•5634 

1-7747 

1 7028 

12 

Dodecagon 

11.196 15 

1 037 

5U6 

L93I9 

1.866 


Example. — What is area of a square (tetragon) when length of its sides is 
7.0710678 inches? 

7.071 067 8 2 = 50, and 50 X 1 = 50 square ins. 


To Compute Length of a Side and Radii of a Regular 

Polygon. 


When Area alone is given. Rule.— Multiply square root of area of poly¬ 
gon by multiplier in column E of the following table for length of side; by 
multiplier in column G for radius of circumscribing circle, and by multiplier 
in column H for radius of inscribed circle or perpendicular. 


No. of 
Sides. 

Polygon. 

E. 

Length of 
Side. 

G. 

Radius of 
Circumscrib¬ 
ing Circle. 

H. 

Radius of 
Inscribed 
Circle. 

Angle. 

Angle of 
Polygon. 

Tangent. 

3 

Trigon 

I- 5 I 97 

.8774 

•4387 

120° 

6o° 

•5774 

4 

Tetragon 

I 

.7071 

•5 

90 

9 ° 

I 

5 

Pentagon 

.7624 

• 6485 

•5247 

72 

108 

I -3764 

6 

Hexagon 

.6204 

• 6204 

•5373 

60 

120 

I- 732 I 

7 

Heptagon 

.5246 

•6045 

•5446 

5! 25.71' 

128 34.29' 

2.0765 

8 

Octagon 

•4551 

•5946 

•5493 

45 

135 

2.4142 

9 

Nonagon 

.4022 

.588 

•5525 

40 

I40 

2-7475 

IO 

Decagon 

,3605 

•5833 

•5548 

36 

144 

3-0777 

II 

Undecagon 

.3268 

•5799 

•5564 

32 43-64' 

147 16.36' 

3-4057 

12 

Dodecagon 

.2989 

•5774 

•5577 

30 

150 

3-7321 


Example i.—A rea of a square (tetragon) is 16 inches; what is length of its side? 
3/16 = 4, and 4X1=4 ins. 

2.—Area of an octagon is 70.698 yards; what is diameter of its circumscribing 
circle? 


f 70.698 X • 5946 = 5, and 5X2 = 10 yards. 




























MENSURATION OF AREAS, LINES, AND SURFACES. 


340 

Additional Uses of foregoing Table. —6th and 7 th columns of table facilitate con¬ 
struction of these 1 figures with aid of a sector. Thus, if it is required to describe an 
octagon, opposite to it in column 6th, is 45; then, with chord of .60 on sector as 
radius, describe a circle, taking length 45 on same line of sector; mark this dis¬ 
tance off on the circumference, which, being repeated around the circle, will give 
points of the sides. 

7th column gives angle which any two adjoining sides of the respective figures 
make with each other; and 8th gives tangent of the angle in column 6th. 

To Compute Radius of Inscribed or Circumscribed 

Circles. 

When Radius of Circumscribing Circle is given. Rule.— Multiply radius 
given by unit in column E, in following Table, opposite to term of polygon 
for which radius is required. 

When Radius of Inscribed Circle is given. Rule.— Multiply radius given 
by unit in column F, in following Table, opposite to term of polygon for 
which radius is required. 

To Compute Area. 

When Radii of Inscribed or Circumscribing Circles are given. Rule.— 
Square radius given, and multiply it by unit in columns G or H, in following 
Table, and opposite to term of polygon for which area is required. 

When Length of a Side is given. Rule. — Square length of side and 
multiply it by unit in column I, in following Table, opposite to term of 
polygon for which area is required. 

To Compute Length, of a Side. 

When Radius of Inscribed Circle is given. Rule.— Multiply radius given 
by unit in column K, in following Table, and opposite to term of polygon for 
which length is required. 


No. of 
Sides. 

PoLYBON. 

E. 

Radius of 
Inscribed 
by Circum¬ 
scribing 
Circle. 

F. 

Radius of 
Circumscrib¬ 
ing by 
Inscribing 
Circle. 

G. 

Area. 

By Radius 
of inscribed 
Circle. 

H. 

Area. 

By Radius 
of Circum¬ 
scribing 
Circle. 

I. 

Area. 

By Length 
of Side. 

K. 

Length of 
Side. 

By' Radius 
of Inscribed 
Circle. 

3 

Trigon 

•5 

2 

5.196 15 

1.299 04 

-433 01 

3.4641 

4 

Tetragon 

.7O7 II 

1.414 21 

4 

2 

I 

2 

5 

Pentagon 

. 809 02 

1.23607 

3.63272 

2.37764 

1.72048 

1.45308 

6 

Hexagon 

. 866 02 

i-i 547 

3.464 1 

2.598 08 

2.59808 

I - I 547 

7 

Heptagon 

.9OO97 

I. IO9 92 

3.37102 

2.73641 

3-633 9 1 

• 9 6 3 15 

8 

Octagon 

.923 88 

1.082 39 

3 - 3 I 37 I 

2.828 42 

4.81843 

.82843 

9 

Nonagon 

.93969 

1.064 18 

3-275 73 

2.89254 

6.182 82 

.727 94 

IO 

Decagon 

.951 06 

1.051 46 

3.2492 

2-93893 

7.694 21 

. 649 84 

II 

Undecagon 

•959 49 

I.O42 22 

3.22989 

2-973 53 

9-36564 

•587 25 

12 

Dodecagon 

• 9 6 5 93 

1.035 28 

3-215 39 

3 

n.19615 

•535 9 


Regular Bodies. 


To Compute Surface or Linear Edge of Regular Body. 

Rule.— Multiply square of linear edge, or radius of circumscribed or in¬ 
scribed sphere, by units in following table, under head of dimension used: 


No. of 
Sides. 

Body. 

Surface by 
Linear Edge. 

Radius of 
Circumscribed 
Sphere. 

Radius of 
Inscribed 
Sphere. 

Linear Edge 
by Surface. 

4 

Tetrahedron 

1-73205 

1.632 99 

4. 898 98 

•759 84 

6 

Hexahedron 

6 

i -154 7 

2 

.40825 

8 

Octahedron 

3.464 1 

I. 414 21 

2.44949 

•537 29 

12 

Dodecahedron 

20.64578 

• 7 1 364 

.898 06 

.220 08 

20 

Icosahedron 

8.660 25 

1.051 46 

1-32317 

•339 8i - 


Example.— What is surface of a hexahedron or cube, having sides of 5 inches? 
5 2 X 6 = 25 X 6 = 150 square ins. 




























MENSURATION OF AREAS, LINES, AND SURFACES. 


341 


To Compute Linear Edge. 

When Surface alone is given. Rule. —Multiply square root of surface, 
by multiplier opposite to term of body under head of Linear Edge by Sur¬ 
face in preceding Table. 

Example.— Wliat is linear edge of a hexahedron, surface being 6 inches? 

-\/6 X .40825 = 1 inch. 

When Radius of an Inscribed or Circumscribed Sphere is given. Rule.— 
Multiply radius given, by multiplier opposite to term of body in preceding 
Table, under bead of the Radius given. 

Example.— Radius of circumscribing sphere of a hexahedron is 10 inches; what 
is its linear edge? 


10 X 1-1547 = 11.547 ins - 


To Compute Surface. 

When Linear Edge is given. Rule. —Multiply square of edge, by multi¬ 
plier opposite to term of body in preceding Table, under head of Surface. 

Example.— Linear edge of a hexahedron is 1 inch; what is its surface? 
i 2 X 6 = 6 square ins. 

Irregular Polygons. 


Definition. —Figures with unequal sides. 

To Compute Area of an Irregular Polygon.—Figs. 15 

and 16. 

Fig. 15. Rule. —Draw diagonals and per- Fig. 16. 

pendiculars, as d f dg , a, and c, Fig. 
j 15, and f d, g d, g 6, g u, and i, 0, r , and 7 
s , Fig. 16, to divide the figures into 
triangles and quadrilaterals: ascer- 
q tain areas of these separately, and 
take their sum. ^ 

Note.— To ascertain area of mixed or compound figures, or 
such as are composed of rectilineal and curvilineal figures to¬ 
gether, compute areas of the several figures of which the whole is composed, then 
add them together, and the sum will give area of compound figure. In this manner 
any irregular surface or field of land may be measured by dividing it into trapeziums 
and triangles, and computing area of each separately. 




When any Part of a Figure is bounded by a Curve the Area may be ascer¬ 
tained as follows: 

Erect any number of perpendiculars upon base, at equal distances, and 
ascertain their lengths. 

Add lengths of the perpendiculars thus ascertained together, and their 
sum, divided by their number, will give mean breadth; then multiply mean 
breadth by length of base. 

To Compute Area of a Long, Irregular Figure.—Fig. IV. 


Fig. 17. 



a 


Rule. —Take mean breadths at several places, at equal 
listances apart, as 1, 2, 3, b d , etc.; add them together, 
iivide their sum by number of breadths for total mean 
breadth, and multiply quotient by length of figure. 

b + b'A-b". etc . 

Or, —- - -= area. 


F F* 












342 MENSURATION OF AREAS, LINES, AND SURFACES. 


To Compute an Area ‘bounded, by a Curve. — Fig. 18 . 


(, Simpson's Rule.) 

F’g- Operation. —Divide line a b into any number of equal parts, 

yS > | "ps. by perpendiculars from base, as i, 2, 3, etc., which will give 

/ \ j \ ; p\ an odd number of points of division. Measure lengths of 

a 12 3 4 5 & these perpendiculars or ordinates, and proceed as follows: 

To sum of lengths of first and last ordinates, add four times sum of lengths of all 

even numbered ordinates and twice sum of odd; multiply their sum by one third 
of distance between ordinates, and product will give area required. 

Illustration.— Water-line of a vessel has a length of 90 feet, and ordinates o, 1, 
1.2,1.5, 2, 1.9, 1.5, 1.1, and o, each 10 feet apart; what is its area? 

Ordinates. 


Even. 

Odd. 

Sums. 

1 

1.2 

first 0 

i -5 

2 

last 0 

1.9 

i -5 

even 22 

1.1 


odd 9.4 

5.5 X 4 = 22. 

4.7 X 2 = 9.4 

3 i -4 


■ 3 = 104.66 square feet 


Circle. 


Diameter is a right line drawn through its centre, bounded by its periphery. 
Radius is a right line drawn from its centre to its circumference. 1 

Circumference is assumed to be divided into 360 equal parts, termed degrees; 
each degree is divided into 60 parts, termed minutes ; each minute into 60 parts, 
termed seconds ; and each second into 60 parts, termed thirds , and so on. 


To Compute Circumference of a Circle. 

Rule.— Multiply diameter by 3.1416. 

Or, as 7 is to 22, so is diameter to circumference. 

Or, as 113 is to 355, so is diameter to circumference. 

Example.— Diameter of a circle is 1.25 inches; what is its circumference? 

1.25 x 3.1416 = 3.927 ins. 

To Compute Diameter of a Circle. 

Rule.— Divide circumference by 3.1416. 

Or, as 22 is to 7, so is circumference to diameter. 

Note. —Divide area by .7854, and square root of quotient will give diameter of circle. 


To Compute Area of a Circle. 

Rule. —Multiply square of diameter by .7854. 

Or, multiply square of circumference by .079 58. 

Or, multiply half circumference by half diameter. 

Or, multiply square of radius by 3.1416. 

Or, p r 2 = area, r representing radius. 

Example. —The diameter of a circle is 8 inches; what is the area of it? 

8 2 = 64, and 64 X .7854 = 50.2656 ins. 

Proportions of a Circle, its Ecuaal, Inscribed, and Cir¬ 
cumscribed Squares. 

CIRCLE. 

1. Diameter X .8862) 0 . . „ 0 

2. Circumference x .2821} = Slde of an E( l ual S( l uare - 

3. Diameter X -7071 ) 

4. Circumference X -2251 > = Side of Inscribed Square. 

5. Area X •9003-7-diam. ) 

6. Diameter X 1-3468 = Side of an Equilateral Triangle. 


SQUARE. 

7. A Side X 1.1442 = Diameter of its Circumscribing Circle. 

8. “ X 4-443 = Circumference of its Circumscribing Circle. 

9. “ X 1.128 = Diameter 1 

10. “ X 3-545 = Circumference [ of an Equal Circle, 

ix. Square inches X 1-273 = Circle inches ) 

Note.— Square described within a circle is one half area of one described without it. 




MENSURATION OF AREAS, LINES, AND SURFACES. 343 

To Compute Side of Greatest Square that can be In¬ 
scribed. in a Circle. 

Rule.—M ultiply diameter by .7071, or take twice square of radius. 
Useful Factors. 

In which p or 7T represents Circumference of a Circle. 

Diameter = 1. 

|- P = 4- lg 8 79+ 
liP = -523598+ 

X P — .392699+ 

T2 P= -261 799-f— 

3X0 P— -008726+ 

Diameter = 10. 

1. Chord of arc of semicircle 

2. Chord of half arc of semicircle 

3. Versed sine of arc of semicircle. 

4. Versed sine of half arc of semicircle 

5. Chord of half arc, of half of arc of semicircle 

6. Half chord, of chord of half arc 

7. Length of arc of semicircle 

8. Length of half arc of semicircle 

9. Square of chord, of half arc of semicircle (2) 

10. Square root of versed sine of half arc (4) 

11. Square of versed sine of half arc (4) 

12. Square of chord of half arc, of half arc of sem 

13. Square of half chord, of chord of half arc (6) 

Note.—I n all computations p is taken at 3.1416, p at .7854, %p at .5236; and 
whenever the decimal figure next to the one last taken exceeds 5, one is added. 
Thus, 3.141 59 for four places of decimals is taken as 3.1416. 

To Compute Length, of an Arc of a Circle.—Ifig. 19 . 

When Number of Degrees and Radius are given. Rule i. — Multiply 
number of degrees in the arc by 3.1416 times the radius, and divide by 180. 

2.—Multiply radius of circle by .01745329, and product by degrees in 
the arc. 

If length is required for minutes, multiply radius by .000 290 889; if for 
seconds, by .000 004 848. 

Example i.—N umber of degrees in an arc, oab, Fig. 19, are 
90, and radius, 0 6, 5 inches; what is length of arc? 

90 X (3-1416 X 5) = 1413.72, which- 4 -180 = 7.854 ins. 

2.—Radius of an arc is 10, and measure of its angle 44 0 30' 
30"; what is length of arc? 

10 X -017 453 29 =. 174 532 q, which X 44 = 7-679 447 6, length 
for 44°. 

10 X .000 290 889 = .002 908 89, which x 30 = .087 266 7, length for 30'. 

10 X .000004 848 = .000048 48, which X 30 = .001454 4, length for 30". 

Then 7.679 447 61 

.087 266 7^=7.768 168 7 ins. 

.0014544) 

Or, reduce minutes and seconds to decimal of a degree, and multiply by it. 

See Rule, page 93. 30' 30" = .5083, and .1745329 from above x 44.5083 = 

7.768 163 ins. 


Fig. 19 . 


c 



o 


p= 3 - 141592653589 + 

2 P= 6 . 283185307179 + 
4 p = 12.566 370 614 359 + 
Vi P — i- 57 ° 796 3 2 6 794 + 
%P — -785398 l6 3 397 + 


Vp~ 

Vl= 

Log. p — 

XzfP = 

36 p — 113.097 335 + 


1-772 453 
.797884 

•4971498 7 

.886 226 + 


= 10 

= 7.071067 
= 5 

= 1.4644 66 
= 3.82683 
= 3-535 533 
= i5-7 0 7 963 
= 7-853 9 81 
= 50 

— 1.210151 
= 2.144664 
circle (5) = 14.64467 
= 12.5 




344 mensuration of areas, lines, and surfaces. 


When Chord of Half Arc and Chord of Arc are given. Rule.—F rom eight 
times chord of half arc subtract chord of arc, and one third of remainder will 
give length nearly. 

8 ^ 

Or,-, c' representing chord of half arc, and c chord of arc. 

Example.— Chord of half arc, a c, Fig. 19, is 30 inches, and chord of arc, ab, 48; 
what is length of arc ? 

30 X 8 = 240 = 8 times chord of half arc ; 240 — 48 = 192, and 192 -1-3 — 64 ins. 


When Chord of Arc and Versed Sine of Arc are given. Rule. — Mul¬ 
tiply square root of sum of square of chord, and four times square of the 
versed sine (equal to twice chord of half arc), by ten times square of versed 
sine ; divide this product by sum of fifteen times square of chord and thirty- 
three times square of versed sine; then add this quotient to twice chord of 
half arc,* and sum will give length of arc very nearly. 


^ s/c 2 -\- 4 v. sin. 2 X 10 v. sin. 2 ... . 

Or, -—--- . — -p 2 c ', v . sm . representing versed sine . 

15 c ~ + 33 v - Sln - 2 

Example.— Chord of an arc is 80, and its versed sine, c r, 30; what is length of arc? 
80 2 = 6400 = square of chord ; 30 2 = 900 = square of versed sine. 

3/(6400-pqjoo x 4) = 100 — square root of square of chord and four limes square 
of versed sine = twice chord of half arc. 

Then 100 X 30 2 X 10 = 900 000 —product of 10 times square of versed sine and root 
above obtained. 

And 80 2 X 15 =96 000 = 15 times square of chord. 

30 2 X 33 = 29 700 = 33 times square of versed sine. 

125 700 

Hence 100 x 9 °° 000 __ and ^,599 _)_ IOO , or twice chord of half arc — 

125 700 

107.1599 length. 


When Diameter and Versed Sine are given. Rule. —Multiply twice chord 
of half the arc by 10 times versed sine; divide product by 27 times versed 
sine subtracted from 60 times diameter, add quotient to twice chord of half 
arc, and the sum will give length of arc very nearly. 


Or, 


2c' X 10 v. sin. 
60 d — 27 v. sin. 


c' — c. 


Example.— Diameter of a circle is 100 feet, and versed sine, cr, of arc 25 ; what 
is length of arc ? 

V25 x 100 = 50 = chord of half arc. See Rule, page 345. 

50 X 2 X 25 x 10 = 25 000 = twice chord of half arc by 10 times versed sine. 
100 X 60 — 25 x 27 = 5325 = 27 times versed sine from 60 times diameter. 

Then ^ °°° = 4.6948, and 4.6948 -f- 50 X 2 = 104.6948 feet. 


To Compute Chord, of air Arc. 

When Chord of Half the Arc and Versed Sine are given. Rule. —From 
square of chord of half arc subtract square of versed sine, and take twice 
square root of remainder. 

Or, f (c' 2 — v. sin. 2 ) X 2 = c. 

Example.— Chord of half arc, a c, is 60, and versed sine, c r, 36: what is length 
of chord of arc? 

60 2 — 36 s = 2304, and f 2304 X 2 = 96. 


* Square root of sum of square of chord and four times square of the versed sine is equal to twice 
chord of half arc. 















MENSURATION OF AREAS, LINES, AND SURFACES. 345 

When Diameter and Versed Sine are given. Multiply versed sine by 2, 
and subtract product from diameter; subtract square of remainder from 
square of diameter, and take square root of that remainder. 

Or, V d 2 — (a! — v. sin. X 2) 2 — c. 

Example. —Diameter of a circle is 100, and versed sine of half arc is 16 ; what is 
length of chord of arc ? 

(36 X 2 — 100) 2 100 2 = 9216, and y/9216 = 96. 

To Compute Cliord. of Half art Arc. 

When Chord, of the Arc and Versed Sine are given. Rule i.— Divide 
square root ot sum of square of chord of the arc and four times square of 
versed sine by two. 

2 -—Take square root of sum of squares of half chord of arc and versed 
sine. 


Or, 


Vc 2 -f 


4 v. sm .- 


= c'. Or, ^ ^ + v. sin . 2 = c'. 


When Diameter and Versed Sine are given. Rule. —Multiply diameter 
by versed sine, and take square root of their product. 

Or, Vd x v. sin. = c'. 

To Compute Diameter. 

Rule i. —Divide square of chord of half arc by versed sine. 

Or, c' 2 -v-v. sin. = diameter. 

2.—Add square of half chord of arc to the square of versed sine, and divide 
this sum by versed sine. 

0r , (c A 


To Compute Versed Sine. 

Rule. —Divide square of chord of half arc by diameter. 

c ' 2 

Or, — v. sin. 
d 

When Chord of the Arc and Diameter are given. Rule.— From square 
of diameter subtract square of chord, and extract square root of remainder; 
subtract this root from diameter, and divide remainder by 2. 

„ d — V d 2 — c 2 

Or,-= v. sm. 

2 

When it is greater than a Semidiameter. Rule. —Proceed as before, but 
add square root of remainder (of squares of diameter and chord) to diam¬ 
eter, and halve the sum. 

d -f- Vd 2 — c 2 

Or, - = v. sm. 

2 

Example.— Diameter of a circle is 100, and chord of arc 97.9796; what is its versed 
sine? 

ioo-|-Vioo 2 — 97.9796 s 100 -1- 20 

- — ~- = DO. 

2 2 

To Compute Ordinate of a Circnlar Curve.-Fig. SO. 
Fig. _o. Vr 2 — x 2 — (r — v) = ordinate. 

f- - A -x—A Illustration.—R adius of circle 5 ins., versed sine 

s-'cb-^ \ 2, and distance x 2; what is length of ordinate 0 ? 


( 


V5 2 — 2 2 — (5 — 2) = 4.58 — 3 = i. 58 ins. 


















34 6 MENSURATION OF AREAS, LINES, AND SURFACES. 


Sector of a Circle. 

Definition.— A part of a circle bounded by an arc and two radii. 

To Compute Area of a Sector of a Circle. 

When Degrees in the Arc are given. —Fig. 21. Rule. —As 360 is to num¬ 
ber of degrees in a sector, so is area of circle of which sector is a part to area 
of sector. 



Or, = area, d representing degrees in arc , and a area 
3°6 

u of circle. 

Example. — Radius of a circle, 0 a, Fig. 21, is 5 ins., and 
number of degrees of sector, a b o, is 22 0 30' ; what is area ? 
o Area of a circle of 5 ins. radius = 78.54 ins. 

Then, as 360° : 22 0 30' :: 78.54 : 4.90875 ins. 

When Length of the Arc , etc ., are given. Rule.— Multiply length of arc 
by half length of radius, and product is area. 

Or, b X r A- 2 = area , b representing arc , and r radius. 

Segment of a Circle. 

Definition. —A part of a circle bounded by an arc and a chord. 


To Compute Area of a Segment of a Circle. 

When Chord and Versed Sine of Arc , and Radius or Diameter of Circle are 

given. 

When Segment is less than a Semicircle , as a b c, Fig. 21. Rule. —Ascer¬ 
tain area of sector having same arc as segment; then ascertain area of tri¬ 
angle formed by chord of segment and radii of sector, and take difference of 
these areas. 

Note. —Subtract versed sine from radius; multiply remainder by one half of 
chord of arc, and product will give area of triangle. 

Or, a — a' = area, a and a' representing areas of sector and triangle. 

When Segment is greater than a Semicircle. Rule. —Ascertain, by pre¬ 
ceding rule, area of iesser portion of circle; subtract it from area of whole 
circle, and remainder will give area. 

Or, a — a' = area, a and a' representing areas of circle and lesser portion. 

See Table of Areas of Segments, page 267. 

Example. — Chord, a c, Fig. 22, is 14.142; diameter, b e , is 20 
ins.; and versed sine, b r, is 2.929; what is area of segment? 

14.142 -3- 2 = 7.071 = half chord of arc. 

V7.071 2 -J-2.929 s = 7.654 — square root of sum of squares of 
half chord of arc and versed sine , which is chord ab of half arc 
a b c. 

By Rule, page 346, 

7.654 X 2 X 2.929 X 10 = 448.371 = twice chord of half arc by 10 
times versed sine. 

20x60 — 2.929X27 = 1120.917 = 60 times diameter subtracted from 27 times 
versed sine. _ 

Then 448.371 -4- 1120.917 = .4, and .4 added to 7.654 X 2 (twice chord of half arc) 
= 15.708 inches , length of arc. 

By Rule above, 15.708 X — = 78.54 = <fte arc multiplied by half length of radius, 
= area of sector. 

IO — 2.929 = 7.071 = versed sine subtracted from a radius , which is height of tri¬ 
angle a oc, and 7.071 X = 50 = area of triangle. 

Consequently, 78.54 — 50 = 28.54. 












MENSURATION OF AREAS, LINES, AND SURFACES. 347 


When the Chords of Arc , and of half of Arc, and Versed Sine are given. 
Rule. —To chord of whole arc add chord of half arc and one third of it 
more; multiply this sum by versed sine, and this product, multiplied by 
.404 26, will give area nearly. 

Or, c + c' -j-- v. sin. X ■ 404 26 = area nearly. 

Example.— Chord of a segment, a c, Fig. 22, is 28 feet; chord of half arc, a b, is 
15; and versed sine, 6?-, 6; what is area of segment? 

28 + 15 + ^ = chord of arc added to chord of half arc and one third of it more. 

48 X 6 — 288 = product of above sum and versed sine. Hence 288 X .404 26 = 116.427 
square feet. 

When the Chord of Arc and Versed Sine only are given. Rule. —Ascer¬ 
tain chord of half arc, and proceed as before. 

To Compute Chord. and Height of a Segment of* a Circle. 

When Area is given. Rule. —Divide area by square of diameter of circle, 
take tab. height for area from table of Areas of Segments of a Circle, p. 267, 
multiply it by diameter, and product will give required height. 

From diameter subtract height, multiply remainder by height, take square 
root of product and multiply it by 2 for required chord. 

Or, = (tab. area for height) xd = h, and V d — /t X ^ X 2 = c. 
Circular Measure. (See Rule, page 113.) 


Sphere. 

Definition. —A figure, surface of which is at a uniform distance from centre. 
To Compute Convex Surface of a Sphere.—Fig. S 3 . 


Fig. 23. 



Rule. —Multiply diameter by circumference, and prod¬ 
uct will give surface. 

Or, 4 pr’ 2 — surface. * Or, p d 2 = surface. 

Example. —What is convex surface of a sphere, Fig. 23, hav¬ 
ing a diameter, a b , of 10 ins? 

10 X 31. 416 = 314.16 square ins. 


Segment of a Sphere. 
Definition. —A section of a sphere. 


To Compute Surface of a Segment of a Sphere.—Fig. 24 r. 

Rule. —Multiply height by the circumference of sphere, and add product 
to the area of base. 



Fig. 24. 


Or, 2 prh — convex surface alone. 

Example. —Height, b 0, of a segment, a b c, Fig. 24, is 36 ins., 
and diameter, b e, of sphere 100; what is convex surface, and 
what whole surface? 

36 X 100 X 3-1416 = ri 309.76 = height of segment multiplied by 
circumference of sphere. 

To ascertain area of base; diameter and versed sine being 
given, diameter of base of segment, being equal to chord of arc, 
is, by Rule, page 347, 

100 — 36 X 2 = 28; V100 2 — 28 2 = 96. 


96 s X -7854 = 7238.2464 = convex surface, and 7238.2464 + 11 309.76 = 18548.0064 
= convex surface added to area of base = square ins. 

Note.—W hen convex surface of a figure alone is required, area or areas of base 
or ends must be omitted. 


* p or rr represents in this, and in all cases where it is used, ratio of circumference of a circle to its 
diameter, or 3.1416. 









348 MENSURATION OF AREAS, LINES, AND SURFACES. 


When the Diameter of Base of Segment and Height of it are alone given. 
Rule. —Add square of half diameter of base to the square of height; divide 
this sum by height, and result will give diameter of sphere. 


Or, d ■— 2 + h 2 -r-li — diameter. 

Spherical Zone (or Frustum of a Sphere). 
Definition.— The part of a sphere included between two parallel chords. 

To Compute Surface of a Spherical Zone.-Fig. 25. 

Rule.—M ultiply height by the circumference of sphere, 
and add product to area of the two ends. 

Or, h c 4 a -f- a' — surface. 

Or, 2 pr h — convex surface alone. 



Example. — Diameter of a sphere, a b, Fig. 25, from which a 
zone, c g , is cut, is 25 inches, and height, eg, is 8; what is convex 
surface ? 


25 X 3-1416 X 8 = 628.32 =z height X circumference of sphere = square ins. 


When the Diameter of Sphere is not given. Rule. —Multiply mean length 
of the two chords by half their difference; divide this product by breadth 
of zone, and to quotient add breadth. To square of this sum add square of 
lesser chord, and square root of their sum will give diameter of sphere. 


Or 


’ \! (i±i x ~-^6+6+r 2 ) =d 


Spheroids or Ellipsoids. 


Definition. —Figures generated by the revolution of a semi-ellipse about one of 
its diameters. 

When revolution is about Transverse diameter they are Prolate, and when it is 
about Conjugate they are Oblate. 


To Compute Surface of a Spheroid.—Fig. 26. 

When Spheroid is Prolate. Rule. — Square diameters, and multiply 
square root of half their sum by 3.1416, and this product by conjugate 


diameter. 



or.y 


d 2 +d’ 2 


X 3-1416 Xd — surface , d and d' represent¬ 


ing conjugate and transverse diameters. 


t Example.— A prolate spheroid, Fig. 26, has diameters, cd 
and a b, of 10 and 14 inches; what is its surface? 


10 2 + i4 z = 296 = sum of squares of diameters. 

296-= 2 = 148, and f 148 '= 12.1655 = square root of half 
sum of squares of diameters. 

12.1655 X 3-1416 X 10= 382.191 ins. —product of root above obtained X 3-1416, 
and by conjugate diameter. 


When Spheroid is Oblate. Rule. —Square diameters, and multiply square 
root of half their sum by 3.1416, and this product by transverse diameter. 

Or,/!+“'* 


X 3-1416 X d' — surface. 


Example. —An oblate spheroid has diameters of 14 and 10 inches; what is its 
surface ? 

12 2 -j- io 2 = 296 = sum of squares of diameters. 

296 -T- 2 = 148, and -f 148 = 12.1655 = square root of half sum of squares of di¬ 
ameter. 


12.1655 X 3-1416 X 14 = 535-0679 ins.=product of root above obtained X 3-1416, 
and by transverse diameter. 














MENSURATION OF AREAS, LINES, AND SURFACES. 349 


To Compute Convex Surface of a Segment of a Splie— 
roid..—Figs. ST and. 28. 

Rule. —Square diameters, and take square root of half their sum; then, 
as diameter from which the segment is cut is to this root, so is the height 
of segment to proportionate height required. Multiply product of other di¬ 
ameter and 3.1416 by proportionate height of segment, and this last product 
will give surface. 


Or, 


Vd 2 -j-d' 2 -h 2 


X h X d' or d X 3- 1416 — surface. 


Fig. 27. 


d or d' 

Example. — Height, a o, of a seg- 


Fig. 28. 



. —]b 



meut, ef of a prolate spheroid, Fig. 

27, is 4 inches, diameters being 10 and 
14; what is convex surface of it? 

Square root of half sum of squares 
of diameters, 12.1655. 

Then 14: 12.1655 :*. 4: 3.4758 — height 
of segment , proportionate to mean of 
diameters , and 10 X 3-1416 X 3-4758 = 109.1957 ins. 

2.—Height, c o, of a segment of an oblate spheroid, Fig. 28, is 4 inches, the diam¬ 
eters being 14 and 10; what is convex surface of it? 214.0272 square ins. 

To Compute Convex Surface of a Frustum or Zone 
of a Splieroid.-—Figs. SO and 30. 

Rule.—P roceed as by previous rule for surface of a segment, and obtain 
proportionate height of frustum; then multiply product of diameter par¬ 
allel to base of frustum and 3.1416 by proportionate height of frustum, and 
it will give surface. 


Fig. 29. 




Example. —Middle frustum, 0 e, of Fig. 30. 

a prolate spheroid, Fig. 29, is 6 inch¬ 
es, diameters of spheroid being 10 
and 14; what is its convex surface? 

^ Mean diameter, as per preceding 
example, is 12.1655. 

Diameter parallel to base of frus¬ 
tum is 10. 

Then 14 : 12.1655:*. 6: 5.2138, and 10 X 31416 X 5.2138= 163.7967 square ins. 

2._Middle frustum of an oblate spheroid, as 0 e , Fig. 30, is 2 inches in height, 

diameters of spheroid, as in preceding examples, being 10 and 14; what is its con¬ 
vex surface? 107.0136 square ins. 

Circnlar Zone. 

Definition. —A part of a circle included between two parallel chords. 

To Compute Area of a. Circmlar Zone. 


Rule. —From area of circle subtract areas of segments. 

Or, see Table of Areas of Zones, page 269. 

When Diameter of Circle is not given. —Multiply mean length of the two 
chords by half their difference; divide this product by breadth of zone, and 
to quotient add the breadth. 

To square of this sum add square of lesser chord, and square root of their 
sum will give diameter of circle. 

Example.— Greater chord, h g, is 90 inches; lesser, a c, is 80; and breadth of zone. 
a 0 , is 72.526; what is its diameter? 

80 ? - X 90 ^ 80 = 85 X 5 = 425, and 425 -~ + 72.526 = 78.385. 

2 2 72.526 

Then V78.385 2 + 80 2 - y/12 544.2 = 112 = diameter. 

Gg 























350 MENSURATION OF AREAS, LINES, AND SURFACES. 


Cylinder. 

Definition.— A figure formed by revolution of a riglit-angled parallelogram around 
one of its sides. 

To Compute Surface of a Cylinder.—Fig. 31 . 

Rule. —Multiply length by circumference, and add product to area of 
the two ends. 


Fig. 31. 



Or, l c -f- 2 a = s, a representing area of end. 

Note.—W hen internal or convex surface alone is wanted, areas of 
ends are omitted. 

Example.— Diameter of a cylinder, b c, Fig. 31, is 30 inches, and its 
length, a b, 50; what is its surface? 

30 x 3-1416 = 94.248, and 94.248 X 50 = 4712.4. 

Then 30 2 X .7854 = 706.86 =area of one end; 706.86 X 2 = 1413.72 
= area of both ends , and 4712.4-1-1413.72 = 6125.12 square ins. 

Prisms. 

Definition. —Figures, sides of which are parallelograms, and ends equal and 
parallel. 

Note. —When ends are triangles, they are termed triangular prisms ; when they 
are square, square or right prisms ; and when they are a pentagon, pentagonal 
prisms, etc. 

To Compute Surface of a Piglat Prism.—Figs. 33 and 33 . 
Rule. — Ascertain areas of ends and sides, and 
add them together. 

Or, 2 a -j- n a' = s, a representing area of ends, a' area 
of sides, and n their number. 

Example. —Side, a b, Fig. 32, of a square prism is 12 
inches, and length, be, 30; what is surface? 

12X12 = 144 = area of one end ; 144 X 2 = 288 = area 
of both ends ; 12 X 30 = 360 = area of one side ; 360 X 4 = 

1440 = area of four sides, and 288 -j- 1440 = 1728 sq. ins. 

To Compute Surface of an OL>liq.ue or Irregular Prism.- 

Pig. 3 - 1 . 




Fig. 34 - 



Rule. —Multiply circumference of one end, by perpenclic- 
a ular height, a o. Or, multiply circumference, c, at a right 
angle to sides by actual length of figure, and add area of ends. 

Example. —Sides, a c, of an oblique hexagonal prism, Fig. 34, are 
10 inches, and perpendicular height, a 0, is 5 feet: what is its sur¬ 
face ? 

10 X 6 = 60 ins. — length of sides. 


60 X 5 X 12 = 3600 square ins.—area of sides, and by table, page 
342, 102 X 2.598 08 X 2 = 519.616 square ins., which added to 3600 = 
4119.6x6 square ins. 

"Wedge. 

Definition.—A wedge is a prolate triangular prism, and its surface is computed 
by rule for that of a right prism. 

To Compute Surface of a Wedge.—Pig. 35 . 

Example. —Back of a wedge, abed, Fig. 35, is 20 by 2 inches, 
and its end, ef 20 by 2; what is its surface ? 

__ 2 

2 ° 2 + 2=1= 401 = sum of squares of half base, a f, and 
height, ef,of triangle, efa. 

V 4 01 — 20.025 = square root of above sum — length of e a. 
Then 20.025X20X2 = 8012= area of sides. 

And 20X2=40 = area of back; and 20 X 2 = 2 X 2 = 40 = 
area of ends. Hence 801 + 40 -f 40 = 881 square ins. 
































MENSURATION OF AREAS, LINES, AND SURFACES. 35 I 



Prismoids. 

Definition. —Figures alike to a prism, having only one pair of sides parallel. 

To Compute Svirface of a Prismoid.— Fig. 36 . 

Rule. — Ascertain area of sides and ends as by rules for 
squares, triangles, etc., and add them together. 

Example.— Ends of a prismoid, efghandabcd , Fig. 36, are ioand 
8 inches square, and its slant height, d h , 25; what is its surface? 

10X10=100 = area of base ; 8 X 8 = 64 = area of top. 

10 8 _ . 

- 1 — X 25 — 225, and 225 X 4 — 900 = area of sides. 

2 

Then 100 -j- 64 -j- 900 = 1064 = square ins. 

To Compute Surface of art ODlique or Irregular Prismoid. 
Proceed as directed for an Oblique or Irregular Prism, page 350. 

XT ngnlas. 

Definition.— Cylindrical ungulas are the parts (including ail or part of the base) 
left by a plane cutting a cylinder through any portion and at any angle. 

To Compute Curved Surface of an Ungula.—Figs. 37 , 

38 , 39 , and 40 . 

When Section is parallel to Axis of the Cylinder , Fig. 37. Rule i.—M ul¬ 
tiply length of arc of one end by height. 

Example.— Diameter of a cylinder, a c, from which an 
ungula, Fig. 37, is cut, is 10 inches, its length, b d, 50, and 
versed sine or depth of ungula is 5 inches; what is curved 
surface ? 

io- 4 - 2 = 5 = radius of cylinder. 

Hence radius and versed sine are equal; the arc, there¬ 
fore, of ungula is one half circumference of the cylinder, 
which is 31.416 4 2 = 15.708, and 15.708 X 5 ° = 785-4 
square ins. 

When Section passes obliquely through opposite Sides of Cyl¬ 
inder , Fig. 38. Rule 2.—Multiply circumference of base of cylinder by 
half sum of greatest and least heights of ungula. 

Example.— Diameter, cd, of a cylindrical ungula, Fig. 38, is 10 inches, and great¬ 
er and less heights, bd and ac, are 25 and 15 inches; what is its curved surface? 

10 diameter — 31.416 circumference ; 25 + 15 = 40, and 40-4- 2 = 20. Hence 31.416 
X 20 = 628.32 square ins. 

When Section passes through Base of Cylinder and one of its Sides , and 
Versed Sine does not exceed Sine , or Base is equal to or less than a Semi¬ 
circle, Fig. 39. Rule 3.— Multiply sine, a d, of half arc, d g, of base, d g f 
by diameter, e g, of cylinder, and from this product subtract product * of arc 
and cosine, a 0. Multiply difference thus found by quotient of height, g c , 
divided by versed sine, a g. 

Note.— The sine of base is half of the longest chord that can be drawn in base. 

Example.— Sine, a d, of half arc of base of an ungula, Fig. 39, is 5, 
diameter of cylinder, eg, is 10, and height, eg, of ungula 10 inches; 



Fig. 38. 



Fig. 39 - 


what is curved surface? 

5 X 10 = 50 = sine of half arc by diameter. 

Length of arc, versed sine and radius being equal, under Rule, page 
346 = 15.708, and as versed sine and radius are equal, cosine is o. 

A,. Hence, when cosine is o, product is o. Therefore 50 — 0 = 50 = dif- 
' ference of product before obtained and product of a 1 c and cosine, and 

50 X 10-7-5 = 50 X 2 = 100 square ins. 


* When the cosine is o, this product is o. 






















352 MENSURATION OF AREAS, LINES, AND SURFACES. 


When Section passes through Base of Cylinder , and Versed Sine , a g, ex¬ 
ceeds Sine , or when Base exceeds a Semicircle , Fig. 40. Rule 4. —Multiply 


Fig. 40. 



sine of half the arc of base by diameter of cylinder, and to this 
product add product of arc and the excess of versed sine over 
the sine of base. Multiply sum thus found by quotient of 
height divided by versed sine. 

Example.— Sine, a d, of half arc of an ungula, Fig. 40, is 12 inches; 
versed sine, a g, is 16; height, c g, 16; and diameter of cylinder, h g, 
25 inches; what is curved surface? 

12 X 25 = 300 = sine of half arc by diameter of cylinder , and length 
of arc of base, Rule, page 344 = arc ofd hf—circumference of base — 
46. 392. 

Then 46.392 X 16 —12.5 = 162.372, and 300-}-162.272 = 462.372; 16 - 4 - 16 = 1, and 
462.372 X 1 = 462.372 square ins. 

Note.—W hen sine of an arc is o, the versed sine is equal to diameter. 

When Section passes obliquely through both Ends of Cylinder , 
Fig. 41. Rule 5.— Conceive section to be continued to m, till it 
meets side of cylinder produced; then, as difference of versed 
sines, a e and d 0, of arcs of two ends of ungula is to versed sine, 
a e , of arc of the less end, so is height of cylinder, a d, to the 
part of side produced. 

Ascertain surface of each of ungulas thus found by Rules 3 
and 4, and their difference will give curved surface. 

Lune. 

Definition.— Space between intersecting arcs of two eccentric circles. 

To Compute Area of a I.une.—Fig. 42 . 

Rule. —Ascertain areas of the two segments from which lune is formed, 
and their difference will give area. 



Fig. 42. 



Example.— -Length of chord ac, Fig. 42, is 20 inches, height 
e d is 3, and eb 2; what is area of lune ? 

By Rule 2, page 345, diameters of circles of which lune is 
formed are thus ascertained: 


For a d c, 


io2 + (3 + 2 ) 


25 - 


For aec. 


! + 2 : 


52 - 


Then, by Rule for Areas of Segments 
of a Circle, page 267, 


Area of adc is 70.5577 sa. ins. 
“ aec “ 27.1638 “ 


Their difference 43.3939 sq. ins. 

Note. — If semicircles be described on the three sides of a right-angled triangle 
as diameters, two lunes will be formed, and their united areas will be equal to that 
of triangle. 

Cycloid. 

Definition. —A curve generated by revolution of a circle on a plane. 

Compute Area, of a Cycloid.—Fig. 43. 

Rule.— Multiply area of generating circle by 3. 

\ Example.— Generating circle of a cycloid, abc , Fig. 43. 

; has an area of 115.45 sq. inches; what is area of cycloid ? 
-''c 115.45 X 3 = 346-35 square ins. 

Compute Length of a Cycloidal Curve. 


Fig. 43 - 



To 


Rule. —Multiply diameter of generating circle by 4. 

Example.— Diameter of generating circle of a cycloid, Fig. 43, is 8 inches; what 
is length of curve d s c ? 

8 X 4 = 32 = product of diameter and 4 = ins. 

Note.—T he curve of a cycloid is line of swiftest descent; that is, a body will fall 
through arc of this curve, from one point to another, in less time than through any 
other path. 
























MENSURATION OP AREAS, LINES, AND SURFACES. 353 


Circnlar Rings. 

Definition. —Space between two concentric circles. 

To Compute Sectional Area of a Circular Ring.—-Fig. 44. 
Rule.—F rom area of greater circle subtract that of less. 


Cylindrical Rings. 

Definition. —A ring formed by curvature of a cylinder. 

To Compute Surface of a Cylindrical Ring.—Fig. 44 . 

Rule. —To diameter of body of the ring add inner diameter of the ring; 
multiply this sum by diameter of the body, and product by 9.8696. 

Or, c X l = surface. 

Example. —Diameter of body of a cylindrical ring, a b , Fig. 44, 
is 2 inches, and inner diameter, b c, is 18; what is surface of it? 

2 + 18 — 20 = thickness of ring added to inner diameter. 

20 X 2 X 9- 8696 — sum above obtained X thickness of ring, and 
that product by 9.8696 = 394.784 ins. 



Link. 


Definition. —An elongated ring. 




To Compute Surface of a Link.— Figs. 45 and. 46. 
Rule. —Multiply length of axis of link by circumference of a section of 
body, a b. 

Or, Z X c = surface. 

To Compute Length, of Axis and Circumference. 

When Ring is Elongated. Rule.— To less diameter add the diameter of 
the body of'the link, and multiply sum by 3.1416; subtract less diameter 
from greater, multiply remainder by 2, and sum of these products is length 
of axis.. 

Example.— Link of a chain, Fig. 45, is 1 inch in diameter 
of body, a b, and its inner diameters, b c and ef are 12.5 
and 2.5 inches; what is its circumference? 

2.5I1X 3.1416= 10.9956 — length of axis of ends. 

12.5-2.5X2 = 20= length of sides of body. t 

Then 10.9956 + 20 = 30.9956 = length of axis of link, and 
30.9956 X 3.1416 (cir. of 1 inch) = 97.3758 square ins. 

When Ring is Elliptical , Fig. 46. Rule.— Square diameters of axes of 
ring, multiply square root of half their sum by 3.1416, and product is length 
of axis. 

Cones. 

Definition. — A figure described by revolution of a right-angled triangle about 
one of its legs. 

For Sections of a Cone, see Conic Sections, page 379. 

To Compute Surface of a Cone.—Fig. 47^. 

Rule. —Multiply perimeter or circumference of base by slant height, or 
side of cone; divide product by 2, and add the quotient to area of the base. 

Or, c X h -4- 2 + a' = surface, c representing perimeter. 

Example. —Diameter, a b , Fig. 47, of base of a cone is 3 feet, 
and slant height, a c, 15; what is surface of cone? 

, 9.4248 X 13 

Circum. of 3 feet = 9.4248, and — - = 70.686 = sur¬ 

face of side; area of base 3=7.068, and 70.686+7.068 = 77.754 
square feet. 

G G* 

















254 MENSURATION OF AREAS, LINES, AND SURFACES. 


To Compute Svirface of the Frustum of a Cone.— 

Fig. 48. 

Rule.—M ultiply sum of perimeters of two ends by slant height of frus¬ 
tum ; divide product by 2, and add it to areas of two ends. 

^ c + c' X h , , . 

Or — 1 -b a + a = surface. 

2 

Example.— Frustum, abed , Fig. 48, has a slant height, c d , of 26 inches, and 
circumferences of its ends are 15.71 and 22 inches respectively; 
what is its surface ? 


Fig. 48 





15 '- 7 — 2 2 * = 490.23 = surface of sides ; * - 7^54 

(———\ x .7854 = 58.119 = areas of ends. Then 490.23 + 
\ 3 -1416/ 

58.119 = 548.349 square ins. 

Pyramid s. 

Definition.— A figure, base of which has three or more sides, and sides of which 
are plane triangles. 

To Compute Surface of a Pyramid.—Figs. 49 and £50. 

Rule.—M ultiply perimeter of base by slant height; divide product by 2, 
and add it to area of base. 

Fig- 49 - o Or a = surface. ^ ° 5 ° 

2 

Example.— Side of a quadrangular pyramid, a b, 

Fig. 49, is 12 inches, and its slant height, a c, 40; 
what is its surface? 

12 X 4 = 48 —perimeter of base. 4 ° = 960 

2 

b area of sides, and 12 X 12 + 960= 1104 square ins. 

To Compute Surface of Frustum of a Pyramid.— 

Fig. SI. 

Rule.—M ultiply sum of perimeters of two ends by slant height; divide 
product by 2, and add it to areas of ends. 

c + c' x h , 

Or,-p a + a = surface. 

2 

Example. —Sides ab,cd , Fig. 51, of frustum of a quadrangular 
pyramid are 10 and 9 inches, and its slant height, a c, 20; what 
is its surface? 

10X4 = 40, and 9 X 4 = 36; 40 + 36 = 76 = sum of perimeters. 

76 x 20 = 1520, and — 760 = area of sides ; 10X10 = 100, 
2 

and 9 X 9 = 81. Then 100+ 81 + 760 = 941 = square ins. 

When Pyramid is Irreyular sided or Oblique. Rule. — The surfaces of 
each of the sides and ends must be computed and added together. 

Helix (Screw). 

Definition.— A line generated by progressive rotation of a point around an axis 
and equidistant from its centre. 

To Compute Length, of a Helix.—Fig. 52. 

Rule.—T o square of circumference described by generating point, add 
square of distance advanced in one revolution, extract square root of their 
sum, and multiply it by number of revolutions of generating point. 



















MENSURATION OF AREAS, LINES, AND SURFACES. 355 


Fig. 52. Or, + (p~ -f -l 2 )n = length , n representing number of revolutions. 

Example. —What is length of a helical lino, Fig. 52, running 3.5 
times around a cylinder of 22 inches in circumference, and advancing 
16 inches in each revolution? 

22 2 4-16 2 = 740 '= sum of squares of circumference and of distance 
advanced. * Then -f 740 X 3- 5 = 95.21 ins. 

To Compute Length of a Revolution of Thread of a 

Screw. 

Rule. —Proceed as above for length and omit number of revolutions. 

Spirals. 

Definition. —Lines generated by the progressive rotation of a point around a 
fixed axis. 

A Plane Spiral is when the point rotates around a central point. 

A Conical Spiral is when the point rotates around an axis at a progressing dis¬ 
tance from its centre, as around a cone. 

To Compute laengtli of a Plane Spiral Line.—Fig. 5S. 

Rule. —Add together greater and less diameters; divide their sum by 2 ; 
multiply quotient by 3.1416, and again by number of revolutions. 

Or, when circumferences are given, take their mean length, and multiply 
it by number of revolutions. 

Or, d -(- d' - 4 - 2 X 3-1416 n = length of line; P x n = radius , and 
pr 2 -r-l —pitch. P representing the pitch. 

Example. —Less and greater diameters of a plane spiral spring, 
as a b, c d , Fig. 53, are 2 and 20 inches, and number of revolutions 
d 10; what is length of it? 

2 + 20=2 = 11 = sum of diameters - 4 - 2; n X 3.1416 = 34.5576 
and 34.5576 X 3-1416. 

Then 34.5576 X 10= 345.576 inches. 

Note.—A bove rule is applicable to winding engines, see page 862, where it is re¬ 
quired to ascertain length of a rope, its thickness, number of revolutions, diameter 
of drum, etc. 

To Compute Length, of a Conical Spiral Line.—Fig. 54r. 

Rule. —Add together greater and less diameters; divide their sum by 
2, and multiply quotient bv 3.1416. 

To square of product of this circumference and number of revolutions of 
spiral, add square of height of its axis, and take square root of the sum. 

_ 2 

Fig. 54. Or, V(d-j-d'-h 2 X 3.1416 n-\-h s ) = length of line. 

Example. —Greater and less diameters of a conical spiral, Fig. 54, are 
20 and 2 inches; its height, c d, 10; and number of revolutions 10; what 
is length of it? 

20+ 2 = 2 = 11 X 3-1416 = 345576 = sum of diameters - 4 - 2, and X 
3.1416; 35-5576 X 10 = 345.576. 

Then V 345.576 s + 10 2 = 345.72 inches. 

Spin, clles. 

Definition. — Figures generated by revolution of a plane area, when the curve is 
revolved about a chord perpendicular to its axis, or about its double ordinate, and 
they are designated by tire name of the arc or curve from which they are generated, 
as Circular, Elliptic, Parabolic, etc. 



Fig- 53 - 




* When the spiral is other than a line, measure diameters of it from middle of body composing it. 






356 MENSURATION OF AREAS, LINES, AND SURFACES 


Circular Spindle. 

To Compute Convex Surface of a Circular Spindle, Zone, 
or Segment of it.— Figs. 55, 56, and 5>T. 

Rule.— Multiply length by radius of revolving arc; multiply this arc by 
central distance, or distance between centre of spindle and centre of revolv¬ 
ing arc; subtract this product from former, double remain¬ 
der, and multiply it by 3.1416. 



Or, L r — (a.^yV 2 — ^ ) 2 p — surface, a representing length 

of arc, and c the spindle chord. 

Example. —What is surface of a circular spindle, Fig. 55, length 
of it, fc, being 14.142 inches, radius of its arc, 0 c, 10, and central 
-*' distance, 0 e, 7.071 ? 

14.142 X 10 = 141.42 = length X radius. Length of arc,fa c, by Rules, page 344 
= 15.708. 

15.708 X 7-071 = 111.0713 = length of arc X central distance ; 141.42 — m.0713 = 
30.3487 = difference of products. Then 30.3487 X 2 X 3-1416 = 190.687 square ins. 


k- 


Zone. 

Example.— What is convex surface of zone of a circular 
spindle, Fig. 56, length of it, i c, being 7.653 inches, radius of 
its arc, 0 g, 10, central distance, o e, 7.071, and length of its 
side or arc, d b, 7.854 inches? 

7.653X10 = 76.53 = length X radius; 7.854 X 7- Q7 1 — 55- 5356 
= length of arc X central distance ; 76.53 — 55-5356 = 20.9944 
= difference of products. 

Then 20.9944 X 2 X 3-1416 = 131.912 square ins. 



\ ' 
V 

O 


Segment. 

Example.— What is convex surface of a segment of a cir¬ 
cular spindle, Fig. 57, length of it, i c, being 3.2495 inches, 
radius of its arc, 0 g, 10, central distance, 0 e, 7.071, and length 
of its side, id, 3.927 inches? 

3.2495 x 10 = 32.495 = length x radius; 3.927 x 7-071 = 27.7678 = length of arc 
X central distance ; 32.495 — 27.7678 = 4.7272 = difference of products. 

Then 4.7272 X 2 X 3.1416 = 29.702 square ins. 



General Formula. —S = 2 (Ir — a c) p = surface, l representing length of spindle, 
segment, or zone, a length of its revolving arc , r radius of generating circle, and c 
central distance. 

Illustration. —Length of a circular spindle is 14.142 inches, length of its revolv¬ 
ing arc is 15.708, radius of its generating circle is 10, and distance of its centre from 
centre of the circle from which it is generated is 7.071; what is its surface? 


2 X (14.142 X 10 —15.708 X 7-071) X 3.1416 = 190.687 square inches. 

Note.—S urface of a frustum of a spindle may be obtained by division of the 
surface of a zone. 

Cycloidal Spindle. 

To Compute Convex Surface of a Cycloidal Spindle.— 

Wig. 58. 

Rule. —Multiply area of generating circle by 64, and divide it by 3. 

Or = surface. 

Example. —Area of generating circle, a b c, of a cycloidal 
spindle, d e, is 32 inches; what is surface of spindle ? 

32 X 64 = 2048 = area of circle X 64 , and 2048 -4- 3 = 
682.667 square ins. 

Note.—A rea of greatest or centre section of a cycloidal spindle is twice area of 
the cycloid. 


Fig- 58, 












MENSURATION OF AREAS, LINES, AND SURFACES. 357 


Ellipsoid., Paraboloid, or Hyperboloid of Rev¬ 
olution. 


Definition.— Figures alike to a cone, generated by revolution of a conic section 
around its axis. 


Note. —These figures are usually known as Conoids. 

When they are generated by revolution of an ellipse, they are termed Ellipsoids, 
and when by a parabola, Paraboloids, etc. 

Revolution of an arc of a conic section around the axis of the curve will give a 
segment of a conoid. 

Ellipsoid. 

To Compute Convex Surface of ail Ellipsoid.—Eig. 50. 

Rule. — Add together square of base and four times square of height; 
multiply square root of half their sum by 3.1416, and this product by radius 
of the base. 



or V 


fc 2 +4/v 


3.1416 r — surface. 


Example. —Base, a b, of an ellipsoid, Fig. 59, is 10 inches, and 
vertical height, cd, 7; what is its surface? 

io 2 -|-7 2 X 4 = 29 6 = sum of square of base and 4 times square 
of height; 296 - 4 - 2 = 148, and f 148 = 12.1655 =square root of half 

above sum. Then 12.1655 X 3-1416 X — = 191.0957 square ins. 


To Compute Convex Surface of a Segment, Frustum, 
or Zone of an Ellipsoid.—Eig. 50. 

See Rules for Convex Surface of a Segment, Frustum, or Zone of a 
Spheroid or Ellipsoid, pages 348-9. 

d or d' X 3.1416 X h — surface , 

, mean. diam. X h , 

and —- -- - = h ; then d x 3. 1416 Xh — surface. 


Earaboloid. 


To Compute Convex Surface of a EaraTooloid.— Eig. 60. 

Rule. —From cube of square root of sum of four times square of height, 
and square of radius of base, subtract cube of radius of base; multiply re¬ 
mainder by quotient of 3.1416 times radius of base divided by six times 
square of height. 

Or, (V4/1 2 -j- r 2 )3 — ? -3 x e'y — surface. 

Example.—A xis, b d, of a paraboloid, Fig. 6o, is 40 inches; ra¬ 
dius, a d , of its base is 18 inches; what is its convex surface? 

40 2 X 4 — 6400 =: 4 times square of height; 6400 + 18 2 m 6724 = 
sum of above product and square of radius of base ; (f 6724)3 — 18 3 
545 53 6 —remainder of cube of radius of base subtracted from cube 
of square root of preceding sum ; 3.1416 X 18 - 4 - (6 X 40 2 ) .005 890 5 

— quotient 0/3.1416 times radius ofbase-h -6 times square of height. 

Then 545 536 X -005 890 5 = 3213.48 square ins. 



Fig. 61. a. 



'T?! 

1 _ 


rf 





Cylinder Sections. 

To Compute Surface of a Cylinder Section. 

—Eig. 61. 

Rule. — From entire surface of cylinder a 0 subtract 
surface of the two ungulas, r 0, 0 c, as per rule, page 351, 
and multiply result by 4. 




























358 MENSURATION OF AREAS, LINES, AND SURFACES. 


Any Figure of Revolution. 

To Ascertain Convex Surface of any Figure of Revolu¬ 
tion.—Figs. 62, 63, and. 64. 

Rule. —Multiply length of generating line by circumference described 
by its centre of gravity. 

Or, l 2 r p = surface, r representing rad ius of centre of gravity. 

Example i.— If generating line, ac, of cylinder, acdf 10 inches 
in diameter, Fig. 62, is 10, then centre of gravity of it will he in b, 
radius of which is b r — 5. 



Hence 10 X 5 X 2 X 3-1416 = 314.16 ins. 

Again, if generating line is e a c g, and it is (e a — 5, a c = 10, 
and eg — 5) = 20, then centre of gravity, o, will be in middle of 
line joining centres of gravity of triangles e ac and ac g — 3.75 
from r. 


Hence 20 X 3.75 X 2 X 3.1416 = 471.24 square ins.—entire surface. 


Verification. 


Fig. 63. 



j Convex surface as above.314.16 

(Area of each end, 10 2 X -7854 X 2 = .157.08 

471.24 inches. 

2.—If generating elements of a cone, Fig. 63, are a d = 10, 
d c = 10, and a c, generating line, = 14.142, centre of gravity of 
which is in 0, and or — 5, 

:444.28s, con- 


Then 14.142 X 5 X 2 X 3.1416: 


Fig. 64. 
a 


vex surface , and 10 x 2 x .7854 = 314.16, area 
of base. 

Hence 444.285 -f- 3x4.16 = 758.445, entire surface. 

3.—If generating elements of a sphere, Fig. 64, are a c = 10, a b c 
will be 15.708, centre of gravity of which is in o, and by Rule, page 
606, or — 3.183. 

Hence 15.708 X 3-183 X 2 X 3.1416 = 314.16 square ins. 



Capillary Tube. 

To Compute Diameter of a, Capillary Tuloe. 

Rule. —Weigh tube when empty, and again when filled with mercury; 
subtract one weight from the other; reduce difference to grains, and divide 
it by length of tube in inches. Extract square root of this quotient, multi¬ 
ply it by .019 224 5, and product will give diameter of tube in inches. 

V w 

— X -019 224 5 = diameter, w representing difference in weights in grains 
and l length of tube. 

Example. —Difference in weights of a capillary tube when empty and when filled 
with mercury is 90 grains, and length of tube is 10 inches; what is diameter of it? 

90 -f-10 = 9 = weight of mercury - 4 - length of tube ; ffg = 3, and 3 x -019 224 5 = 
•°57 673 5 = square root of above quotient X -019 224 5 inches — diameter of tube. 

Proof.— Weight of a cube inch of mercury is 3442.75 grains, and diameter of a 
circular inch of equal area to a square inch is 1.128 (page 342). 

If, then, 3442.75 grains occupy 1 cube inch, 90 grains will require .0261419 cube 
inch, which, - 4 - 10 for height of tube = .002 614 19 inch for area of section of tube. 
Then ff.oo-2 614 19 = .051129 = side of square of a column of mercury of this area. 
Hence .051129 X 1.128 (which is ratio between side of a square and diameter of a 
circle of equal area) = .057 673 5 ins. 

To Ascertain Area of air Irregular Figure. 

Rule. —Take a uniform piece of board or pasteboard, weigh it, cut out 
figure of which area is required, and weigh it; then, as weight of board or 
pasteboard is to entire surface, so is weight of figure as cut out to its surface. 
Or, see rule page 341, or Simpson’s rule, page 342. 



















MENSURATION OF AREAS, LINES, SURFACES, ETC. 359 


To Ascertain Area of any Plane Tfignre. 

Rule. — Divide surfaces into squares, triangles, prisms, etc.; ascertain 
their areas and add them together. 

Redaction, of an Ascending or Descending Line to Hor¬ 
izontal Measurement. 

In Link and Foot. 


Degrees. 

Link. 

Foot. 

Degrees. 

Link. 

Foot. 

Degrees. 

Link. 

Foot. 

I 

. OOO O99 

.00015 

7 

.004 9I7 

.007 45 

13 

.016915 

.025 63 

2 

. OOO 4O3 

. 000 61 

8 

.006 421 

.00973 

14 

.019 602 

.029 7 

3 

. 000 904 

.001 37 

9 

.008125 

.012 31 

15 

.022 486 

.03407 

4 

.001 61 

.002 44 

IO 

.010025 

.015 19 

l6 

.025 569 

.038 74 

5 

.002 515 

.003 81 

II 

.012 I24 

.01837 

W 

.028 925 

•0437 

6 

.OO3 617 

.005 48 

12 

.014 421 

.022 85 

18 

•0323 

.048 94 


Illustration i.—I n an ascending grade of 14 0 , what is reduction in 500 feet? 

14 0 = 500 X .0297 = 14.85 feet— 14 feet 10.2 ins. 

2.—What is reduction in 500 links? 

14 0 = 500 X -019 602 = 9.801 feet = 9 feet 9.6 ins. 

Redaction of Grade of an Ascending or Descending Line 

to Degrees. 

Per 100 Links, Feet , etc. 


Grade. 

Degrees. 

Grade. 

Degrees. 

Grade. 

Degrees. | 

Grade. 

Degrees. 

•25 

/ // 

8 35-2 

I -75 

Of ff 

i 0 10.3 

4-5 

Of ff 

2 34 45-5 

IO 

Of ff 

5 44 20.7 

•5 

17 10.3 

2 

1 8 45.5 

5 

2 51 57-6 

II 

6 18 55.8 

•75 

25 47-6 

2-5 

1 25 57.6 

6 

3 26 22.7 

12 

6 53 3 i 

I 

34 22.7 

3 

1 43 8.3 

7 

4 0 49.6 

13 

7 28 10.3 

1.25 

42 57-9 

3-5 

2 O 20.7 

8 

4 35 18.6 

14 

8 2 51.7 

i -5 

5 i 35-2 

4 

2 17 33 -i 

9 

5 9 49-6 | 

15 

8 37 37-2 


To Riot Angles 'witlioat a Rrotractor. 

On a given line prick off 100 with any convenient scale, and from the 
point so pricked off lay off at right angle with the same scale the natural 
tangent due to the angle (see table of Natural Tangents and Sines) ; or 
strike out a portion of a circle with radius 100 and lay off a chord = 2 sin. 
of half the angle required. 

To Compute Oliord. of an Angle. 

Double sine of half angle. 

Illustration.—W hat is the chord of 21 0 30'? 

21 w qo 

Sine of - : — = io° 45', and sine of io° 45' = .186 52, which, X 2 = .37304 chord. 

2 

To Ascertain Aalue of a Power of a Quantity. 

Rule. —Multiply logarithm of quantity by fractional exponent, and prod¬ 
uct is logarithm of required number. 

Example.— What is the value of 16% ? 

% x log. 16 = % X 1-20412 = .90309. Number for which = 8. 































360 


MENSURATION OF VOLUMES. 


MENSURATION OF VOLUMES. 

Cubes and. IParallelopipedoius. 
Cube. 

Definition. —A volume contained by six equal square sides. 


Fig. 1. 



To Compute Volume of* a Cube.— Fig. 1. 
Rule. —Multiply a side of cube by itself, and that product 
again by a side. 

Or, s 3 = V, s representing length of a side , and V volume. 
Example. —Side, a b, Fig. 1, is 12 inches; what is volume of it? 
12X12X12 = 1728 cube ins. 


Fig. 2. 



IParallelopipedoxi. 

Definition.—A volume contained by six quadrilateral sides, every opposite two 
of which are equal and parallel. 

To Compute Volume of a Farallelopipedon. 

—Fig. 2. 

Rule. —Multiply length by breadth, and that product 
again by depth. 

Or, lb d — V. 

Prisms, IPrismoids, and Wedges. 

Prisms. 

Definition. —Volumes, ends of which are equal, similar, and parallel planes, and 
sides of which are parallelograms. 

Note. —When ends of a prism or prismoid are triangles, it is termed a triangular 
prism or prismoid ; when rhomboids, a rhomboidal prism, and when squares, a 
square prism , etc. 


Fig. 3 . 


4 


To Compute Volume of a Prism.— 
Figs. 3 and. 4. 

Rule. —Multiply area of*base by height. 

Or, a h = V. 

Example. —A triangular prism, a b c, Fig. 4, has sides 
of 2.5 feet, and a length, c b, of 10; what is its volume? 

By Rule, page 339, 2.5 2 X -433 = 2.706 25 = area of 
end a b, and 2.70625 X 10 = 27.0625 cube feet. 


Fig. 4. 
a 


Fig- 5 - 



Fig. 6. 



When a Prism is Oblique or Irregular. 

Rule. — Multiply area of an end by height, as ao; or, 
multiply area taken at a right angle to sides, as at c, by 
actual length. 

To Compute Volume of any Frustum of a 
0 Prism, whether Ftiglat or OUlicLne. — Figs. 
6 and V. 

Rule. —Multiply area of base by perpendicular 
distances between it and centre of gravity of upper 
or other end. 

Or, area at right angle to side as at e by actual length. 

Example. —Area of base, a 0 , of frustum of a rectan¬ 
gular or cylindrical prism, Fig. 6, is 15 inches, and 
height to centre of gravity, c, is 12; what is its volume? 

10 X 12 = 120 cube ins. 


























































































MENSURATION OF VOLUMES. 



Prismoids.* 


To Compute Volume of a JPrismoid.—I^ig. 8. 

Rule.—T o sum of areas of the two ends add four times area of middle 
section, parallel to them, and multiply this sum by one sixth of perpendicu¬ 
lar height. 

Note. — This is the general rule, and known as the Prismoidal Formula , and it 
applies equally to all figures of proportionate or dissimilar ends. 

Or, a -f- a' -)- 4 m X h -1- 6 = V, a and a' representing areas of ends, 
and m area of middle section. 

Example. — What is volume of a rectangular prismoid, Fig. 8, 
lengths and breadths, e g aud g h, a b and b d, of two ends being 
7X6 and 3X2 inches, and height 15 feet ? 

7X64-3X2 = 42 4-6 — 48 = sum of areas of two ends ; 7 + 3 - 4 - 
2 = 5 = length of middle section ; 6-|-2-^2 = 4 = breadth of middle 
section ; 5 X 4 X 4 = 80 =four times area of middle section. 

Then 48 -(- 80 X ^ ^ = 128 X 30 = 3840 cube ins. 

6 

Note i.— Length and breadth of middle section are respectively equal to half 
sum of lengths and breadths of the two ends. 

2.—Prismoids, alike to prisms, derive their designation from figure of their ends, 
as triangular, square, rectangular, pentagonal, etc. 


Fig. 8. 

U b 



When it is Irregular or Oblique and their ends are united by plane or 
curved surfaces, through which and every point of them , a right line may be 
drawn from one of the ends or parallel faces to the other .— Figs. 9,10, and 11. 


Fig. xo. Fig. 11. 



Example.— Areas of ends, a c and 0 r s, Fig. 10, a b c d, and i m n u , Fig. u, and 
a b c e and v x w z. Fig. 9, are each 10 and 30 inches, that of their middle section 
20, and their perpendicular heights j8; what is their volume? 


10 -j- 30 -f- 20 X 4 = 120 

18 . 

120 X — = 360 cube ms. 


— sum of areas .of ends -(- 4 times middle section. 


And 


Wedge. 

To Compute Volume of a Wedge.—Fig. 13. 

Rule.— To length of edge add twice length of back; multiply this sum 
by perpendicular height, and then by breadth of back, and take one sixth 
of product. _ 


Fig. 


Or, (l + l'X2Xhb)-h6=zV. 

Example. —Length of edge of a wedge, e g , is 20 inches, back, 
abed, is 20 by 2, aud its height, ef 20; what is its volume? 

20 -(- 20 X 2 = 60 = length of edge added to twice length of 
back ; 60 X 20 X 2 == 2400 — above sum multiplied by height, and 
that product by breadth of back. 

Then 2400- 4 - 6 = 400 cube ins. 

Note. —When a wedge is a true prism, as represented by 
Fig. 12, volume of it is equal to area of an end multiplied by its length. 



* An excavation or embankment 
g\ilur prismoid. 


of a road, when terminated by parallel cross sections, is a rectan* 


H h 


























3 62 


MENSURATION OF VOLUMES. 


To Compute Frustum of a, "VVedge.—Fig. 13. 

Rule. —To sum of areas of both ends, add 4 times area 
of section parallel to and equally distant from both ends, 
and multiply sum by one sixth of length. 

Or, A + a + 4 a' X j = V. 

Example.—L engths of edge and back of a frustum of a wedge 
a b and c d are 20 X 1 and 20 X 2 ins., and height o r is 20 ins.; 
o a what is its volume? 

——— x 2 + 4 X (20 X — ) X -jr — 60 + 120 X g- = 600 cube ms. 

Note. —When frustum is a true prism, as represented Fig. 13, volume of it is equal 
to mean area of ends multiplied by its length. 

Regular Bodies (RolyliecLroias). 

Definition. —A regular body is a solid contained under a certain number of simi¬ 
lar and equal plane faces,* all of which are equal regular polygons. 

Note i.—W hole number of regular bodies which can possibly be formed is five. 

2.—A sphere may always be inscribed within, and may always be circumscribed 
about a regular body or polyhedron, which will have a common centre. 

Fig. 14. Fig. 15. Fig. 16. Fig. 17. 


1. Tetrahedron, or Pyramid, Fig. 14, which has four triangular faces. 

2. Hexahedron, or Cube, Fig. 1, which has six square faces. 

3. Octahedron, Fig. 15, which has eight triangular faces. 

4. Dodecahedron, Fig. 16, which has twelve pentagonal faces. 

5. Icosahedron, Fig. 17, which has twenty triangular faces. 

To Compute Elements of any Regular Body.—Figs. l4t, 

15, 1G, and. IV. 

To Compute Radius of a Sphere that will Circumscribe a given Regular 
Body , or that may be Inscribed within it. 

When Linear Edge is given. Rule. —Multiply it by multiplier opposite 
to body in columns A and B in following Table, under head of element re¬ 
quired. 

Example.— Linear edge of a hexahedron or cube, Fig. 1, is 2 inches; what are 
radii of circumscribing and inscribed spheres? 

2 X .86602 =; 1.73204 inches =1 radius of circumscribing sphere ; 2 X .5 = 1 inch — 
radius of inscribed sphere. 

When Surface is given. Rule. —Multiply square root of it by multiplier 
opposite to body in columns C and D in following Table, under head of 
element required. 

When Volume is given. Rule. — Multiply cube root of it by multiplier 
opposite to body in columns E and F in following Table, under head of ele¬ 
ment required. 





Angle of adjacent faces of a polygon is termed diedral angle. 


















MENSURATION OF VOLUMES. 


363 


When one of the Radii of Circumscribing or Inscribed Sphere alone is re¬ 
quired , the other being given. Rule. —Multiply given radius by multiplier 
opposite to body in columns G and H in Table, page 364, under head of 
other radius. 

To Compute Linear Edge. 

When Radius of Circumscribing or Inscribed Sphere is given. Rule.— 
Multiply radius given by multiplier opposite to body in columns I and K in 
Table, page 364. 

When Surface is given. Rule. —Multiply square root of it by multiplier 
opposite to body in column L in Table, page 364. 

When Volume is given. Rule. — Multiply cube root of it by multiplier 
opposite to body in column M in Table, page 364. 


To Compute Surface. 

When Radius of Circumscribing Sphere is given. Rule. —Multiply square 
of radius by multiplier opposite to body in column N in Table, page 364. 

When Radius of Inscribed Sphere is given. Rule. —Multiply square of 
radius by multiplier opposite to body in column 0 in Table, page 364. 

When Linear Edge is given. Rule. —Multiply square of edge by multi¬ 
plier opposite to body in column P in Table, page 364. 

When Volume is given. Rule. —Extract cube root of volume, and multi¬ 
ply square of root by multiplier opposite to body in column Q in Table, 
page 364. 

To Compute Volume. 

When Linear Edge is given. Rule. —Cube linear edge, and multiply it 
by multiplier opposite to body in column R in Talkie, page 364. 

When Radius of Circumscribing Sphere is given. Rule. —Multiply cube 
of radius given by multiplier opposite to body in column S in Table, 
page 364. 

When Radius of Inscribed Sphere is given. Rule. — Multiply cube of 
radius given by multiplier opposite to body in column T in Table, page 364. 

When Surface is given. Rule. —Cube surface given, extract square root, 
and multiply the root by multiplier opposite to body in column U in Table, 
page 364. 

Fig. 18. Cylinder. 

To Compute Volume of a Solid Cylinder.— 
Eig. 18. 

Rule. —Multiply area of base by height. 

Example.— Diameter of a cylinder, ft c, is 3 feet, and its length, a 6, 

''''"Till 7 feet; w ^ at * ts v °l ume ? 

C \Tj j|P b Area of 3 feet = 7.068. Then 7.068 X 7 = 49.176 cube fed. 

To Compute Volume of a Hollow Cylinder. 

Rule. —Subtract volume of internal cylinder from that of cylinder. 


Fig. 19. 


Cone. 

To Compute Volume of a Cone;.—Eig. 19. 

Rule. —Multiply area of base by perpendicular height, 
and take one third of product. 

Example. —Diameter, a ft, of base of a cone is 15 inches, and 
height, c e, 32.5 inches; what is its volume? 


Area of 15 inches = 176.7146. Then ? 7 $ —- = 1914.4125 cube ins. 



3 






Units for Elements of th.e Regular Bodies. 


364 


MENSURATION OF VOLUMES 


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Hexahedron . . . 
Octahedron .... 
Dodecahedron . . 
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Tetrahedron .. . 
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054 
























































MENSURATION OF VOLUMES. 


365 


To Compute Volume of Frustum of a Cone.-Fig- 20. 

Rule.—A dd together squares of the diameters or circumferences of greater 
and lesser ends and product of the two diameters or circumferences; mul¬ 
tiply their sum respectively by .7854 and .07958, and this product by 
height then divide this last product by 3. 

Or, d 2 + d ' 2 + dxj' X . 7854 h -4- 3 — V. 

Or, c 2 -(- c ' 2 + c Xc'X -079 58 h-r- 3 = V. 

Example. —What is volume of frustum of a cone, diameters 
of greater and lesser ends, bd,ac, being 5 and 3 feet, and height, 

e °> 9’ _ 

5 2 + 3 2 + 5 X 3 49 i an( l 49 X .7854 = 38.4846 = above sum 

by .7854; and 3 8 - 4846 X 9 _ 453 g cube feet. 


r lg. 20. 



Pyramid. 

Note.— Volume of a pyramid is equal to one third of that of a prism having equal 
bases and altitude. 


Fig. 21. 



a b 


To Compute Volume of a. Pyramid.—Fig. 21. 


Rule.—M ultiply area of base by perpendicular height, and 
take one third of product. 

Example.— What is the volume of a hexagonal pyramid, Fig. 21, 
a side, a b, being 40 feet, and its height, e c, 60? 

40 2 X 2.5981 (tabular multiplier, page 341) = 4156.96 = area of base. 


4156 96 X 60 __ jgg.2 cube feet. 


To Compute 'V'olvime of Frustum of a Pyramid.—Fig. 22. 

Rule. —Add together squares of sides of greater and lesser ends, and 
product of these two sides; multiply sum by tabular multiplier for areas in 
Table, page 341, and this product by height; then divide last product by 3. 

Or, s 2 -f s' 2 + s X s' X tab. mult, x h -4- 3 = V. 

When Areas of Ends are known, or can he obtained without reference to 
a tabular multiplier , use following. 

Or, a + a' + fa X a' X h - 4 - 3 = V. 

Example. —What is the volume of the frustum of a hexagonal 
pyramid, Fig. 22, the lengths of the sides of the greater and lesser 
ends, ab,cd, being respectively 3.75 and 2.5 feet, and its perpen¬ 
dicular height, eo, 7.5? 

3 - 75 2 + 2 . 5 2 = 20.3125 = sum of squares of sides of greater and 
lesser ends; 20.3125-f- 3.75 X 2.5 = 29.6875 — above sum added to 
product of the two sides; 29.6875 X 2.5981 X 7-5 = 578.48 X tab. 
mult., and again by the height, which, -4-3= 192.83 cube feet. 



When Ends of a Pyramid are not those of a Regular Polygon, or when 
Areas of Ends are given 

Rule.—A dd together areas of the two ends and square root of their prod¬ 
uct ; multiply sum by height, and take one third of product. 

Or, a fa' + f a a' X h - 4 - 3 V. 

Example.— What is the volume of an irregular-sided frustum of a pyramid, the 
areas of the two ends being 22 and 88 inches, and the length 20? 

22 + 88 = 1 10 =1 sum of areas of ends ; 22 X 88 = 1936, and f 1936 = 44 = square 
„ 110 + 44X20 , ,, . . 

root of product of areas. Then -= 1026.66 cube ms. 


3 

Hh* 


















366 


MENSURATION OF VOLUMES. 


Spherical Pyramid. 

A Spherical Pyramid is that part of a sphere included within three or more ad¬ 
joining plane surfaces meeting at centre of sphere. The spherical polygon defined 
by these plane surfaces of pyramid is termed the base, and the lateral faces are 
sectors of circles. 

Note. —To compute the Elements of Spherical Pyramids, see Docharty and Hack- 
ley’s Geometry. 

Cylindrical TJngulas. 

Definition.— Cylindrical Ungulas are frusta of cylinders. Conical Ungulas are 
frusta of cones. 


To Compute 'Volume of a Cylindrical Ungula.—Fig;. 23. 

i. When Section is parallel to Axis of Cylinder. Rule.— Multiply area 
Fig. 23. °f base by height of the cylinder. 


Or, ah — V. 

Example.— Area of base, d ef Fig. 23, of a cylindrical ungula is 15.5 
inches, and its height, a e, 20; what is its volume? 

■ I 5 - 5 X 20 = 310 cube ins. 

2. When Section passes Obliquely through opposite sides of 
Cylinder, Fig. 24. Rule. —Multiply area of base of cylinder by 
half sum of greatest and least lengths of ungula. Fig. 24 . 

Or, a x l + 1 ' ~r 2.== V. 

Example.— Area of base, c d, of a cylindrical ungula, Fig. 24, is 25 
inches, and the greater and less heights of it, a c, b d, are 15 and 17; 
what is its volume? 

i 5 4‘ i 7 . ■ 

25 X -— 400 cube ms. c 




3. When Section passes through Base of Cylinder and one of its Sides , 
and Versed Sine does not exceed Sine , or the Base is equal to or less than a 
Semicircle , Fig. 25. Rule. —From two thirds of cube of sine of half arc 
of base subtract product of area of base and cosine * of half arc. Multiply 
difference thus found by quotient arising from height divided by versed sine. 



Or, 


— h 


— a c X 


: V, v. sin. representing versed sine. 


Example.— Sine, a d , of half arc, d ef of base of an ungula, Fig. 25, 
c is 5 inches, diameter of cylinder is 10, and height, eg. of ungula 10; 
what is its volume? 

Two thirds of 53 — 83.333 = two thirds of cube of sine. As versed 
J sine and radius of base are equal, cosine is o. Hence, area of base X 
cosine = 0, and 83.333 — °X 10-1-5 = 166.666 cube ins. 


4. When Section passes through Base of Cylinder , and Versed Sine exceeds 
Radius, or tvhen the Base exceeds a Semicircle , Fig. 26. Rule. —To two 
thirds of cube of sine of half arc of base add product of area of base and 
cosine. Multiply sum thus found by quotient arising from height, divided 
by versed sine. 


Fig. 26. 



Or, 


2 sin. 3 


3 


ocX 



v. sin. 


Example.— Sine, a d, of half arc of an ungula, Fig. 26. is 12 inches, 
versed sine, a g. is 16, height, g c, 10, and diameter of cylinder 25; 
what is its volume? 


Two thirds of 12 3 = 1152 = two thirds of cube of sine of half arc of 
base. Area of base = 331.78; 1152 + 331.78 X 16 —12.5 = 2313.23 = 
sum of two thirds of cube of sine of half the arc of base, and product of 
area of base and cosine. Then 2313.23 X 20=16 = 2891.5375 cube ins. 


* When the cosine is o, the product is o. 































MENSURATION OF VOLUMES. 


367 


5. When Section passes Obliquely through both Ends of Cylinder , Fig. 27. 
Rule. — Conceive section to be continued till it meets side of cylinder 
produced; then, as the difference of versed sines of the arcs of the two ends 
of ungula is to the versed sine of arc of less end, so is the height of cylinder 
to the part of side produced. 

Ascertain volume of each of the ungulas by Rules 3 and 4, and take their 
difference. 


Fig. 27. 

m 



9 


Or, 


v. sin.' h 


. h', v. sin. and v. sin.' representing versed sines 


v. sin. — v. sm. 

of arcs of the two ends , h height of cylinder, and h' height of part pro¬ 
duced. 


Example. —Versed sines, ae,do, and sines, e and 0, of arcs of two 
ends of an ungula, Fig. 27, are assumed to be respectively 8.5 and 25, 
and xi.5 and o inches, length of ungula, b o, within cylinder, cut from 
one having 25 inches diameter, do, is 20 inches; what is height of un¬ 
gula produced beyond cylinder, and what is volume of it? 

25 oj 8.5 : 8.5 :; 20 : 10.303 = height of ungula produced beyond cyl¬ 
inder. 


Greater ungula, sine 0 being o, versed sine = the diameter. Base of ungula being 
a circle of 25 inches diameter, area = 490.875. Versed sine and diameter of base 

2 C 

being equal (25), sine = o. 490.875 X 25 <v — = 6135.9375 = product of area of base 

2 

and cosine , or excess of versed sine over sine of base. 30.303 -5-25 = 1.21212 = quo¬ 
tient of height versed sine. 

Then 6135.9375 x 1.212 12 = 7437.4926 cube inches ; and by Rules 3 and 4, volumes 
of less and greater ungulas '= 515.444, and 6922.0486 = 7437.4926 cube inches. 


Spliere. • 

Definition. —A solid, surface of which is at a uniform distance from the centre. 


Fig. 28. 



To Compute Volume ofa Spliere.— Vig. 28. 
Rule.— Multiply cube of diameter by .5236. 

Or, <P x .5236 = V, d representing diameter. 
Example.— What is volume ofa sphere, Fig. 28, its diameter, 
a b, being 10 inches? 

10 3 = 1000, and 1000 X 5236 = 523.6 cube ins. 


To Compute "Volume of a Hollo'vv Spliere. 

Rule. — Subtract volume of internal space from that of sphere. 

Or, V — v = volume. 

Segment of a Spliere. 

Definition. —A section of a sphere. 

To Compute Volume ofa Segment of a Spliere.—Fig. SO. 

Rule i.— To three times square of radius of its base add square cf its 
height; multiply this sum by height, and product by .5236. 

Fig. 29. ^ Or, 3 r 2 -\- h 2 h X-5236 = V. 

2.—From three times diameter of sphere subtract twice 
height of segment; multiply this remainder by square of 
height, and product by .5236. 

Or, 3 d — 2 h h 2 X • 5236 = V. 

Example.— Segment of a sphere, Fig. 29, has a radius, a e, of 7 
inches for its base, and a height, b 0, of 4; what is its volume ? 

7 2 x 3 4 . 4 z _ j 6 3 _ the sum of three times square of radius and 
square of height; 163 X 4 X •5 2 36 ; =33 I -387 2 cube ins. 






















3 68 


MENSURATION OF VOLUMES. 


Spherical Zone (or Frustum of' a Sphere). 
Definition —Part of a sphere included between two parallel chords. 


To Compute "Volume of" a Splierical Zone.—Fig. 30 . 
Definition. —Part of a sphere included between two parallel planes. 


Rule.— To sum of squares of the radii of the two ends add one third of 
square of height of zone; multiply this sum by height, and again by 1.5708. 


Fig. 30. 



Or, r 2 + r ' 2 + /i 2 -^ 3 /ix 1.5708 = V. 

Example.— What is the volume of a spherical zone, Fig. 30, 
greater and less diameters,//! and de, being 20 and 15 inches, 
and distance between them, or height of zone, c g , being 10 ins.? 

io 2 -j- 7.5 2 —156.25 = sum of squares of radii of the two ends ; 

156.25 + io 2 - 4 - 3 = 189.58 = above sum added to one third of 
square of the height. 

Then 189.58 X 10 X 1.5708 = 2977.9226 cube ins. 


Cylindrical Ring. 

Definition.— A ring formed by the curvature of a cylinder. 

To Compute Volume of a Cylindrical Ifing.— Fig. 31 . 

Rule.— To diameter of body of ring add inner diameter of ring; multi¬ 
ply sum by square of diameter of body, and product by 2.4674. 


Fig- 3 1 - 



Or, d -J- d'd 2 2.4674 = V. 

Or, a l = V, a representing area of section of body , and l length 
of axis of body.. 

Example.— What is volume of an anchor ring, Fig. 31, diameter 
of metal, a b, being 3 inches, and inner diameter of ring, be, 8? 

3 + 8 X 3 2 = 99 = product of sum of diameters and square of di¬ 
ameter of body of ring. 


Then 99 X 2.4674 = 244.2726 cube ins. 


Spheroids (Ellipsoids). 

Definition.— Solids generated by the revolution of a semi-ellipse about one of its 
diameters. When the revolution is about the transverse diameter they are termed 
Prolate, and when about the conjugate they are Oblate. 


To Compute Volume of a Spheroid.—Fig. 32 . 

Rule.— Multiply square of revolving axis by fixed axis, and this product 
by .5236. 

Or, a 2 a' X • 5236 = V, a and a' representing revolving and 
fixed axes. 

Or, 4-^-3 X 3- 14x6 r 2 r'=V, r and r' representing semi-axes. 

Example.— In a prolate spheroid, Fig. 32, fixed axis, ab , is 
14 inches, and revolving axis, cd, 10; what is its volume? 

io 2 X 14 = 1400 = product of square of revolving axis and 
fixed axis. Then 1400 X 5236 = 733.04 cube ins. 

Note. —Volume of a spheroid is equal to % of a cylinder that will circumscribe it. 



Segments of Spheroids. 

To Compute Volume of Segment of a Spheroid.—Fig. 33. 

When Base , e f, is Circular, or parallel to revolving Axis , as c (f Fig. 33, 
or as ef to Axis a 6, Fig. 34. Rule. —Multiply fixed axis by 3, height of 
segment by 2, and subtract one product from the other; multiply remainder 
by square of height of segment, and product by .5236. Then, as square of 
fixed axis is to square of revolving axis, so is last product to volume of 
segment. 
















MENSURATION OF VOLUMES. 


369 



Or, 


: V. 


3 a —2 n h 2 X -5236 X a ' 2 
a 2 

Example. —In a prolate spheroid, Fig. 33, fixed or trans¬ 
verse axis, a b , is 100 inches, revolving or conjugate, c d, 60, 
and height of segment, a 0, 10; what is its volume? 


100 x 3 —10 x 2 = 280 = twice the height of segment sub- 
traded from three times fixed axis; 280 X 10 2 X -5236 = 
14660.8 inches = product of above remainder , square of height , and .5236. Then 
100 2 : 60 2 :: 14669.8 : 5277.888 cube ins. 

When Base , ef is Elliptical , or perpendicular to revolving Axis , a b , Fig. 
33, or as ef to ylx/s c d, Fig. 34. Rule. — Multiply fixed axis by 3, 
and height of segment by 2, and subtract one from the other; multiply re¬ 
mainder by square of height of segment, and product by .5236. Then, as 
fixed axis is to revolving axis, so is last product to volume of segment. 


Fig. 34 - 



Or, 


3 a' — 2 h h 2 X • 5236 X ct 


.V. 


Example. —Diameters of an oblate spheroid, Fig. 34, are 
]5 100 and 60 inches, and height of a segment thereof is 12; 
what is its volume? 

100 X 3 — 12X2 = 276 — twice the height of the segment sub¬ 
tracted from three times the revolving axis ; 276 X 12 2 X -5236 
= 20809.9584 = product of above remainder, the square of height, and .5236. 

Then 100 : 60 :: 20809.9584 : 12485.975 cube ins. 


Frusta of’ Spheroids. 


To Compute Volume of Nticlclle Frxistviira of a Spheroid.— 

Fig. 35. 

When Ends , ef and g h, are Circular, or parallel to revolving Axis , as c d, 
Fig. 35, or a b, Fig. 36. Rule. —To twice square of revolving axis add 
square of diameter of either end; multiply this sum by length of frustum, 
and product by .2618. 

Or, 2 a ’ 2 fd 2 X 1.2618 = V. 

Example. —Middle frustum of a prolate spheroid, i 0, 
Fig. 35, is 36 inches in length, diameter of it being, in 
middle, c d, 50, and at its ends, e f and g h, 40; what is its 
volume ? 

5 ° 2 X 2 -f- 40 2 = 6600 = sum of twice square of middle di¬ 
ameter added to square of diameter of ends. Then 6600 X 
36 X 2618 = 62 203.68 cube ins. 



When Ends , e f and g h , are Elliptical, or perpendicidar to revolving Axis, 
a b , Fig. 35, or e f and g h to Axis, c d, Fig. 36. Rule. —To twice product 
of transverse and conjugate diameters of middle section, add product of 
transverse and conjugate of either end; multiply this sum by length of 
frustum, and’ product by .2618. 

Or, dd'x 2 + did' l X . 2618 = V. 

Example. —In middle frustum of a prolate spheroid, Fig- 
36, diameters of its middle section are 50 and 30 inches, its 
ends 40 and 24, and its length, 0 i, 18; what is its volume? 

50 X 30 X 2 = 3000 = twice product of transverse and con¬ 
jugate diameters; 3000-f-40 X 24 = 3960 = sum of above 
product and product of transverse and conjugate diameters 
of ends. 

Then 3960 X 18 X .2618 = 18 661.104 cube ins. 


















370 


MENSURATION OF VOLUMES. 


Links. 

Definition. —Elongated or Elliptical rings. 

Elongated, or Elliptical Links. 

To Compute Volume of an Elongated, or Elliptical Link. 

— Eigs. 37 and 38. 

Rule.—M ultiply area of a section of the body of link by its length, or 
circumference of its axis. 

Or, a l or c = V. 


Note.— By Rule, page 353, Circumference or length of axis of an Elongated link 
= the sum of 3.1416 times sum of less diameter added to thickness of ring, and 
product of twice remainder of less diameter subtracted from greater. 

Also, Circumference or length of axis of an Elliptical ring = square root of half 
sum of diameters added to thickness of ring or axes squared x s-M* 6 - 
Fig. 37. Example.— Elongated link of a chain, Fig. 37, is 1 inch in diameter 

a of body, a b , and its inner diameters, b c and ef are 10 and 2.5 inches; 
what is its volume? 

Area of 1 inch =. 7854; 2. 5 + 1 X 3-1416 — 10.9956 = 3.1416 times sum 
of less diameter and thickness of ring — length of axis of ends ; 10 — 2.5 
X 2 = 15 = twice remainder of the less diameter subtracted from greater 
— length of sides of body. 

Then 10.9956 +15 = 25.9956 = length of axis of length. 

Hence .7854 X 25.9956 = 20.417 cube ins. 

2.— Elliptical link of a chain, Fig. 38, is of the same dimensions as 
preceding; what is its volume? 



/133-25 
'V 2 


2.5 +1 +10 + 1 = 133.25 = diameter of axes squared 

— 25.643 = square root of half sum of diameters squared X 3.1416: 
cumference of axis of ring. Area of 1 inch = .7854. 

Then 25.643 X .7854 = 20.14 cube ins. 


3-1416 


cir- 



Spherical Sector. 

Definition.— A figure generated by the revolution of a sector of a circle about a 
straight line through the vertex of the sector as an axis. 

Note.— Arc of sector generates surface of a zone, termed base of sector of a 
sphere, and the radii generate surfaces of two cones, having a vertex in common 
with the sector at the centre of the sphere. 

To Compute Volume of a. Spherical Sector.—Eig. 39. 

Rule.— Multiply external surface of zone, which is base of sector, by one 
third of the radius of sphere. 


Or, a r ■— 3 = V, a representing area of base. 

Note.— Surface of a spherical sector := sum of surface of zone and surfaces of the 
two cones. 

Example.— What is volume of a spherical sector, Fig. 
39, generated by sector, c a h, height of zone, abed, be¬ 
ing a o, 12 inches, and radius, g h, of sphere 15 ? 

12 X 94.248 == 1130.976 — height of zone X circumference 
of sphere = external surface of zone (see page 350). 

1130.976 X 15 3 = surface X one third of radius, — 

5654.88 cube ins. 

Spindles. 

Definition. —Figures generated by revolution of a plane area bounded by a curve, 
when the curve is revolved about a chord perpendicular to its axis or about its 
double ordinate, and they are designated by the name of arc from which they are 
generated, as Circular, Elliptic, Parabolic, etc. 


Fig. 39 - 


















MENSURATION OF VOLUMES. 


371 


Circular Spindle. 

To Compute Volume of a Circular Spindle.—Fig. 40. 

Rule. —Multiply central distance by half area of revolving segment; 
subtract product from one third of cube of half length, and multiply re¬ 
mainder by 12.5664. 

Or, ----- -«) X 12.5664 = V, a representing area of revolving segment. 

Example. —What is volume of a circular spindle, Fig. 40, when 
central distance, oe, is 7.071067 inches, length, fc, 14.142 13, and 
radius, 0 c, 10? 

Note. —Area of revolving segment; f e, being = side of square 

that can be inscribed in a circle of 20, is 20 2 X .7854 — 14.142 13 2 
! - 4 - 4 = 28.54 area. 



\ / 7.071067 X 28.54-4-2 = 100.9041 = central distance X half area of 

__ 7.071 67 3 

revolving segment; -100.9041 = 16.947 = remainder of 

above product and one third of cube of half length. 

Then 16.497 X 12.5664 = 212.9628 cube ins. 


Frustum or Zone of a Circular Spindle.* 

To Compute Volume of a Frustum or Zone of a Circular 
Spindle.—Fig. 41. 

Rule. —From square of half length of whole spindle take cne third of 
square of half length of frustum, and multiply remainder by said half length 
of frustum ; multiply central distance by revolving area which generates 
the frustum; subtract this product from former, and multiply remainder by 
6.2832. 


Or, l -4- 2 ■ 


l'- 4 -: 


l' 

X — — (c X a) X 6.2832 = V, l and l' representing lengths of 


spindle and of frustum, and a area of revolving section of frustum. 

Note. — Revolving area of frustum can be obtained by dividing its plane into a 
segment of a circle and a parallelogram. 

Example. —Length of middle frustum of a circular spindle, 
ic, Fig. 41, is 6 inches; length of spindle,/^, is 8; central dis¬ 
tance, oe, is 3; and area of revolving or generating segment 
is 10; what is volume of frustum ? 

(8 4 2) 2 — ^ ~ - = 13, and 13 X 3 = 39 = product of — 

3 2 

length of frustum, and remainder of one third square of half 
length of frustum subtracted from square of half length of 

spindle; 39 — 3X10 = 9= product of central distance and area of segment subtracted 
from preceding product. 

Then 9X6.2832 = 56.5488 cube ins. 



Segment of a Circular Spindle. 

To Compute Volume of a Segment of a Circxxlar 
Spiudle.—Fig. 42. 

Rule. —Subtract length of segment from half length of spindle; double 
remainder, and ascertain volume of a middle frustum of this length. Sub¬ 
tract result from volume of whole spindle, and halve remainder.! 

Or, C — c-4- 2 = V, C and c representing volume of spindle and middle frustum. 


* Middle frustum of a Circular Spindle is one of the various forms of casks. 

t This rule is applicable to segment of any Spindle or any Conoid, volume of the figure and frustum 
being first obtained. 













MENSURATION OF VOLUMES. 


372 

Fig. 42. 



Example. — Length of a circular spindle, i a, Fig. 42, is 
14.14213 inches; central distance, 0 e, is 7.07107; radius of 
arc, o a, is 10; and length of segment, i c, is 3.535 53; what is 
its volume? 

14.14213 


3.535 53 x 2 = 7.071 07 = double remainder of 

Z 

length of segment subtracted from half length of spindle — 
length of middle f rustum. 

Note. —Area of revolving or generating segment of whole spindle is 28.54 inches, 
and that of middle frustum is 19.25. 

The volume of whole spindle is.212.9628 cube ins. 

u “ middle frustum is.162.8982 “ “ 

Hence. 50.0646 -1-2 = 25.°3 2 3 cube ins. 

Cycloidal Spindle.* 

To Compute Volume of a Cycloidal Spindle.—Eig. -4r3. 

Rule. —Multiply product of square of twice diameter of generating circle 
and 3.927 by its circumference, and divide this product by 8. 

_2 

2 d X 3-927 X d X 3.14 16 



Or, 


8 


V, d representing diameter 


of circle, or half width of spindle. 

Example. —Diameter of generating circle, a b c, of a cy¬ 
cloid, Fig. 43, is 10 inches; w 7 hat is volume of spindle, d e? 


10 X 2 X 3 927 = 1570.8 —product of twice diameter squared and 3.927. 

Then 1570.8 X 10 X 3.1416-1-8 = 6168.5316 cube ins. 

Elliptic Spindle. 

To Compute Volume of an Elliptic Spindle. — Fig. 44. 

Rule. — To square of its diameter add square of twice diameter at one 
fourth of its length; multiply sum by length, and product by .1309.! 


Fig. 44. 


Or, d- -|- 2 d' 1 . 1309 = V, d and d' representing diameters as above. 

Example. —Length of an elliptic spindle, a b, Fig. 44, is 
75 inches, its diameter, c d. 35, and diameter, ef at . 25 of its 
length, 25; what is its volume? 



35 2 -f- 25 X 2 = 3725 = sum of squares of diameter of 
spindle and of twice its diameter at one fourth of its length ; 
3725 X 75 = 279 375 = above sum x length of spindle. 

Then 279 375 X • 1309 = 36 570.1875 cube ins. 

Note.—F or all such solid bodies this rule is exact when body is formed by a 
eonic section, or a part of it, revolving about axis of section, and will always be 
very near when figure revolves about another line. 

To Compute Volume of jVliddie Frustum or Zone of 
an Elliptic Spindle.—-Eig. 45. 

Rule. —Add together squares of greatest and least diameters, and square 
of double diameter in middle between the two; multiply the sum by length, 
and product by .13094 


Or, d 2 -f- d'~ -)- 2 d" 1 . 1309 = V, d, d', and d” representing different diameters. 


* Volume of a Cycloidal Spindle is equal to .625 of its circumscribing cylinder, 
t See preceding Note. J See Note above. 



















MENSURATION OF VOLUMES, 


373 


Fig. 45. Example.— Greatest and least diameters, a b and c d, of 

" " the frustum of an elliptic spindle, Fig. 45, are 68 and 50 

inches, its middle diameter, gh , 60, and its length, ef, 75; 
what is its volume? 


68 2 -f- 50 2 -}- 60 x 2 = 21 524 —sum, of squares of greatest 
and least diameters and of double middle diameter. 

Then 21 524 X 75 X -1309 = 211 311.87 cube ins. 

To Compute Volume of a. Segment of an Elliptic Spin¬ 
dle.—Fig. 46. 

Rule. —Add together square of diameter of base of segment and square 
of double diameter in middle between base and vertex; multiply sum by 
length of segment, and product by .1309.* 



__ 2 

Or, d 2 -j- 2 d" l X • 1309 — V,d and d" representing diameters. 


Fig. 46. 



Example.— Diameters, cd and gh, of the segment of an 
elliptic spindle, Fig. 46, are 20 and 12 inches, and length, 
0 e, is 16; what is its volume ? 

20 2 -j-12 x 2 = 976 = sum of squares of diameter at base 
and in middle. 

Then 976 X 116 X .1309 = 2044.134 cube ins. 


IParabolic Spindle. 

To Compute Volume of a Parabolic Spindle.—Fig. 47'. 
Rule i. — Multiply square of diameter bv length, and the product by 

.418884 

Or, d 2 lx. 4i888 = V. 

Rule 2.— To square of its diameter add square of twice diameter at one 
fourth of its length; multiply sum by length, and product by .13094 

Or, d 2 -f- 2 d' l X • 1309 = V. 

Example.— Diameter of a parabolic spindle, a b, Fig. 
47, is 40 ins., and its length, cd, 10; what is its volume? 
40 2 X 10 = 16 000 = square of diameter X length. 
Then 16 000 X .41888 = 6702.08 cube ins. 

Again, If middle diam. at .25 of its length is 30, Then, 

by Rule 2, 4o 2 -)-3o x 2 X 40 X .1309 = 6806.8 cube ins. 

To Compute Volume of IVliddle Frustum of a Eara'bolic 
Spindle.—Fig. 48. 

Rule i. — Add together 8 times square of greatest diameter, 3 times 
square of least diameter, and 4 times product of these two diameters; mul¬ 
tiply sum by length, and product by .052 36. 

Or, d 2 8 -j- d'- 3 -j- d cf X 4 ^ X .052 36 = V. 



Rule 2. — Add together squares of greatest and least diameters and 
square of double diameter in middle between the two; multiply the sum 
by length, and product by .1309. 

Or, d 2 -j- d ' 2 -J- 2 d " 2 l x . 1309 = V, d" representing diameter between the two. 


Fig. 48. 



Example.— Middle frustum of a parabolic spindle, Fig. 
48, has diameters, a b and ef of 40 and 30 inches, and its 
length, c d, is 10; what is its volume? 

40 2 X 8 + 30 2 X 3 + 40X30 X 4 = 20 300 = sum of 8 
times square of greatest diameter, 3 times square of least 
diameter, and 4 times product of these. 

Then 20 300 x 10 X -052 36 = 10629.08 cube ins. 


* See Note, page 372. 


+ 8-15 of .7854. 

I I 


J See Note, page 372. 


























MENSURATION OF VOLUMES. 


To Compute Volume of a Segment of a Parabolic 
Spindle.—Fig. 49. 

Rule. —Add together square of diameter of base of segment and square 
of double diameter in middle between base and vertex; multiply sum by 
height of segment, and product by .1309. 

Or, d 2 + d " 2 lx. 1309 = V. 

Example. —Segment of a parabolic spindle, Fig. 49, has 
diameters, e/and g h, of 15 and 8.75 inches, and height, 
c d, is 2.5; what is its volume? 

_ 2 

i5 2 -f-8.75 X 2 = 531.25 — sum of square of base and of 
double diameter in middle of segment. Then 531.25 X 2.5 
X .1309 = 173.852 cube ins. 


Fig. 49. 





Hyperbolic Spindle. 

To Compute Volume of a Hyperbolic Spindle.—Fig. GO. 

Rule.— To square of diameter add square of double diameter at one 
fourth of its length ; multiply sum by length, and product by .1309.* 



Or, d 2 -f- 2 d' lx - i 309 = V. 

Example. —Length, ab, Fig. 50, of a hyperbolic spindle 
is 100 inches, and its diameters, c d and e f are 150 and 
no; what is its volume? 

_ 2 

150 2 -f- no X 2 X 109 = 7090000 = product of sum of 
squares of greatest diameter and of twice diameter at one 
fourth of length of spindle and length. Then 7 090 000 X 
. 1309 = 928 081 cube inches. 


To Compute Volume of Middle Frustum of a Hyper¬ 
bolic Spindle.—Fig. Gl. 


Rule.— Add together squares of greatest and least diameters and square 
of double diameter in middle between the two; multiply this sum by length, 
and product by .13094 


Tig. 51. 



Or, d 2 -fd' 2 -}-(2 d ") 2 l X .1309 = V. 

Example.— Diameters, a b and c d, of middle frustum of a 
hyperbolic spindle, Fig. 51, are 150 and no inches; diam¬ 
eter, g h , 140; and length, ef 50; what is its volume? 

_ 2 

i5o 2 + no 2 + 140 X 2 = n 3 000 — sum of squares of great¬ 
est and least diameters and of double middle diameter. Then 
113 000 X 50 X -1309 = 739585 cube ins. 


To Compute Volume of a Segment of a Hyperbolic Spin¬ 
dle.—Fig. 52. 


Rule.— Add together square of diameter of base of segment and square 
of double diameter in middle between base and vertex; multiply sum by 
length of segment, and product by .1309. 


Fig. 52. 



Or, d 2 -f- d " 2 lx- 1309 = V. 

Example.— Segment of a hyperbolic spindle, Fig. 52, has 
diameters, e/and gh, of no and 65 inches, and its length, a b, 
25; what is its volume? 

_ 2 

no 2 + 65 x 2 = 29 000 = sum of squares of diameter of base 
and of double middle diameter. 

Then 29 000 X 25 X • 1309 = 94 902.5 cube ins. 


* See Note, page 372. 


t Ibid. 
























MENSURATION OF VOLUMES. 


375 


Ellipsoid, Paraboloid, and Hyperboloid of Revo¬ 
lution* (Conoids). 

Definition.— Figures like to a cone, described by revolution of a conic section 
around and at a right angle to plane of their fixed axes. 

Ellipsoid of Revolution (Spheroid). 

Definition. —An ellipsoid of revolution is a semi-spheroid. (See page 368.) 


Paraboloid of Revolution.) 

To Compute Volume of a Paraboloid, of Revolution.— 

Pig. 53. 

Rule.— Multiply area of base by half height. 

Fig. 53. c Or, a /t — 2 =g V. 

Note.— This rule will hold for any segment of paraboloid, 
whether base be perpendicular or oblique to axis of solid. 

Example.— Diameter, a b , of base of a paraboloid of revolution, 
Fig. 53, is 20 inches, and its height, dc, 20; what is its volume? 

Area of 20 inches diameter of base == 314.16. Then 314.16 X 
20-r- 2 = 3141.6 cube ins. 

Frustum of a Paraboloid of Revolution. 



To Compute Volume of a Frustum of a Paraboloid of 

Revolution.—Pig. 54. 


fig- 54 - 



Rule. — Multiply sum of squares of diameters by 
height of frustum, and this product by .3927. 

Or, cZ 2 -f- d' 2 h X • 3927 = V. 

Example. —Diameters, a b and d c, of the base and vertex 
of frustum of a paraboloid of revolution, Fig. 54, are 20 and 
11.5 inches, and its height, ef 12.6; what is its volume ? 

20 2 -j— 11.5 2 = 532.25 = sum of squares of diameters. Then 
532.25 X 12.6 X - 39 2 7 = 26 33 - 5 8 37 Clibe ins - 


Segment of a Paraboloid of Revolution. 


To Compute Volume of Segment of a Paraboloid of Revo¬ 
lution.—Pig. 55. 



Fig- 55 - 


X 7.442== 384.31 


Rule.— Multiply area of base by half height. 

Or, a X h^-2 = V. 

Note. —This rule will hold for any segment of paraboloid, 
whether base be perpendicular or oblique to axis of solid. 

Example.— Diameter, a b, of the base of a segment of a para¬ 
boloid of revolution, Fig. 55, is 115 inches, and its height, ef is 
7.4; what is its volume? 

Area of 11.5 inches diameter of base = 103.869. Then 103.869 
5 cube ins. 


Hyperboloid of Revolution. 

To Compute Volume of a Hyperboloid of Revolution. 

—Pig. 56. 

Rule.— To square of radius of base add square of middle diameter; mul¬ 
tiply this sum by height, and product by .5236. 


* These figures have been known as Conoids. For the definition of a Conoid, see Haswell’s Men¬ 
suration, page 233. 

+ Volume of a Paraboloid of Revolution is = .5 of its circumference. 











MENSURATION OF VOLUMES. 


376 


Fig. 56- / 



Or, r 2 -f-d 2 hX- 5236=V, d representing middle diameter 
Example.— Base, a 6, of a hyperboloid of revolution, 
Fig. 56, is 80 inches; middle diameter, c d, 66; and height, 
ef 60; what is its volume? 

_ 2 

80 -7- 2 -f- 66 2 = 5956 = sum of square of radius of base and 
middle diam. Then 5956 X 60 X -5236 — 87 113.7 oube ins. 


Segment of a Hyperboloid, of Revolution. 

To Compute Volume of Segment of a Hyperboloid, of 
Revolution, as Fig. 56. 

Rule. —To square of radius of base add square of middle diameter; mul¬ 
tiply this sum by height, and product by .5236. 

Or, r 2 -}- d " 2 h X • 5236 = Y, r representing radius of base. 

Example.— Radius, a e, of base of a segment of a hyperboloid of revolution, as 
Fig. 56, is 21 inches; its middle diameter, c d , is 30; and its height, ef 15; what is 
its volume? 

21 2 -j- 30 2 X 13 = 20115= product of sum of squares of radius of base and middle 
diameter multiplied by height. Then 20115 X . 5236 =: 10 532.2x4 cube ins. 

Frustum, of a Hyperboloid of Revolution. 

To Compute Volume of Frustum of a Hyperboloid of 
Revolution.—Fig. 57. 

Rule. —Add together squares of greatest and least semi-diameters and 
square of diameter in middle of the two; multiply this sum by height, and 
product by .5236. 

/d \ 2 /d '\ 2 

Or, (- ) -f- ( —) + d " 2 h x • 5236 = V, d, d', and d" representing several diameters. 

Example.— Frustum of a hyperboloid of revolution, Fig. 
57, is in height, ei, 50 inches; diameters of greater and 
lesser ends, a b and c d , are no and 42; and that of middle 
diameter, g h, is 80; what is volume? 

110 -4-2 = 55, and 42 -r- 9 = 21. Hence 5s 2 —|— 21 2 —80 2 
= 9866 = sum of squares of semi-diameters of ends and of 
0 middle diam. Then 9866 X 50 X • 5236 = 258 291.88 cube ins. 


2/ \ 2 



Any Figure 
To Compute Volume 


of Revolution. 

of Revolution.— 


of any Figure 
Fig. 58. 

Rule. —Multiply area of generating surface by circumference described 
by its centre of gravity. 

Or, a2rp = V, r representing radius of centre of gravity. 

Fig- 58- Illustration i. — If generating surface, abed, of cylinder, 

A edf Fig. 58, is 5 inches in width and xo in height, then will 
ab — 5 and b d — io, and centre of gravity will be in o, the ra¬ 
dius of which is r o == 5 4 - 2 = 2.5. Hence 10 X 5 — 50 = area 
of generating surface. 

a Fig- 59 - 





Then 50 X 2.5 X 2 X 3-1416 = 785.4 = area 
7_- r J * of generating surface X circumference of its 
.._ - centre of gravity = volume of cylinder. 

Proof. —Volume of a cylinder 10 inches in diameter and 10 
inches in height, io 2 X .7854 = 78.54, and 78.54 X 10 = 785.4. 

2.—If generating surface of a cone, Fig. 59, is ae = 10, de — 
5, then will ad = 11.18, and area of triangle = 10 X 5 = 2 = 25, 
centre of gravity of which is in 0, and o r, by Rule, page 607, 
= 1.666. 



Hence, 25 X 1.666 X 2 X 3-!4 l6 = 261.8 = area of generating surface X circum¬ 
ference of its centre of gravity = volume of cone. 






















MENSURATION OF VOLUMES. 


377 


Fig. 60. a 



3.—If generating surface of a sphere, Fig. 60, is a be, and a c 

= 10, a b c will be — 39.27, centre of gravity of 

\ which is in 0, and by Rule, page 607, or — ■2. 122. 

/ Hence, 39.27 X 2.122 X 2 X 3-1416 = 523.6 = area of generat- 
/ ing surface X circumference of its centre of gravity = volume oj 
sphere. 

Irregmlax* Bodies. 


To Compute Volume of a,n Irregular Body. 


Rule. —Weigh it both in and out of fresh water, and note difference in 
lbs.; then, as 62.5* is to this difference, so is 1728! to number of cube inches 
in body. 

Or, divide difference in lbs. by 62.5, and quotient will give volume in 
cube feet. 

Note.— If salt water is to be used, ascertained weight of a cube foot of it, or 64, is 
to be used for 62.5. 

Example. —An irregular-shaped body weighs 15 lbs. in water, and 30 out; what 
is its volume in cube inches? 

30 — 15 = 15 = difference of weights in and out of water. 

62.5 : 15 ;; 1728 : 4.14.72— volume in cube ins. 

Or, 15 -r- 62.5 = .24, and .24 X 1728 = 414.72 = volume in cube ins. 


CASK GAUGING. 

Varieties of Casks. 

To Compute Volume of a Cask. 

1 st Variety. Ordinary form of middle frustum of a Prolate Spheroid. 

This class comprises all casks having a spherical outline of staves, as Rum 
puncheons, Whiskey barrels, etc. 

Rule. —To twice square of bung diameter add square of head diameter; 
multiply this sum by length of the cask, and product by .2618, and it will 
give volume in cube inches, which, being divided by 231, will give result in 
gallons. 

2 d Variety. Middle frustum of a Parabolic Spindle. 

This class comprises all casks in which curve of staves quickens at the chime, 
as Brandy casks and Provision barrels. 

Rule. —To square of a head diameter add double square of bung diam¬ 
eter, and from sum subtract .4 of square of difference of diameters; multiply 
remainder by length, and product by .2618, which, being divided by 231, 
will give volume in gallons. 

yl Variety. Middle frustum of a Paraboloid. 

This class comprises all casks in which curve of staves quickens slightly at 
bilge, as Wine casks. 

Rule. —To square of bung diameter add square of head diameter; mul¬ 
tiply sum by length, and product by .3927, which, being divided by 231, 
will give volume in gallons. 

4 th Variety. Two equal frustums of Cones. 

This class comprises all casks in which curve of staves quickens sharply at 
bilge, as Gin pipes. 

Rule. —Add square of difference of diameters to three times square of 
their sum ; multiply sum by length, and product by .065 66, and it will give 
volume in cube inches, which, being divided by 231, will give result in 
gallons. 

* Weight of a cube foot of fresh water. 

I I* 


t Number of inches in a cube foot. 






MENSURATION OF VOLUMES. 


378 

Example.— Bung and head diameters of a cask are 24 and 16 inches, and length 
36; what is its volume in gallons ? 


24 —16 —(- (24 -f-16) 2 X 3 = 4864, which x 36 = 175 104, and 175 104 X -065 66 = 
n 497.329, which-r- 231 — 49.77 gallons. 

Generally. 

D M 2 .001 692 L —U.S. gallons , and .001416 2 = Imperial gallons. 

D, d, and M representing interior , head and bung diameters , and L length of cask 
in inches. 

To Ascertain NCean Diameter of a Cask. 

Rule. —Subtract head diameter from bung diameter in inches, and mul¬ 
tiply difference by following units for the four varieties; add product to 
head diameter, and sum will give mean diameter of varieties required. 

1st Variety.7 I 3d Variety.56 

2d Variety.68 | 4th Variety.52 

Example. —Bung and" head diameters of a cask of 1st variety are 24 and 20 inch¬ 
es; what is its mean diameter? 

24 — 20 = 4, and 4 X • 7 = 2.8, which, added to 20, =22.8 ins. 

ULLAGE CASKS. 

To Compute Volume of Ullage Casks. 

When a cask is only partly filled, it is termed an ullage cask , and is con¬ 
sidered in two positions, viz., as lying on its side, when it is termed a Seg¬ 
ment Lying , or as standing on its end, when it is termed a Segment Standing. 

To Ullage a Lying Cask. 

Rule. —Divide wet inches (depth of liquid) by bung diameter ; find quo¬ 
tient in column of versed sines in table of circular segments, page 267, and 
take its corresponding segment; multiply this segment by capacity of cask 
in gallons, and product by 1.25 for ullage required. 

Example.— Capacity of a cask is 90 gallons, bung diameter being 32 inches; what 
is its volume at 8 inches depth ? 

8-f- 32 = .25, tab. seg. of which is . 153 55, which X 90 = 13.8195, and again X 1.25 = 
17.2744 gallons. 

To Ullage a Standing Cask. 

Rule. —Add together square of diameter at surface of liquor, square of 
head diameter, and square of double diameter taken in middle between the 
two; multiply sum by wet inches, and product by .1309, and divide by 231 
for result in gallons. 

To Compute Volume of a Cask by Four Dimensions. 

Rule.— Add together squares of bung and head diameters, and square of 
double diameter taken in middle between bung and head; multiply the sum 
by length of cask, and product by .1309, and divide this product by 231 for 
result in gallons. 

To Compute Volume of any- Cask from Three Dimen¬ 
sions only. 

Rule. —Add into one sum 39 times square of bung diameter, 25 times 
square of head diameter, and 26 times product of the two diameters; mul¬ 
tiply sum by length, and product by .008726; and divide quotient by 231 
for result in gallons. 

For Rules in Gauging in all its conditions and for description and use of 
instruments, see HasweWs Mensuration , pages 307-23. 









CONIC SECTIONS. 


379 


CONIC SECTIONS. 

A Cone is a figure described by revolution of a right-angled triangle 
about one of its legs, or it is a solid having a circle for its base, and 
terminated in a vertex. 

4 

Conic Sections are figures made by a plane cutting a cone. 

If a cone is cut by a plane through vertex and base, section will be a triangle, 
and if cut by a plane parallel to its base, section will be a circle. 

Axis is line about which triangle revolves. Base is circle which is described by 
revolving base of triangle. 


Fig. ii 



An Ellipse is a figure generated by an oblique plane cut¬ 
ting a cone above its base. 




Transverse axis or diameter is longest right line that can be 
drawn in it, as a 6, Fig. i. 

Conjugate axis or diameter is a line drawn 
through centre of ellipse perpendicular to trans¬ 
verse axis, as cd. 

A Parabola is a figure generated by a 
plane cutting a cone parallel to its side, as a b c, Fig. 2. 

Axis is a right line drawn from vertex to middle of base, as b 0. 

Note.—A parabola has not a conjugate diameter. 

A Hyperbola is a figure generated by a plane 
cutting a cone at any angle with base greater than that of 
side of cone, as a b c, Fig. 3. 

Transverse axis or diameter, o 6, is that part of axis, e 6, which, 
if continued, as at 0, would join an opposite cone, o fr. 

Conjugate axis or diameter is a right line drawn through centre, 
g , of transverse axis, and perpendicular to it. 

Straight line through foci is indefinite transverse axis; that part 
of it between vertices of curves, as 0 b , is definite transverse axis. 
Its middle point, g, is centre of curve. 

Eccentricity of a hyperbola is ratio obtained by dividing distance from centre to 
either focus by semi-t ransverse axis. 

Parameter is cord of curve drawn through focus at right angles to axis. 

Asymptotes of a hyperbola are two right lines to which the curve continually ap¬ 
proaches, touches at an infinite distance but does not pass; they are prolongations 
of diagonals of rectangle constructed on extremes of the axes. 

Two hyperbolas are conjugate when transverse axis of one is conjugate of the 
other, and contrariwise. 

Greneral Definitions. 

An Ordinate is a right line from any point of a curve to either of diameters, as 
a e and d 0, Fig. 4, and a b and d f are double ordinates; c 6, Fig. 5, is an ordinate, 
and a b an abscissa. 

An Abscissa is that part of diameter which is contained between 


Fig. 5- 


Fig. 4. <3 ( J , 

— vertex and an ordinate, as c e, g 0 , Fig. 4, and a b, 

Fig. 5- 

Parameter of any diameter is equal to four times 
distance from focus to vertex of curve; parameter 
of axis is least possible, and is termed parameter 
o-f curve. 

Parameter of curve of a conic section is equal 
to chord of curve drawn through focus perpendic¬ 
ular to axis. " 01 

Parameter of transverse axis is least, and is termed parameter of curve. 
Parameter of a conic section and foci are sufficient elements for construction 
of curve. 



e _ 

\f 

J s 

















380 


CONIC SECTIONS. 


A Focus is a point on principal axis where double ordiilate to axis, through point, 
is equal to parameter, as c f Fig. 5. 

It may be determined arithmetically thus: Divide square of ordinate by four 
times abscissa, and quotient will give focal distances, as and s, in preceding figures. 

Directrix of a conic section is a right line at right angles to 
major axis, and it is in such a position that 

f: g:\u-. o. 

Here ad, Fig. 6, is directrix, and 0 is offset to directrix. 

Latus Rectum , or principal parameter, passes through a focus; 
it is a double ordinate, which is a third proportion to the axis. 

Or, A : a :: a : L. 

! ® \ ij A and a representing major and minor axes. (See Haswell's 
» \* Mensuration , page 232.) 


Fig. 6 



r'v Fig. 7. A Conoid is a warped surface generated by a right 

line being moved in such a manner that it will touch 
a straight line and curve, and continue parallel to a 
given plane. Straight line and curve are called di- 
rectrices , plane a plane directrix, and moving line the 
e . . a s y generatrix. 



Thus, let a b a\ Fig. 7, be a circle in a horizontal plane, 
and d d' projection of right lines perpendicular to a ver¬ 
tical plane, r’ be-, if right lines, d a, r s , r' b , r" s, and d' a, 
be moved so as to touch circle and right line dd' and be 
constantly parallel to plane r' b e, it will generate conoid 
dab a'd'. 


Radii vectores are lines drawn from the foci to any point in the curve; hence a 
radius vector is one of these lines. 


Traced angle is angle formed by the radii vectores and the transverse diameter. 

Ellipsoid, Paraboloid , and Hyperboloid of Revolution —Figures generated 
by the revolution of an ellipse, parabola, etc., around their axes. (See Men¬ 
suration of Surfaces and Solids , pages 357-75.) 

Note i.— All figures which can possibly be formed by cutting of a cone are men¬ 
tioned in these definitions, and are five following—viz., a Triangle , a Circle, an El¬ 
lipse , a Parabola , and a Hyperbola ; but last three only are termed Conic Sections. 

2. —In Parabola parameter of any diameter is a third proportional to abscissa 
and ordinate of any point of curve, abscissa and ordinate being referred to that 
diameter and tangent at its vertex. 

3. —In Ellipse and Hyperbola parameter of any diameter is a third proportional 
to diameter and its conjugate. 


To Determine Parameter of siix Ellipse 01 * Hyperbola. 


Rule. — Divide product of conjugate 
diameter, multiplied by itself, by trans¬ 
verse, and quotient is equal to para¬ 
meter. 

In annexed Figs. 8 and 9, of an Ellipse 
and Hyperbola , transverse and conjugate 
diameters, ab, cd, are each 30 and 20. 

Then 30 : 20 ” 20 : 13.333 — parameter. 

Parameter of curve — ef, a double ordinate passing through 
focus, s. 

Ellipse. 

To Eescriloe Ellipses. (See Geometry, page 226.) 
To Compute Terms of a it Ellipse. 



Fig. 9 . 



When any three of four Terms of an Ellipse are given, viz., Transverse 
and Conjugate Diameters , an Ordinate, and its Abscissa, to ascertain remain¬ 
ing Terms. 






















CONIC SECTIONS. 


381 


To Compute Ordinate. 

Transverse and Conjugate Diameters and Abscissa being given. Rule.—A s trans¬ 
verse diameter is to conjugate, so is square root of product of abscissae to ordinate 
wbicli divides them. 


Fig. 10 . 



Example.— Transverse diameter, ab, of an ellipse, Fig. 
10, is 25; conjugate, c d, 16: and abscissa, a i, 7; what is 
length of ordinate, ie? 

25 — 7 = 18 less abscissa j V7 X 18 = 11.225. 

Hence 25 : 16 n.225 : 7-184 ordinate. 


Or, y/c 2 — (fj~j — an V ordinate, c and t representing 

semi-conjugate and transverse diameters, and x distance of ordinate from centre of 
figure. 

To Compute .Abscissae. 


Transverse and Conjugate Diameters and Ordinate being given. Rule.—A s conju¬ 
gate diameter is to transverse, so is square root of difference of squares of ordinate 
and semi-conjugate to distance between ordinate and centre; and this distance be¬ 
ing added to, or subtracted from, semi-transverse, will give abscissae required. 

Example.— Transverse diameter, a b, of an ellipse, Fig. 10, is 25; conjugate, c d, 
16 ; and ordinate, i e, 7.184; what is abscissa, i b ? 


V 8 s 


- 7.184 2 == 3.519 943. Hence, as 16 : 25 :: 3.52 : 5.5. 
Then 25 - 4 - 2 = 12.5, and 12.5 -j- 5.5 = 18 = b i, 

25 - 7-2 = 12.5, and 12.5 — 5.5 


’ ^ h 1 abscissa. 

iff* 7f- a h 1 

To Compute Transverse Diameter. 


Conjugate , Ordinate , and Abscissa being given. Rule.— To or from semi-conju¬ 
gate, according as great or less abscissa is used, add or subtract square root of dif¬ 
ference of squares of ordinate and semi-conjugate. Then, as this sum or difference 
is to abscissa, so is conjugate to transverse. 

Example. — Conjugate diameter, c d, of an ellipse, Fig. 10, is 16; ordinate, i e, 
7.184; and abscissae, b i, i a, 18 and 7; what is length of transverse diameter? 


(16-4-2) 2 — 7. i 8 4 2 = 3-52. 

16-4-23.52 : 18 ;16 : 25; 16-7-2 — 3.52 : 7 ;; 16 : 25 transverse diameter. 

To Compute Conjugate Diameter. 

Transverse , Ordinate , and Abscissa being given. Rule. —As square root of prod¬ 
uct of abscissae is to ordinate, so is transverse diameter to conjugate. 

Example.— Transverse diameter, ab, of an ellipse, Fig. 10, is 25; ordinate, i e, 
7.184; and abscissae, bi and ia, 18 and 7; what is length of conjugate diameter? 

Vi8 X 7 = 11 225. Hence n.225 : 7.184 25 : 16 conjugate diameter. 

To Compute Circumference of an Ellipse. 

Rule.—M ultiply square root of half sum of the squares of two diameters by 
3.1416. 

Example. —Transverse and conjugate diameters, a b and cd, of an ellipse, Fig. 10, 
are 24 an-d 20; what is its circumference? 

24-^ —20^ 

——-= 488, and V488 — 22.09. Hence 22.09 X 3-1416 = 69.398 circumference. 

To Compute Area of aia Ellipse. 

Rule. —Multiply the diameters together, and the product by .7854. Or, multiply 
one diameter by .7854, and the product by the other. 

Example. —The transverse diameter of an ellipse, a b , Fig. 10, is 12, and its con¬ 
jugate, cd, 9; what is its area? 

12 x 9 X .7854 = 84.8232 area. 

Note. —Area of an ellipse is a mean proportional between areas of two circles, 
diameter of one being major axis and of the other minor axis. 

Illustration. —Area of circle of 40 = 1256.64; area of ellipse 40 X 20 = 628.32; 
area of circle of 20 = 314.16, mean proportional of the two circles 1256.64 -f- 314.16 
= 785.4. Therefore the conjugate diameter of an ellipse of an area of 785.4 sq. ins., 
its transverse being 40, is 25 feet, as 40 X 25 X -7854 = 785.4 sq. ins. 










382 


CONIC SECTIONS. 


Segment of an Ellipse. 

To Compute Area of a Segment of an Ellipse. 

When its Base is parallel to either Axis , as e if. Rule. —Divide height of seg¬ 
ment, bi, by diameter or axis, a b, of which it is a part, and find in Table of Areas 
of Segments of a Circle, page 267, a segment having same versed sine as this quo¬ 
tient; then multiply area of segment thus found and the 
axes of ellipse together. 

Example. —Height, bi, Fig. u, is 5, and axes of ellipse are 
30 and 20; what is area of segment? 

5-4-30222.1666 tabular versed sine, the area of which 
(page 267) is .085 54. 

Hence .085 54 X 30 X 20 = 51.324 area. 

To Ascertain Liength of an Elliptic Curve which is less 
tlian Half of entire Figure. 

Let curve of which length is required be A b C, 
Fig. 12. , t 

Extend versed sine b d to meet centre of curve in e. 

Draw line c C, and from e, with distance e b, describo 
b h; bisect h C in i, and from e, with radius e i, de¬ 
scribe k i, and it is equal to half arc A b C. 




To Ascertain Length wHen Carve is greater tlian Half 

entire Eigare. 

Ascertain by above problem curve of less portion of figure; subtract it from cir¬ 
cumference of ellipse, and remainder will be length of curve required. 

Parabola. 

To Describe a Parabola. (See Geometry, page 229.) 


To Compate eitHer Ordiiaate or Abscissa of a Parabola. 

When the other Ordinate and Abscissa ,, or other Abscissa and Ordinates are 
given. Rule. —As either abscissa is to square of its ordinate, so is other abscissa to 
square of its ordinate. 

Or, as square of any ordinate is to its abscissa, so is square of other ordinate to 
its abscissa. 


Fig- I 3- 


a 



Example i. —Abscissa, a b, of parabola, Fig. 13, is 9; its ordi¬ 
nate, b c, 6; what is ordinate, d e, abscissa of which, a d, is 16 ? 

Hence 9 : 6 2 :: 16 : 64, and -^64 == 8 length. 

2.—Abscissae of a parabola are 9 and 16, and their correspond¬ 
ing ordinates 6 and 8; any three of these being taken, it is re¬ 
quired to compute the fourth. 


62 X l6 o ,• , 
1. — =8 ordinate. 

9 


V 


8 2 X 9 _ 


16 


=: 6 ordinate. 


3 - 


16 X 6 2 
8 2 



less abscissa. 


4 - 


9X8 2 

6 2 


2= 16 abscissa. 


Parabolic Curve. 

To Compute Length of Curve of a Parabola cut off by 
a Double Ordinate.—Eig. 13 . 


Rule. —To square of ordinate add — of square of abscissa, and square root of 

3 

this sum, multiplied by two, will give length of curve nearly. 

Example.— Ordinate, d e, Fig. 13, is 8, and its abscissa, ad, 16; what is length of 
curve, fa e ? 


4 X 16 2 


; 4 ° 5 - 333 > an d V 4 ° 5 - 333 X 2 = 40.267 length. 














CONIC SECTIONS. 


383 


Fig. 14. h 



To Compute Area of a Parabola. 

Rule.—M ultiply base by height, and take two thirds of product. 

Corollary .—A parabola is two thirds of its circumscribing par¬ 
allelogram. 

Example.— What is area of parabola, abc, Fig. 14, height, be , 
being 16, and base, or double ordinate, ac, 16 ? 

2 

16 X 16 = 256, and — of 256 = 170.667 area. 


To Compute Area of a Segment of a Parabola. 

Rule. —Multiply difference of cubes of two ends of segment, a c, df by twice its 
height, e 0, and divide product by three times difference of squares of ends. 

Example. —Ends of a segment of a parabola, a c and df Fig. 14, are 10 and 6, and 
height, e 0, is 10; what is its area? 

io 3 ^6’ X 10X2 = 15680, and - 4 - io 2 -v6 2 X 3 = 81.667 area. 

Note.— Any parabolic segment is equal to a parabola of the same height, the base 
of which is equal to base of segment, increased by a third proportional to sum of 
the two ends and lesser end. 


Hyperbola. 

To Describe a Hyperbola. (See Geometry, page 230.) 

To Compute Ordinate of a Hyperbola, 

Transverse and Conjugate Diameters and Abscissce being given. Rule*. —As trans¬ 
verse diameter is to conjugate, so is square root of product of abscissae to ordinate 
required. 

Example. —Hyperbola, abc, Fig. 15, has a transverse 
diameter, a t, of 120; a conjugate, df of 72; and abscissa, 
a e, 40; what is the length of ordinate, e c ? 

40-j- 120 160 greater abscissa, and 

120 : 72 f. y/ (4° X 160) : 48 ordinate. 

Note i. — In hyperbolas lesser abscissa, added to axis 
(the transverse diameter), gives greater. 

2.—Difference of two lines drawn from foci of any hyperbola to any point in curve 
is equal to its transverse diameter. 

To Compute .Abscissae, 

Transverse and Conjugate Diameters and Ordinate being given. Rule. —As con¬ 
jugate diameter is to transverse, so is square root of sum of squares of ordinate and 
semi-conjugate to distance between ordinate and centre, or half sum of abscissa;. 
Then the sum of this distance and semi-transverse will give greater abscissa, and 
their difference the lesser abscissa. 

Example. —Transverse diameter, at, of a hyperbola, Fig. 15, is 120; conjugate, df 
72; and ordinate, e c, 48; what are lengths of abscissae, t e and a e? 

72 ; 120 :: V48 3 -)- (72-4-2) 2 = 6 o ; 100 half sum of abscissce, and 100+ (120- 4 -2):= 
160 greater abscissa, and 100 — (120-4-2) = 40 lesser abscissa. 

To Compute Conjugate Diameter, 

Transverse Diameter, Abscissce, and Ordinate being given. Rule. —As square root 
of product of abscissae is to ordinate, so is transverse diameter to conjugate. 

Example.— Transverse diameter, at, of a hyperbola, Fig. 15, is 120; ordinate, e c, 
48; and abscissae, t e and a e, 160 and 40; what is length of conjugate, df? 

V40 X 160 = 80 : 48 :: 120 : 72 conjugate. 












3§4 


CONIC SECTIONS. 


To Compiate Transverse Diameter, 

Conjugate, Ordinate, and an Abscissa being given. Rule. —Add square of ordinate 
to square of semi-conjugate, and extract square root of their sum. 

Take sum or difference of semi-conjugate and this root, according as greater or 
lesser abscissa is used. Then, as square of ordinate is to product of abscissa and 
conjugate, so is sum or difference above ascertained to transverse diameter required. 

Note. —When the greater abscissa is used, the difference is taken, and con¬ 
trariwise. 

Example. —Conjugate diameter, df of a hyperbola, Fig. 15, is 72; ordinate, e c, 
48; and lesser abscissa, a e, 40; what is length of transverse diameter, at? 

V48 s -j- (72 -r- 2) 2 = 60, and 60 -j- 72 -T- 2 = 96 lesser abscissa, and 40 X 72 = 2880. 

Hence, 48 2 : 2880 :: 96 : 120 transverse diameter. 


To Compute Length of any Arc of a Hyperbola, com¬ 
mencing at Vertex. 

Rule. —To 19 times transverse diameter add 21 times parameter of axis. 

To 9 times transverse diameter add 21 times parameter, and multiply each of 
these sums respectively by quotient of lesser abscissa divided by transverse di¬ 
ameter. 

To each of products thus ascertained add 15 times parameter, and divide former 
by latter; then this quotient, multiplied by ordinate,will give length of arc, nearly. 

Note. —To Compute Parameter , divide square of conjugate by transverse diam¬ 
eter. 



Example. —In hyperbola, abc, Fig. 16, transverse diameter is 120, 
conjugate, 72, ordinate, e c, 48, and lesser abscissa, a e, 40; what is 
length of arc, a b ? 

2 

-— = 43.2 parameter. 120 X 19-1-43.2 X 21 X = 1062.4. 

120 120 

120 X 9 + 43-2 X 21 x = 662.4. Then 1062.4-1-43.2X15^-662.4 

__ 120 

4-43.2X 15 = 1.305, which X 48 = 62.64 length. 


Note.— As transverse diameter is to conjugate, so is conjugate to parameter. 
(See Rule, page 380.) 


To Compute Area of a Flyperbola, 

Transverse, Conjugate , and Lesser Abscissa being given. Rule. —To product of 
transverse diameter and lesser abscissa add five sevenths of square of this abscissa, 
and multiply square root of sum by 21. 

Add 4 times square root of product of transverse diameter and lesser abscissa to 
product last ascertained, and divide sum by 75. 

Divide 4 times product of conjugate diameter and lesser abscissa by transverse 
diameter, and this last quotient, multiplied by former, will give area, nearly. 


Example. —Transverse diameter of a hyperbola, Fig. 16, is 60, conjugate 36, and 
lesser abscissa or height, a e, 20; what is area of figure ? 

60 X 20-}- — of 20 2 = 1485.7143, and y/ 1485.7143 X 21 — 809.43, and V 60 x 20 X 

^ 20 X 4 

4-1-809.43 = 901.02, which-4-75 = 12.0136 and —— 2 -- X 12.0136 = 576.653 area. 

60 

NoTR.-=-For ordinates of a parabola in divisions of eighths and tenths, see page 229. 

Delta JVletal. 


Delta Metal is an improved composition of Aluminium and its alloys; it is 
non-corrosive, capable of being cast, forged, and hot rolled. 


Tensile Strength per Sq. Inch. 

Cast in green sand. 48 380 lbs. I Rolled, annealed. 60 920 lbs. 

Rolled, hard. 75260 “ | Wire, No. 22 W G. 140000 “ 
















PLANE TRIGONOMETRY. 


3»5 


PLANE TKIGONOMETRY. 

By Plane Trigonometry is ascertained how to compute or determine 
four of the seven elements of a plane or rectilinear triangle from the 
other three, for when any three of them are given, one of which being 
a side or the area, the remaining elements may be determined; and 
this operation is termed Solving the Triangle. 

The determination of the mutual relation of the Sines, Tangents, Secants, 
etc., of the sums, differences, multiples, etc., of arcs or angles is also classed 
under this head. 

For Diagram and Explanation of Terms, see Geometry, pp. 219-21. 

Riglit-angled. Triangles. 

For Solution by Lines and Areas , see Mensuration of Areas , Lines , 
and Surf aces, pp. 335-39. 

To Compute a Side., 

When a Side and its Opposite Angle is given. Rule. —As sine of angle 
opposite given side is to sine of angle opposite required side, so is given side 
to required side. 

To Compute an .Angle. 

Rule. —As side opposite to given angle is to side opposite to required 
angle, so is sine of given angle to sine of required angle. 

To Compute Base or Perpendicular in a Biglit-angled 

Triangle. 

When Angles and One Side next Right Angle are given. Rule. —As ra¬ 
dius is to tangent of angle adjacent to given side, so is this side to other side. 

To Compute tile other Side. 

When Two Sides and Included Angle are given. Rule. —As sum of two 
given sides is to their difference, so is tangent of half sum of their opposite 
angles to tangent of.half their difference; add this half difference to half 
sum, to ascertain greater angle; and subtract half difference from half sum, 
to ascertain less angle. The other side may then be ascertained by Rule 
above. 

To Compute Angles. 

When Sides are given. Rule. —As one side is to other side, so is radius 
to tangent of angle adjacent to first side. 

To Compute an Angle. 

When Three Sides are given. Rule i.—S ubtract sum of logarithms of 
sides which contain required angle, from 20; to remainder add logarithm 
of half sum of three sides, and that of difference between this half sum and 
side opposite to required angle. Half the sum of these three logarithms is 
logarithmic cosine of half required angle. The other angles may be ascer¬ 
tained by Rule above, 

2. — Subtract sum of logarithms of two sides which contain required 
angle, from 20, and to remainder add logarithms of differences between 
these two sides and half sum of the three sides. Half result is logarithmic 
sine of half required angle. 

Note.—I n all ordinary cases either of these rules will give sufficiently accurate 
results. Rule 1 should be used when required angle exceeds 90 0 ; and Rule 2 when 
it is less than 90°. 


PLANE TBIGONOMETRY. 


386 

Example. —The sides of a triangle are 3, 4, and 5; what are the angles of the 
hypothenuse ? 

20 — (Log. 4 — .60206 -f- Log. 5 = .69897) = 18.698 97; Log. 3 4 -f- 5 2 — 4 = 

. 30103; and Log. 3 -f 4 + 5 2 — 5 = o. 

Then 18.69897 + .30103 — 19, which-4-2 = 9.5 = log. sin. of half angle = 18 0 26', 
which X 2 = 36° 52' angle. 

Hence 90 0 — 36° 52' = 53 0 8' remaining angle. 

In following figures, 1 and 2: 

A = 90 0 , B — 45 0 , C = 45 0 , Radius = 1, Secant = 1.4142, Cosine = .7071, Sin. 45 0 
= .7071, Tangent = 1, Area = .25. 

By Sin., Tan., Sec., etc., A B, etc., is expressed Sine, Tangent, Secant, etc., of 
angles, A, B, etc. 


To Compute Sides A C and B C. —Figs. 1 and 2. 
When Hyp., Side B A, and Angles B and C are given. 

Sin. B X B A 


Fig- 1. 


B 



Sin. C — AC - 
BAX Cot. C = AC. 
Hyp. X Cos. C = AC. 
Hyp. X Sin. B = AC. 
B A 


Sin. C 
AC 


Sin. B 


= BC. 
= BC. 



A C 
Hyp. 


: Sin. B. 


To Compute Side A C and Angles. 
When Hyp. and Side B A are given. —Fig. 1 and 2. 
B A „ BAX Sin. B 


Hyp. 


; Sin. C. 


Sin. C 


-=AC. 


B C X Sin. B = AC. 


AC 

BA 


To Compute Side B C and Hyp. or Angles. 
When both Sides are given. —Fig. 2. 

B A 


: Tan. B. 


BA 
Sin. C 
BA 
ITc 


= BC. \/A C 2 -j- B A 2 — B C. 


A C 


= Tan. C. 


= Sin. C. 


rd= Si “ B - 


To Compute Sides.—Figs. 3 and 4. 


Fig. 3- c 



When a Side and an Angle are 
given. 

B C X Cos. B = B A. 

BC X Sin. B = AC. 


AB x Sec. B = BC. 

A C X Tan. C ACxSin.C 

Rad. ~ ‘ Sin. B 

A C X Sec. C A C X Rad. 

Rad. ‘ Sin. B 


= B A. 
— B C. 


Fig. 4- B 



In B AC, Fig. 5, a right-angled triangle, C A, is assumed to be radius; 
B A tangent of C, and B C secant to that radius ; Or, dividing each of these 
by base, there is obtained the tangent and secant of C respectively to radius 1. 

























PLANE TRIGONOMETRY. 


387 


Fig- 5- 



Radius C A = 1 
Secant 013 = 1.4142 
Tangent A B = 1 
Co-secant 06 = 1.4142 
Co tangent e B = 1 

VAC 2 + B A 2 = hyp. B C. 
AO-rCos. C = hyp. B C. 

Cos. 


/2 Area , Cos. C 

\ / 7,1 — 7T = Rad. ——- = Cot. C. 
V Tan. C Sin. C 


Sine dg — .7071 

Cosine C g or o d = .7071 
Versed sino gA= .2929 
Co-versed sine 0 e = .2929 
Angle C A B = 90° 

B A-r- Sin. C = hyp. B C. 
1 4- Tan. C = Cot. C. 

B C 2 X Sin. 2 C 


Area. 


B C X Cos. C = Rad. 
B A X Tan. B =; Rad. 
BC4-B A = Sec. B. 
B C X Cos. B = B A. 


Sin. C 
BAX Sec. B = B C. 

BAX Cot. C = Rad. B C X Sin. B = Rad. 

B C X Sin. C = B A. A C X Tan. C = BA. 

1 4- Sin. C = Cosec. C. 1 — Sin. C = Co-ver. sin. 


Cos. CA-Sin. C = Cot. C. C B X Sin. B = AC. 
Trigonometrical Eq_nivalents. 


Perp. 4- hyp. = Sin. C. 
Base 77- hyp. = Cos. C. 
Base 4- hyp. =Sin. B. 
Base 4- perp. = Cotan. C. 


V (1 — sin. 2 ) := Cos. 
Sin. -r- tan. = Cos. 
Sin. X cot. = Cos. 
Sin. 4- cos. m Tan. 
Cos. 4- cot. = Sin. 
Cos. 4- sin. z= Cot. 


Hyp. -r- base = Sec. C. 
Base -7- perp. = Tan. B. 
Perp. -7- hyp. = Cos. B. 
Hyp. — Base = Versin. 

Tan. -7- sin. = Sec. 
Tan. -7- sec. = Sin. 
Tan. X cot. = Rad. 

V (1 —cos. 2 ) = Sin. 

1 -7- cot. == Tan. 


sm. 


Cosec. 


Perp. -7- base = Tan. C. 
Hyp. = perp. = Sec. B. 
Hyp. -7- perp. = Cosec. C. 
Hyp. — Perp. = Co ver, sin. G" 

1 -7- cos. = Sec. 

1 -7- cosec. = Sin. 

1 -4- sec. == Cos. 

1 — cos. =: Versin. 

1 — sin. = Co-ver.sio. 

1 -r- tan. = Cotan. 


Illustrations. —Assume side A B of a right-angled triangle is 100, and angle C 
53 0 8'; what are its elements? 


Fig. 6. 


B 



Obliciiae-aiagled. Triangles. 

To Compote Sides B A and B C. 
When Side A C and Angles are given. 

Sin. C X A C „ . Sin. C X B C 


Fig. 6. 


Sin. B 


-BA. 

Sin. A X A C 
Sin B 


Sin. A 


B A. 


BC. 


To Compote Angles and Side A C. 

When Sides A B, B C, and one of the Angles are given. —Fig. 6. 


B C X Sin. B 
AC 


= Sin. A. 


Sin.CxA C 
B A 

Sin. B X BC 


A B X Sin. B 


AC 


= Sin. C. 


Sin. A 


AC. 



To Compote Sides B A and B C 
When Side A C and Angles are given .—Fi 
Sin. CxBC „ , Sin. Ax AC 

Sin. B 

Fiff, 


BA. 


Sin. A 

When Side B C and Angles are given 


e* 7- 

= BC. 


BC X Sin. C 


= B A. 


Sin. C X AC 
Sin. B 


&• 7- 
BA. 


Sin. (C-f- B) 

Note. — Sine and Cosine of an arc are each equal to sine and cosine of their sup¬ 
plements. 

Spherical Triangles , Right-angled and Oblique. For full formulas see 
Molesworth, Load., 1878, pp. 435-6. 




























388 


PLANE TRIGONOMETRY. 


To Compute Angles and. Side A C. 
When Sides A B, B C, and Angle B are given. —Fig. 7. 


Sin. B X BC 
Sin. A 

A C X Sin. A 
BC 


= AC. 


Sin. B. 


BC X Sin. B 


AC 
BAX Sin. A 


= Sin. A. 


= Sin. C. 


BAX Sin. B 
AC 

BCX Sin. C 


Sin. C. 


= Sin. A. 


BC — AB 

To Compute all the Angles. 

When all the Sides are given , Figs. 6 and 7. Rule.— Let fall a perpen¬ 
dicular, B d, opposite to required angle. Then, as A C : sum of A B, B C :: 
their difference : twice d g, the distance of perpendicular, B d, from middle 
of the base. 

Hence A d, C g are known, and triangle, A B C, is divided into two right- 
angled triangles, B C d , B A d ; then, by rules for right-angled triangles, 


ascertain angle A or 0 . 

Operation.— A C, Fig. 6, . 5014 : A B -f- B C, 1.1174 -f-1.4142 = 2.5316;'. A B co B C, 
1.4142'—1.1174^.2968 : 2 X d g =z 1 4986. 

1.4986 5014 

2 2 


Hence A d = d g — AC^-2 


. 4986, and Cd = Ad-f-AC = i. 


Consequently, triangle B d C, Fig. 6, is divided into two triangles, B A C and B d A. 
To Compute Side A B and Angles. 

When Two Sides and One Angle, or One Side and Two Angles, are given .— 


Fiff. 6. 


A C X Sin. C 


Sin. B 
AC X Sin. C 
AB 


= AB. 


Sin. B, 


B C X Sin. B _ 
A~C 

A B X Sin. B 


Sin. A. 


Sin. C. 


A C X Sin. A 


AB—(ACxCos. A) 

A C X Sin. C 


: Tan. B. 


Tan. B. 



AC BC —(ACxCos. C) 

To Compute Area of a Triangle. — Tig. 8. 

B A X BC X Sin. B ACxBCxSin. C B A X A C X Sin. A 

2 ’ 2 ’ 2 ’ 

Sin. 2C.BC 2 A C 2 , Tan. C „ B A 2 . Cot. C 

Fd— i - , -, and - :- = Area. 

4 2 2 

Note.— For other rules, see Mensuration of Areas, Lines, and 
Surfaces, page 335. 


Area 


To Compute Sides. 

When Areas and Angles are given. —Figs. 6 and 7. 
2 Area 


BC, Sin. C 


AC. 


A C, Sin. A 


= BA. 


2 Area, Sin. A 
V Sin. C, Sin. (A-J-C) 


Fig. 9. 


To Ascertain Distance of Inacces¬ 
sible Objects on a Level Diane.— 
Digs. Q and IO. 

Operation. —Lay off perpendic¬ 
ulars to line A B, Fig. 9, as B c, d e , 
on line A d, terminating on line 
e A. 

Then e d — cB:cB:;Bcl: BA. 

When there are Two Inacces¬ 
sible Objects , as Fig. xo. 

Operation. — Measure a base 
line, A B, Fig. 10, and angles c A B, 




d B A, A c d, B d c, etc. Then pro¬ 
ceed by formulas, page 387, to deduce c d. 

Note.—I f course of cd is required, take difference of angles 
do A and cclB from course A B. 




































PLANE TRIGONOMETRY., 


389 


B Fi §- «• 



When the Objects can be aligned .— 

Fig. 11. 

Operation. —Align c B, Fig. n, at A, 
measure a base line at any angle there¬ 
to, as A o, and angles 0 A c, B A c, and 
B o A. Then proceed as per formula, 
page 386, to deduce c B. 

To Compute Distance from 
a Gfiven Point to an In¬ 
accessible Object. — Dig. 
IS. 



Operation. —Measure a level line, Ac, Fig. 12, and ascertain angles, B Ac, c AB. 
Hence, having side, A c, and two angles, proceed as per formula, page 386, to de¬ 
termine A B. 


To Compute Height of an Elevated. Doint.—Fig. 13 . 


B Fig. 13. 


Operation. — Measure 
distance on a horizontal 
line, A c, Fig. 13; ascertain 
Angle B A c. Then pro¬ 
ceed as per formulas, pp. 
386-8, to ascertain B c. 

When a Horizontal 
Base is not Attainable. 

—Fig. 14. 

Operation.— Measure or 
compute distance Ac, Fig. 
14 ; ascertain angle of depression A 0 c and of elevation 
B A o. Then proceed as per formula, page 386, to ascertain 




Fig. 14. 


0 

Be. 



Operation. —Lay off any suitable and level distance, d d, set up a staff at each ex¬ 
tremity at like elevation from base line d d, and note distances y and x. at which 
the lines of sight of object range with tops of the staffs; deduct height of eye from 
length of staffs, and ascertain heights h. 

Then I> — 4 - h - 4 - s = height, s representing height of line of sight from base d d, 
x—y 

and D length of line d d. 


K K* 


























390 


NATURAL SINES AND COSINES 


jNTatnral Sines and. Cosines. 


Prop. 

parts. 


0 

O 

1° 

2° 

1 


3 ° 

I 

Prop. 

parts. 

29 

/ 

N. sine. 

N. coa. 

N. sine. 

N. cos. 

N. sine. 

| N. cos. 

N. sine. 

N. cos. 

1 

2 

O 

O 

.OOOOO 

I 

•01745 

.99985 

•0349 

.99939 

•05234 

•99863 

60 

2 

O 

I 

. 00029 

I 

.01774 

.99984 

•03519 

.99938 

•05263 

.99861 

59 

2 

I 

2 

.00058 

I 

01803 

.99984 

• 0354 8 

•99937 

.05292 

.9986 

58 

2 

I 

3 

.00087 

I 

.01832 

.99983 

•03577 

•99936 

•05321 

.99858 

57 

2 

2 

4 

.00116 

I 

.01862 

.99983 

.03606 

•99935 

•0535 

• 99 8 57 

56 

2 

2 

5 

.00145 

I 

.01891 

.99982 

•03635 

•99934 

•05379 

• 99 8 55 

55 

2 

3 

6 

.00175 

I 

.0192 

.99982 

.03664 

•99933 

.05408 

.99854 

54 

2 

3 

7 

.00204 

I 

.01949 

99981 

.03693 

•99932 

•05437 

.99852 

53 

2 

4 

8 

.00233 

I 

.01978 

.9998 

• 0 3723 

• 9993 i 

.05466 

.99851 

52 

2 

4 

9 

.00262 

I 

.02007 

.9998 

•03752 

•9993 

•05495 

•99849 

51 

2 

5 

IO 

.00291 

I 

.02036 

.99979 

.03781 

■ 999 2 9 

•05524 

.99847 

50 

2 

5 

II 

.0032 

•99999 

.02065 

•99979 

.0381 

.99927 

•05553 

.99846 

49 

2 

6 

12 

.00349 

•99999 

.02094 

.99978 

•° 3 8 39 

.qqq26 

•05582 

.99844 

48 

2 

6 

!3 

.00378 

•99999 

.02123 

•99977 

.03868 

.99925 

.05611 

.99842 

47 

2 

7 

*4 

.OO4O7 

•99999 

.02152 

•99977 

.03897 

.99924 

■0564 

.99841 

46 

2 

7 

15 

.00436 

•99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

45 

2 

8 

l6 

.00465 

•99999 

.02211 

.99976 

•03955 

.99922 

.05698 

.99838 

44 

I 

8 

*7 

.00495 

•99999 

.0224 

•99975 

•03984 

.99921 

.05727 

.99836 

43 

I 

9 

18 

.00524 

•99999 

.02269 

•99974 

•04013 

.99919 

•05756 

.99834 

42 

I 

9 

*9 

•00553 

.99998 

.02298 

•99974 

.04042 

.99918 

•05785 

• 99 8 33 

4 1 

I 

IO 

20 

.00582 

.99998 

.O2327 

•99973 

. O4O7I 

.99917 

.05814 

.99831 

4 ° 

I 

IO 

21 

.00611 

.99998 

.02356 

.99972 

.O4I 

.99916 

.05844 

.99829 

39 

I 

11 

22 

.0064 

.99998 

.02385 

.99972 

.04129 

■ 999*5 

•05873 

.99827 

38 

I 

II 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

■ 999*3 

.05902 

.99826 

37 

I 

12 

2 4 

.00698 

.99998 

.02443 

•9997 

.04188 

■ 999*2 

•05931 

.99824 

36 

I 

12 

25 

.00727 

•99997 

.02472 

.99969 

.04217 

.99911 

■0596 

.99822 

35 

I 

13 

26 

.00756 

•99997 

.02501 

.99969 

.04246 

.9991 

.05989 

.99821 

34 

I 

13 

27 

.00785 

•99997 

•0253 

.99968 

.04275 

.99909 

.06018 

.99819 

33 

I 

14 

28 

.00814 

•99997 

.0256 

.99967 

•04304 

.99907 

.06047 

.99817 

32 

I 

14 

29 

.00844 

.99996 

.02589 

.99966 

■04333 

.qqqo6 

.06076 

.99815 

3 1 

I 

15 

3 o 

.00873 

.99996 

.02618 

.<39966 

.04362 

•99905 

.06105 

.99813 

3 ° 

I 

15 

3 i 

. 00902 

.99996 

.02647 

.99965 

.04391 

.99904 

•06134 

.99812 

29 

I 

15 

32 

•00931 

.99996 

.02676 

.99964 

.0442 

.gggo2 

.06163 

.9981 

28 

I 

l6 

33 

.0096 

•99995 

•02705 

.99963 

•04449 

.99901 

.06192 

.99808 

27 

I 

l6 

34 

.00989 

•99995 

.02734 

.99963 

.04478 

•999 

.06221 

.qq8o6 

26 

I 

17 

35 

.01018 

•99995 

.02763 

.99962 

.04507 

.99898 

.0625 

.99804 

25 

I 

17 

36 

.01047 

•99995 

.02792 

.99961 

•04536 

.99897 

.06279 

.99803 

24 

I 

18 

37 

.01076 

.99994 

.02821 

.9996 

•04565 

.99896 

.06308 

.99801 

23 

I 

l8 

38 

.01105 

•99994 

.0285 

•99959 

•04594 

.99894 

•06337 

•99799 

22 

I 

19 

39 

.01134 

•99994 

.02879 

•99959 

.04623 

.99893 

.06366 

•99797 

21 

I 

*9 

4 o 

.01164 

•99993 

.02908 

.99958 

•04653 

.99892 

•06395 

•99795 

20 

I 

20 

4 i 

.01193 

•99993 

.02938 

•99957 

.04682 

.9989 

.06424 

•99793 

19 

I 

20 

42 

.01222 

•99993 

.02967 

.99956 

.04711 

.99889 

•06453 

•99792 

18 

I 

21 

43 

.01251 

•99992 

02 QQ 6 

•99955 

•0474 

.99888 

.06482 

•9979 

*7 

I 

21 

44 

.0128 

.99992 

.03025 

•99954 

.04769 

.99886 

.06511 

.99788 

16 

I 

22 

45 

.01309 

• 9999 1 

•03054 

•99953 

.04798 

.99885 

.0654 

.99786 

*5 

I 

22 

46 

•01338 

• 9999 1 

•03083 

.99952 

.04827 

.99883 

.06569 

.99784 

*4 

O 

23 

47 

.01367 

.99991 

.03112 

•99952 

.04856 

.99882 

.06598 

.99782 

*3 

O 

23 

48 

.01396 

•9999 

.03141 

•99951 

.04885 

.99881 

.06627 

•9978 

12 

O 

24 

49 

.01425 

•9999 

.0317 

•9995 

.04914 

.99879 

.06656 

.99778 

II 

O 

24 

5 o 

-01454 

.99989 

.03199 

•99949 

.04943 

.99878 

.06685 

•99776 

IO 

O 

25 

5 i 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

•99774 

9 

O 

25 

52 

•01513 

.99989 

•03257 

•99947 

.05001 

•99875 

•o 6743 

•99772 

8 

O 

26 

53 

.01542 

.99988 

.03286 

.99946 

•0503 

•99873 

.06773 

•9977 

7 

O 

26 

54 

.01571 

.99988 

•03316 

•99945 

•05059 

.99872 

.06802 

.99768 

6 

O 

27 

55 

.016 

.99987 

•03345 

•99944 

.05088 

.9987 

.06831 

.99766 

5 

O 

27 

56 

.01629 

.99987 

•03374 

•99943 

.05117 

.99869 

.0686 

.99764 

4 

O 

28 

57 

.01658 

.99986 

•03403 

•99942 

•05146 

.99867 

.06889 

.99762 

3 

O 

28 

58 

.01687 

.99986 

•03432 

.99941 

•05175 

.99866 

.06918 

■9976 

2 

O 

29 

59 

.01716 

.99985 

.03461 

•9994 

•05205 

.99864 

.06947 

.99758 

I 

O 

29 

60 

•01745 

.99985 

•0349 

•99939 

•05234 

.99863 

.06976 

•99756 

O 

O 



N. cos. 

N. sine. 

N. cos. | 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 1 



1 

89° 

88° 

87° 

86° 






















































NATURAL SINES AND COSINES. 39 1 


p.,2 

0 t, : 

*- a 1 
Sh p. 

2 9 

/ 

4 

N. sine. 

0 

N. cos. 

5 ° 

N. sine. | N. cos. 

6 

N. sine. 

0 

N. cos. 

7 

N. sine. 

ro 

N. cos. 


Prop. 
■*“• parts. 

O 

0 

.06976 

•99756 

.o 87 i 6 

.99619 

•10453 

•99452 

.12187 

• 99 2 55 

60 

4 

O 

1 

.07005 

•99754 

•08745 

.99617 

.10482 

•99449 

.122x6 

.99251 

59 

4 

I 

2 

.07034 

•99752 

.08774 

.99614 

.10511 

.99446 

.12245 

.99248 

58 

4 

I 

3 

.07063 

•9975 

.08803 

.99612 

.1054 

•99443 

.12274 

.99244 

57 

4 

2 

4 

.O7O92 

.99748 

.08831 

.99609 

.10569 

•9944 

.12302 

•9924 

56 

4 

2 

5 

.07121 

•99746 

.0886 

.99607 

• 10597 

•99437 

•I 233 I 

•99237 

55 

4 

3 

6 

•0715 

•99744 

.08889 

.99604 

.10626 

•99434 

.1236 

• 99 2 33 

54 

4 

3 

7 

.07179 

•99742 

.08918 

.99602 

•10655 

• 9943 1 

•I2389 

• 99 2 3 

53 

4 

4 

8 

.07208 

•9974 

.08947 

•99599 

.10684 

.99428 

.12418 

.99226 

S 2 

3 

4 

9 

.07237 

•99738 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

5 i 

3 

5 

IO 

.07266 

•99736 

.09005 

•99594 

• 10742 

• 9942 i 

.12476 

.99219 

50 

3 

5 

II 

.07295 

•99734 

•09034 

• 9959 1 

•IO77I 

.99418 

.12504 

•99215 

49 

3 

6 

12 

•07324 

• 9973 1 

.09063 

•99588 

. 108 

• 994 I 5 

•12533 

.99211 

48 

3 

6 

*3 

•07353 

•99729 

.O9O92 

.99586 

.10829 

.99412 

.12562 

.99208 

47 

3 

7 

*4 

.07382 

•99727 

.09121 

•99583 

.10858 

• 99+09 

•I 259 I 

.QQ204 

46 

3 

7 

i 5 

.O74II 

•99725 

.09x5 

•9958 

.10887 

.99406 

. 1262 

.992 

45 

3 

8 

l6 

•0744 

•99723 

.09179 

•99578 

.10916 

• 9 Q 402 

.12649 

.99197 

44 

3 

8 

1 7 

.07469 

.99721 

.09208 

•99575 

• io 945 

•99399 

.I2678 

• 99*93 

43 

3 

9 

18 

.07498 

.99719 

•09237 

■99572 

•10973 

.99396 

.12706 

.99189 

4 2 

3 

9 

x 9 

.07527 

•99716 

.09266 

•9957 

. 11002 

•99393 

• I2 735 

.99186 

4 1 

3 

IO 

20 

•07556 

.99714 

■09295 

•99567 

.11031 

•9939 

.12764 

.99182 

4 ° 

3 

IO 

21 

■07585 

.99712 

.09324 

.99564 

.1106 

•99386 

• I2 793 

.99178 

39 

3 

II 

22 

.07614 

.9971 

•09353 

.99562 

.11089 

•99383 

.12822 

• 99*75 

3 « 

3 

II 

2 3 

.07643 

.99708 

.09382 

•99559 

.11118 

•9938 

.12851 

• 99 * 7 I 

.37 

2 

12 

2 4 

.07672 

•99705 

.O94II 

•99556 

.11147 

•99377 

.1288 

• 99*67 

36 

2 

12 

2 5 

.O77OI 

•99703 

•0944 

•99553 

.1x176 

■99374 

.12908 

• 99*63 

35 

2 

13 

26 

•0773 

.99701 

.09469 

• 9955 i 

.11205 

•9937 

•12937 

.9916 

34 

2 

13 

2 7 

•07759 

.99699 

.09498 

•99548 

.11234 

•99367 

.12966 

• 99*56 

33 

2 

14 

28 

.07788 

.99696 

.09527 

•99545 

.11263 

•99364 

.12995 

• 99*52 

3 2 

2 

14 

29 

.07817 

.99694 

.09556 

.99542 

.II291 

•9936 

.13024 

.99148 

3 * 

2 

15 

3 o 

.07846 

.99692 

•09585 

•9954 

• 1132 

■99357 

•13053 

• 99*44 

3 ° 

2 

15 

3 i 

•07875 

.99689 

.09614 

•99537 

•11349 

•99354 

.13081 

.9914 1 

29 

2 

15 

32 

.07904 

.99687 

.09642 

•99534 

.11378 

■99351 

• 13x1 

• 99*37 

28 

2 

l6 

33 

•07933 

.99685 

.09671 

• 9953 i 

.11407 

•99347 

•13139 

• 99*33 

27 

2 

l6 

34 

.07962 

•99683 

•097 

•99528 

•11436 

•99344 

.13168 

■99129 

2 6 

2 

17 

35 

• 0799 1 

.9968 

.O9729 

•99526 

.11465 

• 9934 1 

•I 3 I 97 

.99*25 

25 

2 

17 

36 

.0802 

.99678 

•09758 

•99523 

.11494 

•99337 

.13226 

.99122 

24 

2 

18 

37 

.08049 

.99676 

.09787 

•9952 

■11523 

•99334 

•13254 

.99118 

23 

2 

18 

38 

.08078 

•99673 

.09816 

•99517 

•11552 

• 9933 1 

•13283 

• 99**4 

22 

I 

*9 

39 

.08107 

.99671 

.09845 

•99514 

• 1158 

•99327 

•13312 

• 99 11 

21 

I 

J 9 

40 

.08136 

.99668 

.09874 

■99511 

.11609 

•99324 

•I 334 I 

.99106 

20 

I 

20 

4 i 

.08165 

.99666 

•09903 

.99508 

.11638 

•9932 

•1337 

.99102 

19 

I 

20 

42 

.08194 

.99664 

.09932 

•99506 

.11667 

•99317 

•13399 

.99098 

l8 

I 

21 

43 

.08223 

.99661 

.09961 

•99503 

.11696 

• 993 M 

•13427 

•99094 

17 

I 

21 

44 

.08252 

.99659 

.0999 

•995 

.11725 

• 993 i 

•13456 

•99091 

16 

'I 

22 

45 

.08281 

•99657 

.IOOI9 

■99497 

•11754 

■99307 

•13485 

.99087 

15 

I 

22 

46 

.0831 

.99654 

.10048 

•99494 

.11783 

•99303 

•I 35 I 4 

.99083 

14 

I 

23 

47 

•08339 

.99652 

.IOO77 

.99491 

.11812 

•993 

•13543 

•99079 

13 

I 

23 

48 

.08368 

•99649 

.10106 

.99488 

.1x84 

.99297 

•13572 

•99075 

12 

I 

24 

49 

.08397 

.99647 

•10135 

•99485 

.11869 

.99293 

•136 

.99071 

11 

I 

24 

50 

.08426 

.99644 

.10164 

.00482 

.11898 

.9929 

•13629 

.99067 

10 

I 

25 

5 i 

•08455 

.99642 

.IOI92 

•99479 

.II927 

.99286 

•13658 

•99063 

9 

I 

25 

52 

.08484 

.99639 

.1022I 

•99476 

.11956 

.99283 

•13687 

■99059 

8 

I 

26 

53 

•08513 

•99637 

.1025 

•99473 

•11985 

.99279 

.13716 

•99055 

7 

O 

26 

54 

.08542 

• 99 6 35 

.IO279 

•9947 

.12014 

.99276 

•13744 

.99051 

6 

O 

27 

55 

•08571 

■99632 

.10308 

•99467 

.12043 

.99272 

•13773 

•99047 

5 

O 

27 

56 

.086 

•9963 

•10337 

.99464 

.12071 

.99269 

.13802 

•99043 

4 

O 

• 28 

57 

.08629 

.99627 

.10366 

.99461 

. 121 

.99265 

•13831 

•99039 

3 

O 

28 

58 

.08658 

.99625 

•10395 

•99458 

.I2I29 

.99262 

.1386 

•99035 

2 

O 

29 

59 

.08687 

.99622 

IO424 

•99455 

.12158 

.99258 

•13889 

.99031 

1 

O 

29 

60 

.08716 

.99619 

•10453 

.99452 

.12187 

• 99 2 55 

• I 39 I 7 

.99027 

0 

O 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 




85° 

84° 

83° 

82° 


















































39 2 


NATURAL SINES AND COSINES. 


cL * 

© >• 
> c 5 
Ph p, 

28 

/ 

8 

N. sine. 

0 

N. cos. 

9 

N. sine. 

0 

N. cos. 

1 

N. sine. 

30 

N. cos. 

1 

N. sine. 

10 

N. cos. 


Prop. 

1 O' parts. 

O 

O 

•I 39 I 7 

.QQ027 

■15643 

.98769 

•17365 

.98481 

M 

VO 

0 

GO 

M 

.98163 

60 

6 

O 

I 

•13946 

•99023 

.15672 

.98764 

•17393 

.98476 

.I9IO9 

• 98 i 57 

59 

6 

I 

2 

•13975 

.99019 

.15701 

.9876 

. 17422 

.98471 

.19138 

.98152 

58 

6 

I 

3 

.I4OO4 

.99015 

•1573 

•98755 

•I 745 I 

.98466 

.19167 

.98146 

57 

6 

2 

4 

•14033 

.99011 

•15758 

•98751 

•17479 

.98461 

•i 9!95 

.9814 

56 

6 

2 

5 

.14061 

.99006 

■15787 

.98746 

.17508 

•98455 

.I9224 

•98135 

55 

6 

3 

6 

.1409 

. 99002 

.15816 

.98741 

•17537 

•9845 

.19252 

.98129 

54 

5 

3 

7 

.14119 

.98998 

•15845 

•98737 

•17565 

•98445 

. 19281 

.98124 

53 

5 

4 

8 

.14148 

•98994 

•15873 

.98732 

•17594 

■9844 

•19309 

.98118 

52 

5 

4 

9 

.14177 

.9899 

.15902 

.98728 

.17623 

•98435 

•19338 

.98112 

5 i 

5 

5 

IO 

.14205 

.98986 

•I 593 I 

■98723 

.17651 

•9843 

.19366 

.08107 

5 o 

5 

5 

II 

.14234 

.98982 

•15959 

.98718 

. 1768 

.98425 

•19395 

.98101 

49 

5 

6 

12 

.14263 

.98978 

.15988 

.98714 

.17708 

.9842 

.19423 

.98096 

48 

5 

6 

13 

. 14292 

•98973 

.16017 

.98709 

•17737 

.98414 

.19452 

.9809 

47 

5 

7 

14 

.1432 

.98969 

.16046 

.98704 

.17766 

.98409 

.19481 

.98084 

46 

5 

7 

15 

•14349 

.98965 

.16074 

.987 

•17794 

.98404 

•19509 

.98079 

45 

5 

7 

16 

•14378 

.98961 

.16103 

•98695 

.17823 

•98399 

•19538 

.98073 

44 

4 

8 

17 

.14407 

•98957 

.16132 

.9869 

•17852 

.98394 

• 19566 

.98067 

43 

4 

8 

18 

.14436 

•98953 

.1616 

.98686 

. 1788 

•98389 

•19595 

.98061 

42 

4 

9 

19 

.14464 

.98948 

.16189 

.98681 

.17909 

•98383 

• 19623 

.98056 

4 i 

4 

9 

20 

.14493 

.98944 

.16218 

.98676 

•17937 

.98378 

.19652 

.9805 

40 

4 

IO 

21 

.14522 

.9894 

.16246 

.98671 

.17966 

•98373 

. 1968 

.98044 

39 

4 

IO 

22 

•I 455 I 

.98936 

.16275 

.98667 

•17995 

.98368 

.19709 

.98039 

38 

4 

II 

23 

•1458 

.98931 

.16304 

.98662 

.18023 

.98362 

•19737 

•98033 

37 

4 

II 

24 

.14608 

.98927 

•16333 

•98657 

.18052 

•98357 

.19766 

.98027 

36 

4 

12 

25 

•14637 

•98923 

.16361 

.98652 

.18081 

•98352 

.19794 

.98021 

35 

4 

12 

26 

.14666 

.98919 

.1639 

.98648 

.18109 

•98347 

.19823 

.98016 

34 

3 

13 

27 

.14695 

.98914 

.16419 

.98643 

.18138 

.98341 

.19851 

.9801 

33 

3 

13 

28 

•14723 

.9891 

•16447 

.98638 

.18166 

• 9 8 336 

. 1988 

.98004 

32 

3 

14 

2 9 

•14752 

.98906 

.16476 

•98633 

.18195 

•98331 

. 19908 

.97988 

3 i 

3 

14 

3 ° 

.14781 

.98902 

• 16505 

.98629 

.18224 

•98325 

•19937 

•97992 

3 ° 

3 

14 

31 

. 1481 

.98897 

•16533 

.98624 

.18252 

•9832 

•19965 

.97987 

29 

3 

15 

32 

.14838 

.98893 

.16562 

.98619 

.18281 

•98315 

• 19994 

.97981 

28 

3 

is 

33 

.14867 

.98889 

.16591 

.98614 

.18309 

•9831 

.20022 

•97975 

27 

3 

16 

34 

.14896 

.98884 

. 1662 

.98609 

•18338 

•98304 

.20051 

.97969 

26 

3 

16 

35 

•14925 

.9888 

.16648 

.98604 

.18367 

.98299 

.2OO79 

.97963 

25 

3 

17 

3 b 

•14954 

.98876 

. 16677 

.986 

•i 8395 

.98294 

-20108 

•97958 

24 

2 

17 

37 

.14982 

.98871 

.16706 

•98595 

.18424 

.98288 

.20136 

•97952 

23 

2 

18 

3 8 

.15011 

.98867 

•16734 

•9859 

.18452 

.98283 

.20165 

.97946 

22 

2 

l8 

39 

.1504 

.98863 

.16763 

.98585 

.18481 

.98277 

.20193 

•9794 

21 

2 

19 

4 ° 

.15069 

.98858 

.16792 

.9858 

.18509 

.98272 

.20222 

■97934 

20 

2 

19 

4 i 

•15097 

.98854 

. 1682 

•98575 

.18538 

.98267 

.2025 

.97928 

19 

2 

20 

42 

.15126 

.98849 

. 16849 

•9857 

.18567 

.98261 

.20279 

.97922 

18 

2 

20 

43 

•I 5 I 55 

.98845 

.16878 

•98565 

•i 8595 

.98256 

.20307 

• 979 l6 

17 

2 

21 

44 

.15184 

.98841 

.16906 

.98561 

.18624 

.9825 

• 20336 

.9791 

16 

2 

21 

45 

.15212 

.98836 

•16935 

•98556 

.18652 

•98245 

.20364 

• 979°5 

15 

2 

21 

4 6 

.15241 

.98832 

.16964 

•98551 

.18681 

.9824 

•20393 

•97899 

14 

I 

22 

47 

•1527 

.98827 

.16992 

.98546 

. 1871 

•98234 

.20421 

•97893 

13 

I 

22 

48 

.15299 

.98823 

.17021 

.98541 

.18738 

.98229 

• 2045 

.97887 

12 

I 

23 

49 

•15327 

.98818 

•1705 

•98536 

.18767 

.98223 

.20478 

.97881 

11 

I 

23 

50 

•15356 

.98814 

. 17078 

•98531 

•i 8795 

.98218 

• 20507 

•97875 

10 

I 

24 

5 i 

•15385 

.98809 

.I7IO7 

.98526 | 

.18824 

.98212 

•20535 

.97869 

9 

I 

24 

52 

•I 54 I 4 

.98805 

.17136 

.98521 

.18852 

.98207 

.20563 

•97863 

8 

I 

25 

53 

.15442 

.988 

.17164 

.98516 

.18881 

.98201 

.20592 

•97857 

7 

I 

25 

54 

•I 547 I 

.98796 

•I 7 I 93 

.98511 

.1891 

.98196 

.2062 

•97851 

6 

I 

26 

55 

•155 

.98791 

. 17222 

.98506 

.18938 

.9819 

.20649 

•97845 

5 

I 

26 

56 

•15529 

.98787 

■1725 

.98501 

.18967 

.98185 

.20677 

•97839 

4 

O 

27 

57 

•15557 

.98782 

.17279 

.98496 

.18995 

.98179 

.20706 

•97833 

3 

O 

27 

58 

•15586 

.98778 

.17308 

.98491 

. 19024 

.98174 

.20734 

■97827 

2 

O 

28 

59 

•15615 

•98773 

•17336 

.98486 

• 19052 

.98168 

•20763 

.97821 

I 

O 

28 

60 

•15643 

.98769 

•17365 

.98481 

.19081 

.98163 

.20791 

•97815 

O 

O 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 




810 

80 ° 

79° 

<1 

00 

0 






















































NATURAL SINES AND COSINES 


393 


Prop. 

parts. 


12° 

1 

13 ° 

140 

15 ° 


Prop. 

parts. 

27 

1 ' 

N. sine. 

1 N. cos. 

.1 

X. sine. 

] N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 


9 

O 

0 

.20791 

•97815 

•22495 

•97437 

.24192 

■9703 

.25882 

•96593 

60 

9 

O 

I 

.2082 

.97809 

.22523 

•9743 

.2422 

.97023 

.2591 

.96585 

59 

9 

I 

2 

.20848 

.97803 

•22552 

•97424 

. 2424 Q 

•97015 

.25938 

.96578 

58 

9 

I 

3 

.20877 

•97797 

.2258 

■ 974 U 

.24277 

.97008 

.25966 

•9657 

57 

9 

2 

4 

.20905 

.97791 

.22608 

.97411 

•24305 

.97001 

•25994 

.96562 

56 

8 

2 

5 

.20933 

.97784 

.22637 

.97404 

•24333 

.96994 

.26022 

•96555 

55 

8 

3 

6 

.20962 

.97778 

.22665 

•97398 

• 24362 

.96987 

.2605 

•96547 

54 

8 

3 

7 

. 2099 

.97772 

.22693 

• 9739 1 

•2439 

.9698 

.26079 

•9654 

53 

8 

4 

8 

.21019 

.97766 

. 22722 

•97384 

.24418 

•96973 

.26107 

.96532 

52 

8 

4 

9 

.21047 

.9776 

.2275 

•97378 

.24446 

.96966 

•26135 

.96524 

5 i 

8 

5 

<IO 

.21076 

•97754 

.22778 

• 9737 i 

•24474 

.96959 

.26163 

•96517 

50 

8 

5 

. II 

.21104 

.97748 

.22807 

•97365 

•24503 

.96952 

• 26191 

.96509 

49 

7 

5 

12 

.21132 

.97742 

•22835 

•97358 

•24531 

.96945 

.26219 

.96502 

48 

7 

6 

13 

.21161 

•97735 

.22863 

• 9735 i 

•24559 

•96937 

.26247 

•96494 

47 

7 

6 

14 

.21189 

.97729 

.22892 

•97345 

.24587 

•9693 

•26275 

.96486 

46 

7 

7 

15 

.21218 

•97723 

• 2292 

•97338 

•24615 

.96923 

• 26303 

•96479 

45 

7 

7 

l6 

.21246 

•97717 

.22948 

•97331 

.24644 

.96916 

•26331 

.96471 

44 

7 

8 

J 7 

.21275 

.97711 

•22977 

•97325 

.24672 

.96909 

•26359 

•96463 

43 

6 

8 

l8 

.21303 

•97705 

.23005 

• 973 i 8 

•247 

.96902 

•26387 

•96456 

42 

6 

9 

19 

•21331 

.97698 

•23033 

• 973 H 

.24728 

.96894 

•26415 

.96448 

4 1 

6 

9 

20 

.2136 

.97692 

.23062 

•97304 

•24756 

.96887 

•26443 

•9644 

4 ° 

6 

9 

21 

.21388 

.97686 

.2309 

.97298 

•24784 

.9688 

.26471 

•96433 

39 

6 

IO 

22 

.2x417 

.9768 

.23118 

.97291 

•24813 

•96873 

•265 

.96425 

38 

6 

IO 

23 

.21445 

•97673 

•23146 

.97284 

.24841 

.96866 

.26528 

.96417 

37 

6 

II 

24 

.21474 

.97667 

•23175 

.97278 

.24869 

.96858 

•26556 

.9641 

S 6 

5 

II 

25 

.21502 

.97661 

.23203 

.97271 

•24897 

.96851 

.26584 

.96402 

35 

5 

12 

26 

•2153 

•97655 

.23231 

•97264 

■24925 

.96844 

.26612 

.96394 

34 

5 

12 

27 

• 2 X 559 

.97648 

. 2326 

•97257 

•24954 

.96837 

.2664 

.96386 

33 

5 

43 

28 

.21587 

.97642 

.23288 

• 9725 i 

.24982 

.96829 

.26668 

•96379 

32 

5 

13 

2 9 

.21616 

.97636 

•23316 

.97244 

.2501 

96822 

.26696 

•96371 

3 1 

5 

14 

30 

.21644 

•9763 

•23345 

•97237 

• 25038 

•96815 

.26724 

•96363 

3 ° 

5 

14 

3 i 

.2x672 

.97623 

■23373 

•9723 

.25066 

.96807 

.26752 

•96355 

29 

4 

14 

32 

.21701 

•97617 

■23401 

.97223 

.25094 

.968 

.2678 

•96347 

28 

4 

15 

33 

.21729 

.97611 

.23429 

.97217 

.25122 

•96793 

.26808 

•9634 

27 

4 

15 

34 

.21758 

.97604 

.23458 

.9721 

•25151 

.96786 

.26836 

.96332 

26 

4 

l6 

35 

.21786 

•97598 

.23486 

.97203 

•25179 

.96778 

.26864 

.96324 

25 

4 

l6 

36 

.21814 

•97592 

•23514 

.97196 

.25207 

• 9 6 77 i 

.26892 

.96316 

24 

4 

17 

37 

.21843 

•97585 

•23542 

.97189 

•25235 

.96764 

.2692 

.96308 

23 

3 

17 

3 « 

.21871 

•97579 

•23571 

.97182 

.25263 

.96756 

.26948 

.96301 

22 

3 

18 

39 

.21899 

•97573 

•23599 

.97176 

•25291 

.96749 

.26976 

.96293 

21 

3 

18 

40 

.21928 

.97566 

.23627 

.97x69 

•2532 

•96742 

.27004 

.96285 

20 

3 

18 

4 i 

.21956 

•9756 

•23656 

.97162 

.25348 

•96734 

. 27032 

.96277 

19 

3 

r 9 

42 

.21985 

•97553 

.23684 

•97155 

•25376 

.96727 

.2706 

.96269 

18 

3 

r 9 

43 

.22012 

•97547 

.23712 

.97148 

.25404 

.96719 

.27088 

.96261 

17 

3 

20 

44 

.22041 

•97541 

•2374 

.97141 

•25432 

.96712 

.27116 

.96253 

l6 

2 

20 

45 

.2207 

•97534 

.23769 

•97134 

•2546 

.96705 

.27144 

.96246 

15 

2 

21 

46 

.22098 

.97528 

•23797 

• 97 I2 7 

.25488 

.96697 

.27172 

.96238 

*4 

2 

21 

47 

.22126 

•97521 

•23825 

.9712 

•25516 

.9669 

.272 

.9623 

13 

2 

22 

48 

•22155 

•97515 

•23853 

• 97 II 3 

•25545 

.96682 

.27228 

.96222 

12 

2 

22 

49 

.22183 

.97508 

.23882 

.97106 

•25573 

.96675 

.27256 

.96214 

II 

2 

23 

50 

.22212 

.97502 

.2391 

.971 

.25601 

.96667 

.27284 

.96206 

IO 

2 

23 

5 i 

.2224 

.97496 

•23938 

•97093 

.25629 

.9666 

.27312 

.96198 

9 

I 

23 

52 

.22268 

.97489 

.23966 

.97086 

•25657 

.96653 

• 2 734 

.9619 

8 

I 

24 

53 

.22297 

•97483 

•23995 

•97079 

•25685 

.96645 

.27368 

.96182 

7 

I 

24 

54 

.22325 

.97476 

.24023 

.97072 

•25713 

.96638 

.27396 

.96174 

6 

I 

25 

55 

•22353 

•9747 

.24051 

.97065 

•25741 

■9663 

.27424 

.96166 

5 

I 

25 

56 

.22382 

•97463 

.24079 

.97058 

.25769 

.96623 

.27452 

•96158 

4 

I 

26 

57 

.2241 

•97457 

.24108 

•97051 

•25798 

.96615 

.2748 

.9615 

3 

O 

26 

5 « 

.22438 

•9745 

.24136 

.97044 

.25826 

.96608 

.27508 

.96142 

2 

O 

27 

59 

.22467 

•97444 

.24164 

•97037 

•25854 

.966 

•27536 

.96134 

1 

O 

27 

60 

.22495 

•97437 

.24192 

•9703 

.25882 

•96593 

.27564 

.96126 

0 




N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 




770 

76 ° I 

750 

740 | 









































































394 


NATURAL SINES AND COSINES 


o u 

*"■ C 3 

Ph p. 

27 

/ 

l< 

N. sine. 

5° 

N. cos. 

1?0 

N". sine. | N. cos. 

1 

N. sine. 

8° 

N. cos. 

190 

N. sine. | N. cos. 


a.® 
£ £ 
A 13 

Ph 0. 

9 

0 

0 

.27564 

.96126 

•29237 

•9563 

. 30Q02 

.95106 

•32557 

•94552 

60 

9 

0 

1 

.27592 

.96118 

.29265 

.95622 

•30929 

•95097 

•32584 

.94542 

59 

9 

I 

2. 

.2762 

.9611 

.29293 

•95613 

•30957 

.95088 

.32612 

•94533 

58 

9 

I 

3 

.27648 

.96102 

.29321 

•95605 

.30985 

•95079 

•32639 

•94523 

57 

9 

2 

4 

.27676 

.q 6 oq 4 

.29348 

•95596 

.31012 

•9507 

.32667 

'94514 

56 

8 

2 

5 

.27704 

.96086 

.29376 

•95588 

.3104 

.95061 

.32694 

•94504 

55 

8 

3 

6 

•27731 

.96078 

.29404 

•95579 

.31068 

•95052 

.32722 

•94495 

54 

8 

3 

7 

•27759 

.9607 

.29432 

• 9557 i 

•31095 

•95043 

•32749 

•94485 

53 

8 

4 

8 

.27787 

.96062 

.2946 

•95562 

• 3 II2 3 

•95033 

•32777 

.94476 

52 

8 

4 

9 

•27815 

.96054 

•29487 

•95554 

•31151 

.95024 

.32804 

.94466 

5 i 

8 

5 

IO 

•27843 

.96046 

•29515 

•95545 

.31178 

•95015 

.32832 

•94457 

50 

8 

5 

II 

.27871 

.96037 

•29543 

•95536 

.31206 

.95006 

•32859 

•94447 

49 

7 

5 

12 

.278QQ 

.96029 

•29571 

•95528 

•31233 

•94997 

.32887 

•94438 

48 

7 

6 

13 

.27027 

.96021 

•29599 

•95519 

.31261 

.94988 

.32914 

.94428 

47 

7 

6 

14 

•27955 

.96013 

.29626 

• 955 ii 

.31289 

•94979 

.32942 

.94418 

46 

7 

7 

15 

■ 279 8 3 

.96005 

.29654 

•95502 

• 3 i 3 i 6 

•9497 

•32969 

•94409 

45 

7 

7 

l6 

.28011 

•95997 

.29682 

■95493 

•31344 

.94961 

.32997 

•94399 

44 

7 

8 

17 

•28039 

.95989 

.2971 

•95485 

•31372 

•94952 

•33024 

•9439 

43 

6 

8 

18 

.28067 

• 959 Sl 

•29737 

•95476 

•31399 

•94943 

•33051 

•9438 

42 

6 

9 

J 9 

.28095 

.95972 

.29765 

•95467 

•31427 

•94933 

•33079 

•9437 

4 1 

6 

9 

20 

.28123 

.95964 

•29793 

•95459 

• 3*454 

.94924 

.33106 

.94361 

4 ° 

6 

9 

21 

•2815 

•95956 

.29821 

•9545 

.31482 

• 949*5 

• 33 L 34 

9435 i 

39 

6 

IO 

22 

. 28178 

.95948 

.29849 

• 9544 i 

• 3 i 5 i 

.94906 

• 33 i 6 i 

.94342 

38 

6 

IO 

23 

.28206 

•9594 

.29876 

•95433 

•31537 

.94897 

•33189 

•94332 

37 

6 

IX 

24 

•28234 

• 9593 i 

. 29904 

•95424 

•31565 

. 94888 

.33216 

.94322 

36 

5 

II 

25 

.28262 

• 959 2 3 

•29932 

•95415 

•31593 

.94878 

•33244 

•94313 

35 

5 

12 

26 

.2829 

• 959*5 

.2996 

•95407 

.3162 

.94869 

•33271 

•94303 

34 

5 

12 

27 

.28318 

■ 959°7 

.29987 

•95398 

.31648 

.9486 

•33298 

•94293 

33 

5 

13 

28 

.28346 

.95898 

• 3 OOI 5 

•95389 

•31675 

.94851 

.33326 

.94284 

32 

5 

13 

29 

•28374 

•9589 

• 30043 

•9538 

•31703 

.94842 

■33353 

•94274 

3 i 

5 

M 

30 

.28402 

.95882 

.30071 

•95372 

• 3 i 73 

.94832 

• 3338 i 

.94264 

3 ° 

5 

14 

31 

.28429 

•95874 

.30098 

■95363 

•31758 

•94823 

•33408 

•94254 

29 

4 

M 

32 

.28457 

•95865 

.30126 

•95354 

.31786 

.94814 

•33436 

•94245 

28 

4 

15 

33 

.28485 

•95857 

•30154 

•95345 

•31813 

.94805 

•33463 

•94235 

27 

4 

15 

34 

•28513 

.95849 

.30182 

•95337 

.31841 

•94795 

•3349 

.94225 

26 

4 

l6 

35 

.28541 

.95841 

.302oq 

•95328 

.31868 

.94786 

• 335 i 8 

.94215 

25 

4 

l6 

36 

.28569 

•95832 

•30237 

•95319 

.^i8q6 

•94777 

•33545 

.94206 

24 

4 

*7 

37 

.28597 

.95824 

.30265 

• 953 i 

• 3 i 9 2 3 

.94768 

•33573 

.94196 

23 

3 

17 

38 

.28625 

95816 

.30292 

•95301 

• 3 * 95 * 

•94758 

•336 

.94186 

22 

3 

18 

39 

.28652 

.95807 

•3032 

•95293 

•31979 

•94749 

•33627 

.94176 

21 

3 

18 

40 

.2868 

•95799 

•30348 

.95284 

.32006 

•9474 

•33655 

.94167 

20 

3 

18 

41 

.28708 

• 9579 1 

•30376 

•95275 

•32034 

•9473 

.33682 

•94157 

19 

3 

19 

42 

.28736 

.95782 

•30403 

.95266 

.32061 

.94721 

• 337 i 

.94147 

18 

3 

*9 

43 

.28764 

•95774 

•30431 

•95257 

.32089 

• 947 12 

•33737 

•94137 

*7 

3 

20 

44 

.28792 

.95766 

•30459 

.95248 

.32116 

.94702 

•33764 

.94127 

l6 

2 

20 

45 

.2882 

•95757 

.30486 

•9524 

.32144 

•94693 

•33792 

.94118 

i 5 

2 

21 

46 

.28847 

•95749 

•30514 

•95231 

.32171 

94684 

•33819 

.94108 

M 

2 

21 

47 

.28875 

•9574 

•30542 

.95222 

.32199 

•94674 

•33846 

.94098 

13 

2 

22 

48 

.28903 

•95732 

•3057 

•95213 

.32227 

•94665 

•33874 

.94088 

12 

2 

22 

49 

.28931 

•95724 

• 30597 

.95204 

•32254 

.94656 

• 339 01 

.94078 

II 

2 

23 

50 

.28959 

•95715 

•30625 

■95195 

.32282 

.94646 

•33929 

.94068 

IO 

2 

23 

51 

.28987 

•95707 

•30653 

.95186 

.32309 

•94637 

•33956 

.94058 

9 

1 

23 

52 

.29015 

.95698 

.3068 

•95177 

•32337 

.94627 

•33983 

•94049 

8 

1 

24 

53 

. 29042 

•9569 

• 30708 

.95168 

•32364 

.94618 

.34011 

•94039 

7 

r 

24 

54 

.2907 

.95681 

•30736 

•95159 

.32392 

. Q460Q 

•34038 

. Q 4 - 02 Q 

6 

1 

25 

55 

. 29098 

•95673 

• 30763 

• 95 i 5 

.32419 

•94599 

.34065 

.94019 

5 

r 

25 

56 

.29126 

•95664 

.30791 

.95142 

•32447 

•9459 

• 34093 

. Q4.OOQ 

4 

1 

26 

57 

.29154 

•95656 

.30819 

•95133 

•32474 

•9458 

.3412 

•93999 

3 

0 

26 

58 

.29182 

•95647 

.30846 

• 95 I2 4 

•32502 

•94571 

•34147 

■93989 

2 

0 

27 

59 

.29209 

•95639 

.30874 

• 95 ii 5 

•32529 

• 9456 i 

•34175 

•93979 

I 

0 

27 

60 

.29237 

•9563 

.30902 

.95106 

•32557 

•94552 

. 34202 

•93969 

O 

0 



N. cos. 

N. sine. 

N. cos. 

N, sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 




730 

72° 

71° 

700 


















































NATURAL SINES AND COSINES 


Pi W 
£ u 


20 ° 

210 

22 ° 

23 ° 

27 

f 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

0 

0 

.34202 

.93969 

•35837 

•93358 

•3746 i 

.92718 

•39073 

•9205 

0 

I 

.34229 

•93959 

•35864 

•93348 

•37488 

.92707 

•39 1 

•92039 

I 

2 

•34257 

•93949 

•35891 

•93337 

•37515 

.92697 

•39 I2 7 

.92028 

I 

3 

.34284 

■93939 

•359 l8 

•93327 

•37542 

.92686 

•39*53 

.92016 

2 

4 

•343H 

•93929 

•35945 

• 933 i 6 

•37569 

.92675 

•39 l8 

.92005 

2 

5 

•34339 

•939 I 9 

•35973 

■933o6 

•37595 

.92664 

•39207 

.91994 

3 

6 

•34366 

•93909 

•36 

•93295 

.37622 

•9 2 653 

•39234 

.giq82 

3 

7 

•34393 

•93899 

•36027 

•93285 

•37649 

92642 

.3926 

■91971 

4 

8 

.34421 

•93889 

•36054 

•93274 

•37676 

.92631 

.39287 

•91959 

4 

9 

.34448 

•93879 

.36081 

.93264 

•37703 

.9262 

•39314 

.91948 

5 

IO 

•34475 

.93869 

.36108 

•93253 

•3773 

.92609 

•3934i 

.9x936 

5 

II 

•34503 

•93859 

•36135 

•93243 

•37757 

.92598 

39367 

.91925 

5 

12 

•3453 

•93849 

.36162 

.93232 

•37784 

•92587 

•39394 

.91914 

6 

13 

•34557 

■93839 

.3619 

.Q3222 

.37811 

.92576 

.39421 

.91902 

6 

h 

•34584 

.93829 

.36217 

.93211 

•37838 

•92565 

•39448 

.91891 

7 

15 

.34612 

.93819 

.36244 

•93201 

•37865 

•92554 

•39474 

.9x879 

7 

l6 

•34639 

.93809 

.36271 

•93i9 

•37892 

•9 2 543 

• 395 oi 

.91868 

8 

17 

.34666 

•93799 

.36298 

.9318 

•379 r 9 

•92532 

•39528 

.9x856 

8 

18 

.34694 

•93789 

•36325 

.93169 

•37946 

.92521 

•39555 

.91845 

9 

19 

•34721 

•93779 

•36352 

•93159 

•37973 

.9251 

•3958 x 

•9 i8 33 

9 

20 

•34748 

•93769 

•36379 

.93148 

•37999 

•92499 

.39608 

.91822 

9 

21 

•34775 

•93759 

.36406 

•93137 

.38026 

.92488 

•39635 

.9181 

IO 

22 

•34803 

•93748 

•36434 

•93127 

•38053 

.92477 

.39661 

.9x799 

IO 

23 

•3483 

•93738 

.36461 

•93 ij 6 

.3808 

.92466 

.39688 

.91787 

II 

24 

34857 

•93728 

.36488 

93 iq 6 

.38107 

•92455 

•39715 

•9*775 

II 

25 

.34884 

•937 l8 

■36515 

•93095 

•38134 

•92444 

•39741 

.91764 

12 

26 

•349 12 

.93708 

•36542 

.93084 

.38161 

.92432 

.39768 

•9*752 

12 

27 

•34939 

•93698 

•36569 

•93074 

.38188 

.92421 

•39795 

.9x741 

13 

28 

.34966 

.93688 

•36596 

•93063 

•38215 

.9241 

.39822 

.91729 

13 

29 

• 34993 

•93677 

• 36623 

•93052 

.38241 

.92399 

•39848 

.91718 

14 

30 

.35021 

.93667 

•3665 

93042 

.38268 

.92388 

•39875 

.91706 

14 

31 

•35048 

•93657 

•36677 

•93031 

•38295 

•92377 

.39902 

.91694 

14 

32 

•35075 

•93647 

•36704 

.9302 

.38322 

.92366 

.39928 

.91683 

15 

33 

.35102 

•93637 

•36731 

,9301 

•38349 

•92355 

•39955 

.91671 

15 

34 

•35i3 

.93626 

•36758 

.92999 

•38376 

•92343 

.39982 

.9166 

l6 

35 

•35157 

.93616 

•36785 

.92988 

•38403 

•92332 

.40008 

.91648 

16 

3 6 

•35184 

.Q3606 

.36812 

.92978 

-3843 

.92321 

•40035 

•9 i6 3 6 

17 

37 

•352ii 

•93596 

•36839 

.92967 

•38456 

.9231 

.40062 

.91625 

17 

38 

•35239 

•93585 

.36867 

.92956 

•38483 

.92299 

.40088 

.9x613 

18 

39 

.35266 

•93575 

•36894 

.92945 

•3851 

.92287 

.40115 

.91601 

18 

40 

•35293 

•93565 

.36921 

•92935 

•38537 

.92276 

.4OI4I 

■9 1 59 

18 

4 1 

•3532 

•93555 

•36948 

.92924 

•38564 

.92265 

.40168 

•9*57 8 

*9 

42 

•35347 

•93544 

•36975 

•9 2 9 x 3 

-3859 1 

•92254 

.40195 

.9x566 

X 9 

43 

•35375 

•93534 

.37002 

.92902 

.38617 

.92243 

.40221 

•9 I 555 

20 

44 

.35402 

•93524 

• 37029 

.92892 

.38644 

.92231 

.40248 

•9 I 543 

20 

45 

•35429 

■93514 

•37056 

.92881 

.38671 

.9222 

•40275 

•9*53* 

21 

46 

•35456 

•93503 

•37083 

.9287 

.38698 

.92209 

.40301 

•9 I 5 I 9 

21 

47 

•35484 

•93493 

•37 11 

.92859 

•38725 

.92198 

■40328 

.91508 

22 

48 

•355H 

•93483 

•37137 

.92849 

•38752 

.92186 

•40355 

.91496 

22 

49 

•35538 

•93472 

•37164 

.92838 

•38778 

•92175 

.40381 

.91484 

23 

50 

•35565 

.93462 

•37I9 1 

.92827 

.38805 

.92164 

.40408 

.91472 

23 

5i 

•3559 2 

•93452 

.37218 

.928x6 

.38832 

.92x52 

.40434 

.91461 

23 

52 

•35619 

•9344i 

•37245 

92805 

•38859 

.92141 

.40461 

•9*449 

24 

53 

•35647 

•93431 

.37272 

.92794 

.38886 

.9213 

.40488 

•9*437 

24 

54 

•35674 

•9342 

.37299 

.92784 

.38912 

.92119 

.40514 

.9x425 

25 

55 

•35701 

•934i 

•37326 

•9 2 773 

•38939 

.92107 

.40541 

.914x4 

25 

56 

•35728 

•934 

•37353 

.92762 

.38966 

.92096 

•40567 

.91402 

26 

57 

•35755 

•93389 

•3738 

•9 2 75i 

•38993 

.92085 

•40594 

•9*39 

26 

58 

•35782 

•93379 

•37407 

.9274 

.3902 

•92073 

.40621 

•9 I 37 8 

27 

59 

•358 i 

•93368 

•37434 

.92729 

.39046 

.92062 

.40647 

.91366 

27 

60 

•35837 

•93358 

•3746 i 

.92718 

•39073 

•9205 

.40674 

•9*355 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 



es° 

68° 

67° 

66 0 


























































NATURAL SINES AND COSINES 


30 


Prop. 

parts. 


240 

1 

2. 

50 

2 ( 

30 

2' 

7 ° 


Prop. 

parts. 

26 

/ 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 


14 

O 

O 

.40674 

• 9*355 

.42262 

.90631 

•43837 

.89879 

•45399 

.89101 

60 

14 

O 

I 

.407 

• 9!343 

.42288 

.90618 

.43863 

.89867 

•45425 

89087 

59 

14 

I 

2 

.40727 

•91331 

•42315 

.90606 

.43889 

.89854 

•45451 

. 89074 

58 

14 

I 

3 

•40753 

■ 9 I 3 1 9 

•42341 

.90594 

.43916 

.89841 

•45477 

.89061 

57 

13 

2 

4 

.4078 

• 9 I 3°7 

.42367 

.90582 

■43942 

.89828 

•45503 

.89048 

56 

13 

2 

5 

. 40806 

.91295 

.42394 

.90569 

.43968 

.89816 

45529 

.89035 

55 

13 

3 

6 

•40833 

.91283 

.4242 

• 9°557 

•43994 

.89803 

•45554 

.89021 

54 

13 

3 

7 

.4086 

.91272 

.42446 

• 9°545 

.4402 

8979 

4558 

.89008 

53 

12 

3 

8 

.40886 

.9126 

•42473 

.90532 

.44046 

.89777 

.45606 

.88995 

52 

12 

4 

9 

.40913 

.91248 

.42499 

.9052 

.44072 

.89764 

.45632 

.88981 

5 i 

12 

4 

IO 

•40939 

• 9 I2 3 6 

•42525 

• 9 0 507 

.44098 

.89752 

•45658 

.88968 

50 

12 

5 

II 

.40966 

.91224 

•42552 

■ 90495 

.44124 

•89739 

.45684 

•88955 

49 

II 

5 

12 

.40QQ2 

.91212 

.42578 

.90483 

• 44 i 5 i 

.89726 

■ 457 i 

.88942 

48 

II 

6 

13 

.4IOI9 

.912 

.42604 

.9047 

•44177 

• 8 97 i 3 

■45736 

.88928 

47 

II 

6 

14 

.41045 

.91188 

.42631 

.90458 

.44203 

.897 

•45762 

.88915 

46 

II 

7 

is 

.41072 

.91176 

.42657 

■00446 

.44229 

.89687 

•45787 

.88902 

45 

II 

7 

16 

.41098 

.91164 

.42683 

• 9°433 

•44255 

.89674 

•45813 

.88888 

44 

IO 

7 

17 

.41125 

,91152 

.42709 

.9O42I 

.44281 

.89662 

•45839 

.88875 

43 

IO 

8 

18 

.41151 

.9114 

.42736 

.90408 

•44307 

.89649 

•45865 

.88862 

42 

IO 

8 

19 

.41178 

.91128 

.42762 

.90396 

•44333 

.89636 

.45891 

.88848 

4 i 

IO 

9 

20 

.41204 

.91116 

.42788 

•90383 

■44359 

.89623 

• 459*7 

.88835 

40 

9 

9 

21 

,41231 

.9IIO4 

.42815 

■ 9 0 37 i 

•44385 

. 8961 

.45942 

.88822 

39 

9 

IO 

22 

•41257 

.9IO92 

.42841 

•90358 

.44411 

•89597 

.45968 

.88808 

38 

9 

IO 

23 

.41284 

.9108 

.42867 

.90346 

•44437 

.89584 

•45994 

.88795 

37 

9 

IO 

24 

• 4 i 3 i 

.91068 

.42894 

• 9°334 

•44464 

89571 

4602 

.88782 

S 6 

8 

II 

25 

•41337 

.9x056 

.4292 

.90321 

•4449 

•89558 

.46046 

.88768 

35 

8 

II 

261 

•41363 

.91044 

.42946 

.90309 

.44516 

•89545 

.46072 

•88755 

34 

8 

12 

27 

•4139 

.91032 

.42972 

.90296 

•44542 

•89532 

46097 

.88741 

33 

8 

12 

28 

.41416 

.9102 

•42999 

.90284 

.44568 

.89519 

.46123 

.88728 

32 

7 

13 

2 9 

•41443 

.91008 

.43025 

90271 

•44594 

.89506 

.46149 

.88715 

3 1 

7 

13 

3 ° 

.41469 

.90996 

• 43 0 5 i 

.90259 

.4462 

.89493 

•46175 

.88701 

3 ° 

7 

13 

3 i 

.41496 

.90984 

•43077 

.90246 

.44646 

.8948 

46201 

.88688 

29 

7 

h 

32 

41522 

.90972 

•43104 

• 9° 2 33 

.44672 

.89467 

.46226 

.88674 

28 

7 

14 

33 

'41549 

.9096 

4313 

.90221 

• 446 q 8 

.89454 

.46252 

.88661 

27 

6 

15 

34 

•41575 

.90948 

•43156 

.90208 

•44724 

.89441 

.46278 

.88647 

26 

6 

15 

35 

.41602 

.90936 

.43182 

.90196 

•4475 

.89428 

•46304 

.88634 

25 

6 

l6 

36 

.41628 

.QOQ 24 

.43209 

.90183. 

.44776 

.89415 

•4633 

.8862 

24 

6 

l6 

37 

• 4 i 655 

.9O9II 

•43235 

.9OI7I 

44802 

.89402 

•46355 

.88607 

23 

5 

l6 

3 * 

.41681 

.90899 

■43261 

90158 

.44828 

.89389 

.46381 

•88593 

22 

5 

17 

39 

.41707 

.90887 

.43287 

.90146 

.44854 

•89376 

.46407 

.8858 

21 

5 

17 

40 

•41734 

.90875 

■43313 

9 OI 33 

.4488 * 

•89363- 

•46433 

.88566 

20 

5 

l8 

4 i 

.4176 

.90863 

•4334 

.9012 

.44906 

•8935 

•46458 

•88553 

19 

4 

18 

42 

.41787 

.90851 

.43366 

.90108 

•44932 

•89337 

.46484 

•88539 

18 

4 

19 

43 

.41813 

.90839 

• 4339 2 

.90095 

.44958 

.89324 

.4651 

.88526 

17 

4 

J 9 

44 

.4184 

.90826 

.43418 

90082 

• 449 8 4 

.89311 

•46536 

.88512 

l6 

4 

20 

45 

.41866 

.90814 

•43445 

.9OO7 

.4501 

.89298 

.46561 

.88499 

15 

4 

20 

46 

.41892 

.90802 

• 4347 i 

.90057 

.45036 

.89285 

.46587 

.88485 

14 

3 

20 

47 

• 4 I 9 I 9 

.9079 

•43497 

.90045 

.45062 

.89272 

•46613 

.88472 

13 

3 

21 

48 

•41945 

.90778 

•43523 

.Q0032 

.45088 

.89259 

•46639 

.88458 

12 

3 

21 

49 

.41972 

.90766 

•43549 

.90019 

•45114 

.89245 

.46664 

.88445 

II 

3 

22 

50 

.41998 

•90753 

■43575 

.9OOO7 

• 45 i 4 

.89232 

.4669 

•88431 

IO 

2 

22 

5 i 

.42024 

.90741 

.43602 

89994 

.45166 

.89219 

.46716 

.88417 

9 

2 

23 

52 

.42051 

.9O729 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

8 

2 

23 

53 

.42077 

.90717 

• 436.54 

.89968 

45218 

.89193 

•46767 

.8839 

7 

2 

23 

54 

.42104 

.90704 

4368 

• 89956 

•45243 

.8918 

•46793 

.88377 

6 

T 

24 

55 

.4213 

.90692 

.43706 

.89943 

.45269 

. 89167 

.46819 

.88363 

5 

I 

24 

56 

.42156 

.9068 

■43733 

■8993 

•45295 

•89153 

.46844 

.88349 

4 

I 

25 

57 

.42183 

.90668 

•43759 

.899x8 

4532 i 

.8914 

.4687 

.88336 

0 

I 

25 

58 

.42209 

.90655 

• 437 8 5 

• 89905 

•45347 

.89x27 

.46896 

.88322 

2 

O 

26 

! 59 

•42235 

.90643 

.43811 

.89892 

•45373 

.89114 

.46921 

88308 

I 

O 

20 

! 60 

.42262 

.90631 

•43837 

.89879 

•45399 

.89101 

•46947 

88295 

O 

O 


I 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 



1 

65° 

64° 

63° 

62° 















































NATURAL SINES AND COSINES 


397 


pL,.eo 

> C 3 

P-. 

25 

/ 

2? 

N. sine. 

jo 

N. cos. 

2! 

N. sine. 

jo 

N. cos. 

Ip 

3 ( 

N. sine. 

N. cos. 

33 

N. sine. 

L° 

N. cos. 


m Prop. 
+• parts. 

O 

O 

.46947 

.88295 

.48481 

.87462 

•5 

.86603 

•51504 

•85717 

60 

H 

O 

I 

.46973 

.88281 

.48506 

.87448 

• 50025 

.86588 

•51529 

.85702 

59 

H 

I 

2 

.46999 

.88267 

•48532 

•87434 

• 5005 

•86573 

•51554 

.85687 

58 

I 4 

I 

3 

.47024 

.88254 

•48557 

.8742 

.50076 

.86559 

•51579 

.85672 

57 

*3 

2 

4 

• 47°5 

. 8824 

.48583 

.87406 

.50101 

.86544 

.51604 

•85657 

56 

*3 

2 

5 

.47076 

. 88226 

.48608 

•87391 

.50126 

•8653 

.51628 

.85642 

55 

*3 

3 

6 

.47101 

.88213 

.48634 

•87377 

•50151 

•86515 

•51653 

.85627 

54 

J 3 

3 

7 

.47127 

.88199 

.48659 

.87363 

.50176 

.86501 

.51678 

.85612 

53 

12 

3 

8 

• 47 I 53 

.88185 

.48684 

• 8 7349 

.50201 

.86486 

■51703 

■85597 

52 

12 

4 

9 

.47178 

.88172 

.4871 

•87335 

.50227 

.86471 

.51728 

•85582 

5 i 

12 

4 

IO 

.47204 

.88158 

•48735 

•87321 

• 50252 

.86457 

• 5 U 53 

•85567 

5 o 

12 

5 

II 

.47229 

.88144 

.48761 

• 87306 

.50277 

.86442 

• 5 I 77 8 

•85551 

49 

II 

5 

12 

•47255 

.8813 

.48786 

.87292 

• 50302 

.86427 

.51803 

•85536 

48 

II 

5 

*3 

.47281 

.88117 

.48811 

.87278 

•50327 

.86413 

.51828 

• 8552 X 

47 

II 

6 

H 

47306 

.88103 

.48837 

.87264 

•50352 

.86398 

.51852 

.85506 

46 

II 

6 

15 

■47332 

.88089 

.48862 

.8725 

•50377 

.86384 

•51877 

.85491 

45 

II 

7 

l6 

•47358 

.88075 

.48888 

■87235 

• 50403 

.86369 

■ 5 J 902 

.85476 

44 

IO 

7 

J 7 

•47383 

.88062 

.48913 

.87221 

.50428 

•86354 

•51927 

.85461 

43 

IO 

8 

18 

.47409 

.88048 

.48938 

.87207 

•50453 

.8634 

•51952 

.85446 

42 

IO 

8 

*9 

•47434 

.88034 

.48964 

.87193 

•5047 8 

•86325 

•51977 

•85431 

4 i 

IO 

8 

20 

.4746 

.8802 

•48q8q 

.87178 

• 50503 

.8631 

.52002 

.85416 

40 

9 

9 

21 

.47486 

.88006 

.49014 

.87164 

.50528 

.86295 

.52026 

.85401 

39 

9 

9 

22 

• 475 II 

• 8 7993 

•4904 

•8715 

•50553 

.86281 

•52051 

•85385 

38 

9 

IO 

23 

•47537 

.87979 

.49065 

.87136 

•50578 

.86266 

•52076 

■8537 

37 

9 

IO 

24 

.47562 

.87965 

•4909 

.87121 

.50603 

.86251 

.52101 

•85355 

36 

1 8 

IO 

25 

.47588 

•87951 

.49116 

.87107 

.50628 

.86237 

.52126 

•8534 

35 

8 

II 

26 

.47614 

•87937 

• 49 I 4 I 

87093 

•50654 

,86222 

•52151 

•85325 

34 

8 

II 

27 

• 47 6 39 

.87923 

.49166 

.87079 

.50679 

.86207 

•52175 

•8531 

33 

8 

12 

28 

.47665 

.87909 

.49192 

.87064 

• 50704 

.86192 

.522 

.85294 

32 

7 

12 

29 

.4769 

.87896 

.49217 

.8705 

• 50729 

.86178 

•52225 

.85279 

3 1 

7 

13 

30 

.47716 

.87882 

.49242 

.87036 

• 50754 

.86163 

•5225 

.85264 

3 ° 

7 

J 3 

31 

•47741 

.87868 

.49268 

.87021 

■50779 

.86148 

•52275 

.85249 

29 

7 

1 3 

32 

•47767 

.87854 

.49293 

.87007 

■ 50804 

•86133 

.52299 

•85234 

28 

7 

*4 

33 

•47793 

.8784 

.49318 

.86993 

.50829 

.86119 

•52324 

.85218 

27 

6 

M 

34 

.47818 

.87826 

•49344 

.86978 

•50854 

.86104 

•52349 

.85203 

26 

6 

i 5 

35 

■47844 

. 87812 

.49369 

.86964 

.50879 

.86089 

•52374 

.85188 

25 

6 

i 5 

36 

.47869 

.87798 

•49394 

.86949 

• 50904 

.86074 

•52399 

•85173 

24 

6 

15 

37 

• 47 8 95 

.87784 

.49419 

.86935 

.50929 

.86059 

•52423 

■85157 

23 

5 

l6 

38 

•4792 

.8777 

•49445 

.86921 

•50954 

.86045 

.52448 

.85142 

22 

5 

l6 

39 

.47946 

•87756 

•4947 

.86906 

• 50979 

.8603 

•52473 

•85127 

21 

5 

J 7 

40 

• 4797 1 

•87743 

• 49495 

.86892 

• 51004 

.86015 

.52498 

.85112 

20 

5 

1 7 

4 1 

•47997 

.87729 

.49521 

.86878 

.51029 

.86 

•52522 

.85096 

19 

4 

18 

42 

.48022 

•87715 

.49546 

.86863 

•51054 

• 859 8 5 

•52547 

.85081 

18 

4 

18 

43 

.48048 

.87701 

•49571 

.86849 

•51079 

•8597 

•52572 

.85066 

J 7 

4 

18 

44 

.48073 

.87687 

.49596 

.86834 

.5x104 

•85956 

•52597 

•85051 

16 

4 

*9 

45 

.48099 

.87673 

.49622 

.8682 

.5x129 

.85941 

.52621 

•85035 

15 

4 

J 9 

46 

.48124 

.87659 

.49647 

.86805 

• 5 II 54 

.85926 

.52646 

.8502 

14 

3 

20 

47 

.4815 

.87645 

.49672 

.86791 

• 5 H 79 

.85911 

•52671 

.85005 

13 

3 

20 

48 

•48170 

•87631 

•49697 

.86777 

,51204 

.85896 

.52696 

.84989 

12 

3 

20 

49 

48201 

.87617 

•49723 

.86762 

.51229 

.85881 

.5272 

.84974 

11 

3 

21 

50 

.48226 

.87603 

.49748 

.86748 

•51254 

.85866 

•52745 

.84959 

10 

2 

21 

5 i 

.48252 

.87589 

•49773 

•86733 

.51279 

•85851 

•5277 

•84943 

9 

2 

22 

52 

.48277 

•87575 

.49798 

.86719 

•51304 

.85836 

•52794 

.84928 

8 

2 

22 

53 

.48303 

•87561 

.49824 

.86704 

•51329 

,85821 

.52819 

.849x3 

7 

2 

23 

54 

.48328 

.87546 

• 49849 

.8669 

•51354 

,85806 

.52844 

.84897 

6 

1 

2 3 

55 

•48354 

•87532 

.49874 

.86675 

•51379 

•85792 

.52869 

„84882 

5 

1 

23 

56 

•48379 

•87518 

.49899 

.86661 

.51404 

•85777 

•52893 

.84866 

4 

1 

24 

57 

.48405 

•87504 

•49924 

.86646 

.51429 

,85762 

.52918 

.84851 

3 

1 

24 

58 

•4843 

.8749 

■4995 

.86632 

• 5 M 54 

•85747 

•52943 

.84836 

2 

0 

25 

59 

.48456 

.87476 

•49975 

.86617 

■51479 

•85732 

.52967 

.8482 

I 

0 

25 

60 

.48481 

.87462 

•5 

.86603 

•51504 

•85717 

.52992 

.84805 

O 

0 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 




61 ° 

60 ° 

59 ° 

58 ° 




Ll 


















































398 


NATURAL SINES AND COSINES. 


di 

O j_ 

* a 
Pm q- 

23 

/ 

3J 

N. sine. 

JO 

N. cos. 

3: 

N. sine. 

jo 

N. cos. 

340 

N. sine. | N. cos. 

3, 

N. sine. 

50 

N. cos. 


m Prop. 
M parts. 

O 

O 

• 52992 

. 84805 

•54464 

•83867 

•559*9 

.82904 

•57358 

•81915 

60 

l6 

O 

I 

•53017 

.84789 

• 54488 

•83851 

•55943 

.82887 

•5738 i 

.81899 

59 

l6 

I 

2 

•53041 

.84774 

•545*3 

•83835 

• 55968 

.82871 

•57405 

.81882 

58 

*5 

I 

3 

• 53°66 

•84759 

•54537 

.83819 

•5599 2 

•82855 

•57429 

.81865 

57 

*5 

2 

4 

•53091 

•84743 

•5456 i 

.83804 

.56016 

.82839 

•57453 

. 81848 

56 

*5 

2 

5 

•53**5 

.84728 

•54586 

.83788 

.5604 

.82822 

•57477 

.81832 

55 

*5 

2 

6 

•5314 

.847x2 

.5461 

.83772 

. 56064 

.82806 

•575oi 

.81815 

54 

*4 

3 

7 

•53164 

.84697 

•54635 

•83756 

. 56088 

.8279 

•57524 

.817Q8 

53 

*4 

3 

8 

•53189 

.84681 

• 54659 

•8374 

.56112 

.82773 

•57548 

.81782 

52 

*4 

3 

9 

•53214 

.84666 

•54683 

.83724 

•56136 

.82757 

■57572 

•81765 

5* 

*4 

4 

IO 

•53238 

.8465 

• 54708 

•83708 

.5616 

.82741 

•57596 

.81748 

50 

*3 

4 

II 

•53263 

•84635 

•54732 

.83692 

•56184 

.82724 

•57619 

.81731 

49 

*3 

5 

12 

•53288 

.84619 

•54756 

•83676 

.56208 

.82708 

•57643 

.81714 

48 

*3 

5 

13 

•53312 

.84604 

■5478 i 

.8366 

.56232 

.82692 

•57667 

.81698 

47 

*3 

5 

14 

•53337 

.84588 

•54805 

.83645 

.56256 

.82675 

•5769* 

.81681 

46 

12 

6 

15 

•5336 i 

•84573 

•54829 

•83629 

.5628 

.82659 

■577*5 

.81664 

45 

12 

6 

l6 

•53386 

•84557 

•54854 

•83613 

•56305 

.82643 

•57738 

.81647 

44 

12 

7 

17 

•534” 

.84542 

.54878 

•83597 

•56329 

.82626 

.57762 

.81631 

43 

II 

7 

18 

•53435 

.84526 

•549° 2 

.83581 

•56353 

.8261 

.57786 

.81614 

42 

II 

7 

19 

•5346 

•845x1 

•549 2 7 

•83565 

•56377 

•82593 

•5781 

•8i597 

4* 

II 

8 

20 

•53484 

•84495 

•5495* 

•83549 

.56401 

.82577 

•57833 

.8158 

4° 

II 

8 

21 

•53509 

. 844^ 

■54975 

•83533 

•56425 

.82561 

■57857 

•81563 

.39 

IO 

8 

22 

•53534 

.84464 

•54999 

•835*7 

•56449 

.82544 

.57881 

•81546 

3« 

IO 

9 

23 

•53558 

.84448 

■55024 

•83501 

•56473 

.82528 

• 57904 

•8153 

37 

IO 

9 

24 

•53583 

•84433 

•55048 

•83485 

•56497 

•82511 

.57928 

•815*3 

36 

IO 

IO 

25 

•53607 

.84417 

• 55072 

.83469 

•56521 

.82495 

■57952 

.81496 

35 

9 

IO 

26 

•53632 

.84402 

•55097 

•83453 

•56545 

.82478 

•57976 

.81479 

34 

9 

IO 

27 

•53656 

.84386 

-55121 

•83437 

• 56569 

.82462 

•57999 

.81462 

33 

9 

II 

28 

•5368 i 

•8437 

■55*45 

•83421 

•56593 

.82446 

.58023 

•8x445 

3 2 

9 

II 

29 

•53705 

•84355 

•55*69 

•83405 

.56617 

.82429 

• 58047 

.81428 

3* 

8 

12 

30 

•5373 

•84339 

•55*94 

•83389 

•56641 

•82413 

.5807 

.81412 

3° 

8 

12 

31 

•53754 

.84324 

•55218 

•83373 

.56665 

.82396 

.58094 

•8x395 

29 

8 

12 

32 

•53779 

.84308 

•55242 

•83356 

.56689 

.8238 

.58118 

.81378 

28 

7 

13 

33 

•53804 

.84292 

•55266 

•8334 

•567*3 

.82363 

.58141 

.81361 

27 

7 

*3 

34 

.53828 

.84277 

•55291 

•83324 

•56736 

.82347 

•58165 

.81344 

26 

7 

13 

35 

•53853 

.84261 

•553*5 

•83308 

•5676 

•8233 

.58189 

.81327 

25 

7 

*4 

36 

•53877 

.84245 

•55339 

.83292 

.56784 

.82314 

.58212 

•8131 

24 

6 

*4 

37 

•539°2 

.8423 

•55363 

.83276 

.56808 

.82297 

.58236 

.81293 

23 

6 

15 

38 

.53926 

.84214 

•55388 

.8326 

•56832 

.82281 

.5826 

.81276 

22 

6 

15 

39 

•53951 

.84198 

•554*2 

•83244 

.56856 

.82264 

•58283 

.81259 

21 

6 

*5 

40 

•53975 

.84182 

•55436 

.83228 

.5688 

.82248 

• 58307 

.81242 

20 

5 

l6 

41 

•54 

.84167 

•5546 

.83212 

•56904 

.82231 

•5833 

.81225 

*9 

5 

l6 

42 

• 54024 

.84151 

•55484 

•83195 

.56928 

.82214 

•58354 

.81208 

18 

5 

l6 

43 

•54049 

•84135 

•55509 

•83*79 

•56952 

.82198 

■58378 

.81191 

*7 

5 

U 

44 

•54073 

.8412 

•55533 

•83163 

•56976 

.82181 

.58401 

.81174 

l 6 

4 

17 

45 

•54097 

.84104 

•55557 

•83*47 

•57 

.82165 

•58425 

.81157 

*5 

4 

18 

46 

.54122 

.84088 

•5558 i 

•83*31 

.57024 

.82148 

.58449 

. 8114 

*4 

4 

18 

47 

.54146 

.84072 

•55605 

•83**5 

■57047 

.82132 

•58472 

.81123 

*3 

3 

18 

48 

•54*7* 

•84057 

•5563 

.83098 

-57071 

.82115 

.58496 

.81106 

12 

0 

0 

*9 

49 

•54195 

.84041 

•55654 

.83082 

•57095 

.82098 

•585*9 

.81089 

II 

3 

*9 

50 

•5422 

• 84025 

•55678 

.83066 

•57**9 

.82082 

•58543 

.81072 

IO 

0 

j 

20 

51 

■54244 

.84009 

•55702 

•8305 

•57*43 

.82065 

•58567 

•81055 

9 

2 

20 

52 

•54269 

.83994 

•55726 

•83034 

■57*67 

.82048 

•5859 

.81038 

8 

2 

20 

53 

•54293 

•83978 

•5575 

•83017 

■57*9* 

.82032 

.58614 

.81021 

7 

2 

21 

54 

•54317 

83962 

•55775 

.83001 

•572*5 

.82015 

•58637 

.81004 

6 

2 

21 

55 

•54342 

.83946 

•55799 

•82985 

•57238 

.81999 

.58661 

.80987 

5 

1 

21 

56 

.54366 

•8393 

•55823 

.82969 

.57262 

.81982 

.58684 

.8097 

4 

1 

22 

57 

•5439 1 

■ 839*5 

•55847 

•82953 

.57286 

.81965 

.58708 

•80953 

3 

1 

22 

58 

•54415 

.83899 

■55871 

.82936 

•573* 

.81949 

•5873* 

.80936 

2 

1 

23 

59 

•5444 

.83883 

•55895 

.8292 

•57334 

.81932 

•58755 

•8oqiq 

I 

0 

23 

60 

.54464 

•83867 

•559*9 

.82904 

•57358 

.81915 

•58779 

.80902 

O 

0 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

~~r 




57° 

560 

550 

54° 












































NATURAL SINES AND COSINES. 


399 


Prop. 

parts. 


360 

.1 

31 

ro 

31 

3 ° 

39 ° 


Prop. 

parts. 

23 

/ 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 


18 

O 

O 

•58779 

. 80902 

.60182 

.79864 

.61566 

.78801 

.62932 

•77715 

60 

18 

O 

I 

.58802 

. 80885 

.60205 

.79846 

•61589 

.78783 

•62955 

.77606 

59 

18 

1 

2 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62077 

.77678 

58 

17 

I 

3 

.58849 

.8085 

.60251 

.79811 

•61635 

.78747 

•63 

.7766 

57 

17 

2 

4 

•58873 

.80833 

.60274 

•79793 

.61658 

.78729 

.63022 

.77641 

56 

*7 

2 

5 

.58896 

.80816 

.60298 

•79776 

.61681 

.78711 

•63045 

.77623 

55 

17 

2 

6 

.5892 

.80799 

.60321 

•79758 

.61704 

.78694 

.63068 

.77605 

54 

l6 

3 

7 

•58943 

.80782 

•60344 

• 7974 i 

.61726 

.78676 

.6309 

•77586 

53 

l6 

3 

8 

.58967 

•80765. 

.60367 

• 797 2 3 

.61749 

.78658 

•63113 

.77568 

52 

l6 

3 

9 

•5899 

.80748 

.6039 

.79706 

.61772 

.7864 

•63135 

•7755 

5 i 

15 

4 

IO 

.59014 

.8073 

.60414 

.79688 

• 6 i 795 

.78622 

•63158 

•77531 

5 o 

15 

4 

II 

• 59°37 

.80713 

.60437 

.79671 

.61818 

.78604 

•6318 

•77513 

49 

15 

5 

12 

.59061 

.80696 

.6046 

•79653 

.61841 

78586 

■63203 

•77494 

48 

14 

5 

13 

• 59084 

.80679 

.60483 

•79635 

.61864 

.78568 

.63225 

•77476 

47 

14 

5 

14 

.59108 

.80662 

.60506 

.79618 

.61887 

•7855 

.63248 

•77458 

46 

14 

6 

15 

•sg^i 

.80644 

.60529 

•796 

.61909 

• 7853 2 

.63271 

•77439 

45 

14 

6 

l6 

• 59 I 54 

.80627 

•60553 

79583 

61932 

•78514 

.63293 

.77421 

44 

13 

7 

17 

•59178 

.8061 

.60576 

■79565 

• 6 i 955 

.78496 

.63316 

.77402 

43 

i 3 

7 

18 

.59201 

•80593 

•60599 

•79547 

.61978 

.78478 

•63338 

•77384 

42 

13 

7 

J 9 

•59225 

.80576 

.60622 

•7953 

.62001 

.7846 

■63361 

•77366 

4 1 

12 

8 

20 

.59248 

.80558 

.60645 

•79512 

,62024 

.78442 

•63383 

•77347 

4 ° 

12 

8 

21 

•59272 

.80541 

.60668 

•79494 

.62046 

.78424 

.63406 

•77329 

39 

12 

8 

22 

•59295 

.80524 

.60691 

•79477 

.62069 

.78405 

.63428 

• 773 i 

38 

II 

9 

23 

• 593 i 8 

•80507 

.60714 

•79459 

.62092 

•78387 

•63451 

. 772 Q 2 

37 

II 

9 

24 

•59342 

.80489 

•60738 

.79441 

.62115 

.78369 

•63473 

•77273 

36 

II 

IO 

25 

•59365 

.80472 

.60761 

.79424 

.62138 

• 7 8 35 i 

63496 

■77255 

35 

11 

IO 

26 

•59389 

•80455 

.60784 

.79406 

.6216 

•78333 

.63518 

.77236 

34 

IO 

10 

27 

•59412 

.80438 

.60807 

.79388 

.62183 

•78315 

•6354 

.77218 

33 

IO 

II 

28 

•59436 

.8042 

.6083 

• 7937 i 

.62206 

.78207 

•63563 

.77199 

32 

10 

II 

29 

•59459 

.80403 

•60853 

■79353 

.62229 

.78279 

•63585 

.77181 

3 1 

9 

12 

30 

.59482 

.80386 

.60876 

•79335 

.62251 

.78261 

.63608 

.77162 

30 

9 

12 

31 

•59506 

.80368 

.60899 

• 793 i 8 

.62274 

.78243 

•6363 

•77144 

29 

9 

12 

32 

•59529 

•80351 

.60922 

•793 

.62297 

.78225 

•63653 

•77125 

28 

8 

13 

33 

•59552 

• 8 0334 

.60945 

.79282 

.6232 

.78206 

•63675 

.77107 

27 

8 

13 

34 

■59576 

.80316 

. 60968 

.79264 

62342 

.78188 

.63698 

.77088 

26 

8 

13 

35 

•59599 

.80299 

.60991 

.79247 

•62365 

.7817 

.6372 

.7707 

25 

8 

14 

36 

.59622 

.80282 

.61015 

. 7 Q 22 Q 

.62388 

78152 

•63742 

•77051 

24 

7 

14 

37 

• 59646 

.80264 

.61038 

.79211 

.62411 

• 78 i 34 

•63765 

•77033 

23 

7 

15 

38 

.59669 

.80247 

61061 

• 79*93 

•62433 

78116 

•63787 

• 77 OI 4 

22 

7 

15 

39 

•59693 

.8023 

.61084 

- 79 * 7 6 

.62456 

.78098 

•6381 

.76996 

21 

6 

15 

40 

• 597 i 6 

.80212 

.61107 

•79158 

.62479 

.78079 

•63832 

.76977 

20 

6 

16 

41 

•59739 

.80195 

.6113 

.7914 

.62502 

.78061 

•63854 

•76959 

19 

6 

16 

42 

•59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.7694 

l8 

5 

l6 

43 

.59786 

. 8016 

.61176 

• 79 io 5 

.62547 

.78025 

•63899 

.76921 

17 

5 

*7 

44 

• 59809 

•80143 

.61199 

.79087 

.6257 

.78007 

.63922 

.76903 

l6 

5 

*7 

45 

•59832 

.80125 

.61222 

.7Qo6q 

62592 

.77088 

•63944 

.76884 

15 

5 

18 

46 

•59856 

.80108 

.61245 

• 79 ° 5 I 

.62615 

•7797 

.63966 

.76866 

14 

4 

18 

47 

• 59 8 79 

.80091 

.61268 

• 79°33 

.62638 

•77952 

.63989 

.76847 

13 

4 

18 

48 

• 599 ° 2 

• 80073 

.61291 

.79016 

.6266 

•77934 

.64011 

.76828 

12 

4 

*9 

49 

• 59926 

.80056 

.61314 

.78998 

.62683 

• 779 l6 

•64033 

. 7681 

II 

3 

19 

50 

• 59949 

.80038 

• 6 i 337 

. 7898 

.62706 

■77897 

. 64056 

76791 

IO 

3 

20 

51 

•59972 

.80021 

■6136 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

3 

20 

52 

•59995 

.80003 

•61383 

•78944 

.62751 

.77861 

, 641 

•76754 

8 

2 

20 

53 

.60019 

.79986 

61406 

.78926 

.62774 

•77843 

.64123 

•76735 

7 

2 

21 

54 

.60042 

• 70 q 68 

.61429 

.78908 

.62796 

•77824 

64 M 5 

.76717 

6 

2 

21 

55 

.60065 

•79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

2 

21 

56 

.60089 

•79934 

.61474 

.78873 

62842 

.77788 

.6419 

.76679 

4 

I 

22 

57 

.60112 

• 799 16 

.61497 

•78855 

.62864 

•77769 

.64212 

.76661 

3 

I 

22 

58 

•60135 

.79899 

.6152 

.78837 

.62887 

•77751 

.64234 

.76642 

2 

I 

23 

59 

.60158 

.79881 

• 6 i 543 

78819 

.62909 

•77733 

.64256 

.76623 

1 

O 

23 

60 

.60182 

.79864 

.61566 

.78801 

.62932 

•77715 

.64279 

.76604 

0 

O 


j 

N. cos. 

1 NT. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine. 

/ 



1 

53 ° 

52 ° 

51 ° 

50 ° 












































400 


NATURAL SINES AND COSINES 


P -.2 

O fc- 
> C 3 

Ph p. 

22 

/ 

40 ° 

N. sine. N. cos. 

4 

N. sine. 

l 6 

N. cos. 

420 

N. sine, j N. cos. 

4 

j N. sine. 

3 ° 

N. cos. 


ft 
a. 1 

*9 

O 

O 

.64279 

.76604 

.65606 

• 7547 1 

.66913 

•74314 

.682 

•73135 

60 

*9 

O 

I 

.64301 

.76586 

.65628 

•75452 

•66935 

•74295 

1 .68221 

.73116 

59 

19 

I 

2 

•64323 

.76567 

•6565 

•75433 

.66956 

.74276 

1 .68242 

.73096 

5 « 

18 

I 

3 

.64346 

.76548 

•65672 

•75414 

.66978 

74256 

.68264 

.73076 

57 

18 

I 

4 

.64368 

•7653 

•65694 

- 75395 

.66999 

•74237 

.68285 

•73056 

56 

18 

2 

5 

•6439 

.76511 

•65716 

•75375 

.67021 

.74217 

.68306 

• 73036 

55 

17 

2 

6 

.64412 

.76492 

■65738 

75356 

.67043 

74198 

•68327 

.73016 

54 

17 

3 

7 

■64435 

•76473 

•65759 

•75337 

.67064 

.74178 

.68349 

.72996 

53 

17 

o 

D 

8 

•64457 

76455 

65781 

• 753 i 8 

.67086 

•74159 

.6837 

.72976 

52 

l6 

3 

9 

.64479 

76436 

•65803 

•75299 

.67107 

•74139 

•68391 

•72957 

51 

l6 

4 

IO 

•64501 

.76417 

■65825 

.7528 

.67129 

.7412 

.68412 

•72937 

50 

l6 

4 

ii 

•64524 

.76398 

.65847 

.75261 

.67151 

.741 

.68434 

• 7 2 9 I 7 

49 

l6 

4 

12 

.64546 

.7638 

.65869 

75241 

.67172 

.7408 

.68455 

.72897 

48 

15 

5 

13 

.64568 

.76361 

•65891 

• 75 222 

.67194 

.74061 

.68476 

.72877 

47 

15 

5 


•6459 

.76342 

• 659 r 3 

• 75 2 o 3 

•67215 

.74041 

.68497 

.72857 

46 

15 

6 

15 

.64612 

•76323 

•65935 

•75184 

.67237 

.74022 

.68518 

.72837 

45 

14 

6 

l6 

.64635 

.76304 

65956 

•75165 

.67258 

.74002 

•68539 

.72817 

44 

14 

6 

17 

.64657 

.76286 

.65978 

■75146 

.6728 

•73983 

.68561 

•72797 

43 

14 

7 

l8 

.64679 

.76267 

.66 

.75126 

•67301 

• 73963 

.68582 

•72777 

42 

13 

7 

x 9 

.64701 

.76248 

.66022 

•75107 

•67323 

•73944 

.68603 

.72757 

4 1 

13 

7 

20 

.64723 

.76229 

.66044 

.75088 

•67344 

•73924 

.68624 

•72737 

4 ° 

13 

8 

21 

.64746 

. 7621 

.66066 

.75069 

•67366 

•73904 

.68645 

• 7 2 7 I 7 

39 

12 

8 

22 

.64768 

.76192 

.66088 

•7505 

.67387 

•73885 

.68666 

.72697 

38 

12 

8 

2 3 

.6479 

•76173 

.66109 

•7503 

.67409 

■73865 

.68688 

72677 

37 

12 

9 

2 4 

.64812 

•76154 

.6613I 

.75011 

•6743 

.73846 

.687O9 

.72657 

3 6 

II 

9 

2 5 

.64834 

•76135 

.66153 

.74992 

.67452 

.73826 

•6873 

■71637 

35 

II 

IO 

26 

.64856 

.76116 

•66175 

•74973 

•67473 

.73806 

•68751 

.72617 

34 

II 

IO 

27 

.64878 

.76097 

.66197 

•74953 

•67495 

•73787 

.68772 

.72597 

33 

IO 

IO 

28 

.64901 

.76078 

.662l8 

•74934 

•67516 

•73767 

•68793 

‘ 7 2 577 

32 

IO 

II 

29 

.64923 

.76059 

.6624 

• 749 I 5 

•67538 

•73747 

.68814 

•72557 

3 1 

IO 

II 

30 

.64945 

.76041 

.66262 

.74896 

•67559 

.73728 

•68835 

•72537 

3 ° 

IO 

II 

3 i 

.64967 

.76022 

.66284 

.74876 

•6758 

■73708 

•68857 

•72517 

29 

9 

12 

3 2 

.64989 

■ 76003 

.66306 

•74857 

.67602 

•73688 

.68878 

•72497 

28 

9 

12 

33 

.65011 

•75984 

•66327 

.74838 

.67623 

.73669 

.68899 

•72477 

27 

9 

12 

34 

■65033 

■75965 

.66349 

.74818 

•67645 

•73649 

.6892 

•72457 

26 

8 

i 3 

35 

■65055 

•75946 

.66371 

• 74799 

.67666 

.73629 

.6894I 

•72437 

25 

8 

13 

36 

.65077 

• 759 2 7 

.66393 

.7478 

.67688 

.7361 

.68962 

.72417 

24 

8 

14 

37 

.651 

.75908 

.66414 

•7476 

.67709 

•7359 

.68983 

72397 

23 

7 

14 

38 

.65122 

.75889 

.66436 

7474 i 

•6773 

•7357 

.69OO4 

•72377 

22 

7 

i 4 

39 

.65144 

•7587 

.66458 

• 747 22 

.67752 

• 7355 i 

•69025 

•72357 

21 

7 

i 5 

40 

.65166 

■75851 

.6648 

•74703 

•67773 

• 7353 i 

.69O46 

•72337 

20 

6 

i 5 

4 i 

.65188 

75832 

665OI 

.74683 

■67795 

• 735 H 

.69067 

•72317 

J 9 

6 

i 5 

42 

•6521 

•75813 

.66523 

.74664 

.67816 

• 7349 1 

.69088 

. 722 Q 7 

18 

6 

16 

43 

.65232 

•75794 

.66545 

74644 

.67837 

•73472 

.69IO9 

.72277 

17 

5 

16 

44 

•65254 

•75775 

.66566 

.74625 

.67859 

•73452 

.6913 

•72257 

16 

5 

*7 

45 

.65276 

•75756 

66588 

. 74606 

.6788 

•73432 

.69151 

.72236 

15 

5 

*7 

46 

•65298 

•75738 

.6661 

.74586 

.67901 

• 734 I 3 

.69I72 

.72216 

14 

4 

*7 

47 

■6532 

•75719 

.66632 

•74567 

.67923 

•73393 

.69193 

.72196 

13 

4 

18 

48 

•65342 

•757 

.66653 

•74548 

•67944 

•73373 

.69214 

.72176 

12 

4 

18 

49 

•65364 

■7568 

•66675 

•74528 

•67965 

•73353 

•69235 

72156 

11 

3 

j8 

50 

.65386 

.75661 

.66697 

•74509 

.67987 

•73333 

.69256 

.72136 

10 

3 

J 9 

5 i 

.65408 

•75642 

.66718 

.74489 

.68008 

• 733 H 

.69277 

.72116 

9 

3 

*9 

5 2 

•6543 

•75623 

.6674 

7447 

.68029 

•73294 

.69298 

72095 

8 

3 


53 

•65452 

.75604 

.66762 

•74451 

.68051 

•73274 

•69319 

72075 

7 

2 

20 

54 

.65474 

•75585 

•66783 

• 7443 i 

.68072 

•73254 

■6934 

•72055 

6 

2 

20 

55 

.65496 

•75566 

.66805 

.74412 

.68093 

•73234 

.69361 

•72035 

5 

2 

21 

56 

•65518 

•75547 

.66827 

•74392 

.68115 

•73215 

.69382 

.72OI5 

4 

1 

21 

57 

•6554 

•75528 

.66848 

•74373 

.68136 

•73195 

69403 

•71995 

3 

1 

21 

58 

•65562 

•75509 

.6687 

•74353 

68157 

•73175 

.69424 

•71974 

2 

1 

22 

59 

■65584 

•7549 

.66891 

•74334 

68179 

•73155 

.69445 

•71954 

1 

0 

22 

60 

.65606 

• 7547 i 

•66913 

•74314 

.682 

•73135 

69466 

•71934 

0 

0 



N. cos. 

N. sine. 

N. cos. 

N. sine. 

N. cos. 

N. sine, i 

N. cos. 

N. sine. 

/ 



1 

490 

48 ° 

470 

46 ° 
































































NATURAL SINES AND COSINES. 


401 


d.2 

0 u 
> c5 

CL 

II 

/ 

4 4 

N. sine. 

0 

N. cos. 


di 

0 t- 

P-t CL 

*9 

d.2 

O Jm 

> 05 

P* CL 

22 

/ 

44 

N. sine. 

0 

N. cos. 


Prop. 
0 parts. 

O 

0 

.69466 

•71934 

DO 

*9 

II 

3* 

.70112 

•7*305 

29 

9 

O 

I 

.69487 

.719x4 

59 

*9 

12 

32 

.70132 

.71284 

28 

9 

I 

2 

.69508 

.71894 

5 8 

18 

12 

33 

•7°*53 

.71264 

27 

9 

I 

3 

.69529 

•7* 8 73 

57 

18 

12 

34 

.70174 

•71243 

26 

8 

I 

4 

.69549 

•7 i8 53 

56 

18 

*3 

35 

•7°*95 

.71223 

25 

8 

2 

5 

•6957 

•7 i8 33 

55 

*7 

*3 

36 

.70215 

.7*203 

24 

8 

2 

6 

.69591 

.71813 

54 

*7 

*4 

37 

.70236 

.71182 

23 

7 

3 

7 

.69612 

.71792 

53 

*7 

*4 

3 8 

.70257 

.71162 

22 

7 

3 

8 

.69633 

.71772 

52 

l6 

*4 

39 

.70277 

.71141 

21 

7 

3 

9 

.69654 

•7W52 

5* 

l6 

*5 

40 

.70298 

.71121 

20 

6 

4 

IO 

.69675 

•7 I 73 2 

50 

l6 

*5 

4* 

•7°3*9 

•7** 

*9 

6 

4 

II 

.69696 

.71711 

49 

l6 

*5 

42 

•7°339 

.7108 

18 

6 

4 

12 

.69717 

.71691 

48 

*5 

16 

43 

.7036 

•7*059 

*7 

5 

5 

13 

•69737 

.71671 

47 

*5 

l6 

44 

.7038* 

■7*039 

l6 

5 

5 

14 

.69758 

•7 i6 5 

46 

*5 

*7 

45 

.7O4OI 

.7IOI9 

*5 

5 

6 

15 

.69779 

■7 i6 3 

45 

*4 

*7 

46 

.70422 

.70998 

*4 

4 

6 

l6 

.698 

.7x61 

44 

*4 

*7 

47 

•7°443 

.70978 

*3 

4 

6 

w 

.69821 

•.7159 

43 

*4 

18 

48 

.70463 

•70957 

12 

4 

7 

18 

.69842 

• 7*569 

42 

*3 

18 

49 

.70484 

•7°937 

II 

3 

7 

19 

.69862 

•7*549 

4* 

*3 

18 

50 

•7 0 505 

.70916 

IO 

3 

7 

20 

.69883 

•7*529 

40 

*3 

*9 

5* 

•70525 

.70896 

9 

3 

8 

21 

.69904 

.71508 

39 

12 

*9 

52 

.70546 

.70875 

8 

3 

8 

22 

.69925 

.71488 

3 8 

12 

*9 

53 

.70567 

•7 o8 55 

7 

2 

8 

23 

.69946 

.71468 

37 

12 

20 

54 

.70587 

.70834 

6 

2 

9 

24 

.69966 

•7*447 

36 

II 

20 

55 

.70608 

.708*3 

5 

2 

9 

25 

.69987 

•7*427 

35 

II 

21 

56 

.70628 

•70793 

4 

I 

IO 

26 

.70008 

.7*407 

34 

II 

21 

57 

.70649 

.70772 

3 

I 

IO 

27 

.70029 

.71386 

33 

IO 

21 

5« 

.7067 

•70752 

2 

I 

IO 

28 

.70049 

.7*366 

32 

IO 

22 

59 

.7069 

•70731 

I 

O 

II 

29 

.7007 

•7*345 

3* 

IO 

22 

60 

707 II 

.7O7II 

O 

O 

II 

3° 

.7OO9I 

•7*325 

30 

IO 









N. cos. 

N. sine. 

f 




I N.cos. 

| N. sine. 

/ 




45 ° 


1 



45 ° 




Preceding Table contains Natural Sine and Cosine for every minute 
of the Quadrant to Radius 1. 

If Degrees are taken at head of columns, Minutes, Sine, and Cosine must 
be taken from head also; and if they are taken at foot of column, Minutes, 
etc., must be taken from foot also. 

Illustration. —.3173 is sine of 18 0 30', and cosine of 71 0 30'. 


To Compute Sine or Cosine for Seconds. 

When Angle is less than 45 0 . Rule. —Ascertain sine or cosine of angle 
for degrees and minutes from Table; take difference between it and sine 
or cosine of angle next below it. Look for this difference or remainder,* 
if Sine is required,'at head of column of Proportional Parts , on left side; 
and if Cosine is required, at head of column on right side; and in these 
respective columns, opposite to number of seconds of angle in column, is 
number or correction in seconds to be added to Sine, or subtracted from 
Cosine of angle. 


Illustration i.—W hat is sine of 8° 9' 10" P 


Sine of 8° 9', 
Sine of 8° 10', 


per Table =. 141 77;) 
“ =.14205;} 


.00028 difference. 


In left side column of proportional parts, under 28, and opposite to io', is 5, cor¬ 
rection for io', which, being added to .141 77 = .141 82 Sine. 


* The table in some instances will give a unit too much, but this, in general, is of little importance. 

L L* 









































402 


NATURAL SINES AND COSINES. 


2.—What is cosine of 8° 9' 10"? 

Cosine of 8° f per Table = .98990;} difference. 

Cosine of 8° 10 , =.99986;) ^ M 

In right-side column of proportional parts, under 4, and opposite to io', is 1, the 
correction for 10', which, being subtracted from .989 90 = .989 89 cosine. 

When Angle exceeds 45 0 . Rule.—A scertain sine or cosine for angle in 
degrees and minutes from Table, taking degrees at the foot of it; then take 
difference between it and sine or cosine of angle next above it. Look for re¬ 
mainder, if Sine is required, at head of column of Proportional Parts , on right 
side; and if Cosine is required, at head of column on left side; and in these 
respective columns, opposite to seconds of angle, is number or correction in 
seconds to be added to Sine, or subtracted from Cosine of angle. 

Illustration. —What is the Sine and Cosine of 8i° 50' 50"? 

Sine of 810 so ;, per Table = .989 86;} _ Q difference. 

Sine of 8i° 51 , “ =.9899; j ^ M 

In right-side column of proportional parts, and opposite to 50', is 3, which, added 
to .98986 = .989 89 Sine. 


A 


00025 difference. 


Cosine of 8i° 50', per Table =. 142 05; 

Cosine of 8i° 51', “ =.14177; 

In left-side column of proportional parts , and opposite to 50', is 24, which, sub¬ 
tracted from . 142 05 =. 141 81 Cosine. 

To Ascertain or Compute Number of Degrees, Minutes, 
and. Seconds of a given Sine or Cosine. 


When Sine is given. Rule. —If given sine is in Table, the degrees of it 
will be at top or bottom of page, and minutes in marginal column, at left or 
right side, according as sine corresponds to an angle less or greater than 45°. 

If given sine is not in Table, take sine in Table which is next less than the 
one for which degrees, etc., are required, and note degrees, etc., for it. Sub¬ 
tract this sine from next greater tabular sine, and also from given sine. 

Then, as tabular difference is to difference between given sine and tabu¬ 
lar sine, so is 60 seconds to seconds for sine given. 

Example.— What are the degrees, minutes, and seconds for sine of .75? 

Next less sine is .74992, arc for which is 48° 35'. Next greater sine is .75011, 
difference between which and next less is .75011—.749 92 = .00019. Difference be¬ 
tween less tabular sine and one given is .75 —. 749 92 = 8. 

Then 19 : 8 ;; 60 : 25-f, which, added to 48° 35' = 48° 35' 25". 

When Cosine is given. Rule. —If given cosine is found in Table, degrees 
of it will be found as in manner specified when sine is given. 

If given cosine is not in Table, take cosine in Table which is next greater 
than one for which degrees, etc., are required, and note degrees, etc., for it. 
Subtract this cosine from next less tabular cosine, and also from given cosine. 

Then, as tabular difference is to difference between given cosine and tabu¬ 
lar cosine, so is 60 seconds to seconds for cosine given. 

Example. —What are the degrees, minutes, and seconds for cosine of .75? 

Next greater cosine is .750 n, arc for which is 41 0 24'. Next less cosine is .749 92, 
difference between which and next greater is .750 11 — .749 92 = .000 19. Difference 
between greater tabular cosine and one given is . 75011 —.75000 = 11. 

Then 19 : u ;; 60 : 35—, which, added to 41 0 24'= 41° 24' 35'". 


To Compute Versed Sine of an Angle. 


Subtract cosine of angle from 1. 

Illustration.— What is the versed sine of 21 0 30'? 

Cosine of 21 0 30' is .93042, which, — 1 = .06958 versed sine. 


To Compute Co-versed Sine of an Angle. 
Subtract sine of angle from 1. 

Illustration. —What is the co-versed sine of 21 0 30'? 

The sine of 21 0 30' is .3665, which, —1 ==.6335 co-versed sine. 


NATURAL SECANTS AND CO-SECANTS 


403 


UNTatnral Secants and. Co-secants. 



0 ° 

1 ° 

2 ° 

3 ° 


f 

Secant. 

Co-secant. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

r 

0 

1 

Infinite. 

1.0001 

57-299 

1.0006 

28.654 

1.0014 

19.107 

60 

I 

1 

3437-7 

.0001 

6-359 

.0006 

8.417 

.0014 

9.002 

59 

2 

1 

1718.9 

.0002 

5-45 

.0006 

8.184 

.0014 

8.897 

58 

3 

1 

145-9 

.0002 

4-57 

.0006 

7-955 

.0014 

8.794 

57 

4 

1 

859.44 

.0002 

3-718 

.0006 

7-73 

.0014 

8.692 

56 

5 

1 

687.55 

1.0002 

52.891 

1.0007 

27.508 

1.0014 

18.591 

55 

6 

1 

572.96 

.0002 

2.09 

.0007 

7.29 

.0015 

8.491 

54 

7 

1 

491.11 

.0002 

*•313 

.0007 

7-075 

.0015 

8-393 

53 

8 

1 

29.72 

.0002 

0.558 

.0007 

6.864 

.0015 

8.295 

52 

9 

1 

3 8i -97 

.0002 

49.826 

.0007 

6.655 

.0015 

8.198 

51 

IO 

1 

343-77 

1.0002 

49 - II 4 

1.0007 

26.45 

1.0015 

18.103 

50 

II 

1 

12.52 

.0002 

8.422 

.0007 

6.249 

.0015 

8.008 

49 

12 

1 

286.48 

.0002 

7-75 

.0007 

6.05 

.0016 

7 - 9*4 

48 

13 

1 

64.44 

.0002 

7.096 

.0007 

5-854 

.0016 

7.821 

47 

14 

1 

45-55 

0002 

6.46 

.0008 

5.661 

.0016 

7-73 

46 

15 

1 

229. l8 

1.0002 

45-84 

1.0008 

25-471 

1.0016 

17-639 

45 

l6 

1 

14.86 

.0002 

5-237 

.0008 

5.284 

.0016 

7-549 

44 

17 

1 

02.22 

.0002 

4-65 

.0008 

5-1 

.0016 

7.46 

43 

l8 

1 

190.99 

.0002 

4.077 

.0008 

4.918 

.0017 

7-372 

42 

*9 

1 

80.73 

.0003 

3-52 

.0008 

4-739 

.0017 

7.285 

4 1 

20 

1 

I7I.89 

1.0003 

42.976 

1.0008 

24.562 

1.0017 

17. I98 

4 ° 

21 

1 

63-7 

.0003 

2-445 

,0008 

4-358 

.0017 

7 - IJ 3 

39 

22 

1 

56.26 

.0003 

1.928 

.0008 

4.2l6 

.0017 

7.028 

38 

23 

1 

49-47 

.0003 

1.423 

.0009 

4.047 

.0017 

6.944 

37 

24 

1 

43- 2 4 

.0003 

4°-93 

.0009 

3.88 

.0018 

6.861 

3 6 

25 

1 

137-51 

1.0003 

40.448 

1.0009 

23.716 

1.0018 

16.779 

35 

26 

1 

32.22 

.0003 

39.978 

.0009 

3-553 

.0018 

6.698 

34 

2 7 

1 

27.32 

.0003 

9.518 

.0009 

3-393 

.0018 

6.617 

33 

28 

1 

22.78 

.0003 

9.069 

.0009 

3235 

.0018 

6-538 

3 2 

29 

1 

18.54 

.0003 

8.631 

.0009 

3-079 

.0018 

6.459 

3 1 

30 

1 

H 4-59 

1.0003 

38.201 

1.0009 

22.925 

1.0019 

16.38 

3 ° 

3 i 

1 

IO.9 

.0003 

7.782 

.001 

2-774 

.0019 

6.303 

29 

32 

1 

07-43 

.0003 

7 - 37 1 

.001 

2.624 

.0019 

6.226 

28 

33 

1 

04.17 

.0004 

6.969 

.001 

2.476 

.0019 

6.15 

27 

34 

1 

OI. II 

.0004 

6.576 

.001 

2.33 

.0019 

6.075 

26 

35 

1 

98.223 

1.0004 

36.191 

1.001 

22.186 

1.0019 

l6 

25 

36 

1 

5-495 

.0004 

5.814 

.001 

2.044 

.002 

5.926 

24 

37 

1 

2 - 9 J 4 

.0004 

5-445 

.001 

I- 9 Q 4 

.002 

5-853 

23 

38 

1.0001 

2.469 

.0004 

5.084 

.001 

1-765 

.002 

5-78 

22 

39 

.0001 

88.149 

.0004 

4 . 7 2 9 

.0011 

1.629 

.002 

5.708 

21 

40 

1.0001 

85.946 

1.0004 

34.382 

1.0011 

21.494 

1.002 

15-637 

20 

4 i 

.0001 

3-849 

.0004 

4.042 

.0011 

1.36 

.0021 

5.566 

i 9 

42 

.0001 

1-853 

.0004 

3.708 

.0011 

1.228 

.0021 

5-496 

18 

43 

.0001 

79-95 

.0004 

3 - 38 i 

.0011 

1.098 

.0021 

5 - 4 2 7 

i 7 

44 

.0001 

8-133 

.0004 

3.06 

.0011 

20.97 

.0021 

5-358 

16 

45 

1.0001 

76.396 

1.0005 

32-745 

1.0011 

20.843 

1.0021 

15.29 

i 5 

46 

.0001 

4-736 

.0005 

2.437 

.0012 

0.717 

.0022 

5.222 

14 

47 

.0001 

3.146 

.0005 

2.134 

.0012 

o .593 

.0022 

5-155 

13 

48 

.0001 

1.622 

.0005 

1.836 

.0012 

0 - 47 1 

.0022 

5.089 

12 

49 

.0001 

I. l6 

.0005 

i -544 

.0012 

0.35 

.0022 

5.023 

11 

50 

1.0001 

68.757 

1.0005 

3 I - 2 57 

1.0012 

20.23 

1.0022 

14.958 

10 

5 i 

.0001 

7.409 

.0005 

30.976 

.0012 

0.112 

.0023 

4-893 

9 

52 

.0001 

6.113 

.0005 

O.699 

.0012 

19.995 

.0023 

4.829 

8 

53 

.0001 

4.866 

.0005 

O.428 

.0013 

9.88 

.0023 

4-765 

7 

54 

.0001 

3.664 

.0005 

O. l6l 

.0013 

9.766 

.0023 

4.702 

6 

55 

1.0001 

62.507 - 

1.0005 

29.899 

1.0013 

19-653 

1.0023 

14.64 

5 

56 

.0001 

I * 39 I 

.0006 

9.641 

.0013 

9 - 54 1 

.0024 

4-578 

4 

57 

.0001 

I * 3 I 4 

.0006 

9.388 

.0013 

9-431 

.0024 

4 - 5 i 7 

3 

58 

.0001 

59- 2 74 

.0006 

9 - *39 

.0013 

9.322 

.0024 

4-456 

2 

59 

.0001 

8.27 

.0006 

8.894 

.0013 

9.214 

.0024 

4-395 

1 

60 

1.0001 

57-299 

1.0006 

28.654 

1.0014 

1 

I 9 . IO 7 

1.0024 

14-335 

0 

f 

Co-sec’t 

Secant. 

Co-sec't 

Secant. 

1 Co-sec’t 

Secant. 

Co-sec’t. 

Secant. 

9 



89 ° 

88° 

87° 

860 

















































































404 


NATURAL SECANTS AND CO-SECANTS, 



40 

50 

6° 

70 


/ 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

C'o-sec’t. 

Secant. 

I Co-sec’t. 

/ 

o 

1.0024 

14-335 

1.0038 

11.474 

1-0055 

9.5668 

1.0075 

8.2055 

60 

I 

.0025 

4.276 

.0038 

1.436 

•0055 

•5404 

.0075 

.1861 

59 

2 

.0025 

4.217 

.0039 

I -39 8 

.0056 

•5i4i 

.0076 

. 1668 

58 

3 

.0025 

4-*59 

.0039 

1.36 

.0056 

.488 

.0076 

.1476 

57 

4 

.0025 

4. IOI 

.0039 

1-323 

.0056 

.462 

.0076 

.1285 

56 

5 

1.0025 

14.043 

1.0039 

11.286 

1.0057 

9.4362 

1.0077 

8.1094 

55 

6 

.0026 

3.986 

.004 

1.249 

•0057 

.4105 

.0077 

•0905 

54 

7 

.0026 

3-93 

.004 

1-213 

•0057 

•385 

.0078 

.0717 

53 

8 

.0026 

3-874 

.004 

1.176 

•0057 

•359 6 

.0078 

.0529 

52 

9 

.0026 

3.818 

.004 

1.14 

.0058 

•3343 

.0078 

•0342 

5i 

io 

1.0026 

i3-7 6 3 

I. 0041 

II. IO4 

1.0058 

9.3092 

I.OO79 

8.0156 

50 

II 

.0027 

3.708 

.0041 

1.069 

.0058 

.2842 

.OO79 

7.9971 

49 

12 

.OO27 

3-654 

.0041 

1-033 

.0059 

•2593 

.OO79 

.9787 

48 

13 

.OO27 

3-6 

.OO4I 

0.988 

.0059 

•2346 

.008 

.9604 

47 

14 

.OO27 

3-547 

.0042 

0.963 

• 0059 

. 21 

.008 

.9421 

46 

15 

I.OO27 

13-494 

1.0042 

IO.929 

1.006 

9-1855 

1.008 

7.924 

45 

l6 

.0028 

3-441 

.0042 

0.894 

.006 

. 1612 

0081 

•9°59 

44 

0 

.0028 

3-389 

.0043 

0.86 

.006 

•137 

.0081 

.8879 

43 

18 

.0028 

3-337 

.0043 

0 

00 

to 

ON 

.0061 

.1129 

.0082 

.87 

42 

r 9 

.0028 

3.286 

.0043 

O. 792* 

.0061 

.089 

0082 

.8522 

4 1 

20 

I.OO29 

I3-235 

X.0043 

10.758 

1.0061 

9.0651 

1.0082 

7-8344 

40 

21 

.OO29 

3- i8 4 

.OO44 

0.725 

.0062 

.0414 

.0083 

.8168 

39 

22 

.OQ29 

3-134 

.OO44 

0.692 

.0062 

.0179 

.0083 

.7992 

38 

23 

.0029 

3.084 

.OO44 

0.659 

.0062 

8.9944 

.0084 

.7817 

37 

24 

.OO29 

3-034 

.OO44 

0.626 

.0063 

.97x1 

.0084 

.7642 

36 

25 

1.003 

12.985 

1.0045 

10.593 

1.0063 

8.9479 

1.0084 

7.7469 

35 

26 

.003 

2-937 

.0045 

0.561 

.0063 

.9248 

.0085 

.7296 

34 

27 

.003 

2.888 

.0045 

0.529 

.0064 

.9018 

.0085 

.7124 

33 

28 

.003 

2.84 

.0046 

0.497 

.0064 

•879 

.0085 

•6953 

32 

29 

.0031 

2-793 

.0046 

0.465 

.0064 

•8563 

.0086 

.6783 

3i 

30 

I.OO3I 

12.745 

1.0046 

10.433 

1.0065 

8-8337 

1.0086 

7.6613 

30 

31 

.0031 

2.698 

.0046 

O. 402 

.0065 

.8112 

.0087 

.6444 

29 

32 

.0031 

2.652 

.0047 

o-37i 

.0065 

.7888 

.0087 

.6276 

28 

33 

.OO32 

2.606 

.0047 

o-34 

.0066 

.7665 

.OC87 

.6108 

27 

34 

.OO32 

2.56 

.0047 

0.309 

.0066 

•7444 

.0088 

•5942 

26 

35 

I.OO32 

12.514 

1.0048 

10.278 

1.0066 

8.7223 

1.0088 

7-5776 

25 

39 

.OO32 

2.469 

.0048 

0.248 

.0067 

.7004 

.0089 

.5611 

24 

37 

.OO32 

2.424 

.0048 

O. 217 

.0067 

.6786 

.0089 

•5440 

23 

38 

•0033 

2-379 

.0048 

0.187 

.0067 

• 6569 

.0089 

.5282 

22 

39 

•0033 

2-335 

.0049 

o-i57 

.0068 

•6353 

.009 

•5119 

21 

40 

1.0033 

12.291 

I.OO49 

IO. 127 

1.0068 

8.6138 

I.OO9 

7-4957 

20 

41 

•0033 

2.248 

.0049 

0.098 

.0068 

•5924 

.OO9 

•4795 

l 9 

42 

.0034 

2.204 

.005 

0.068 

.0069 

•57 11 

.OO9I 

•4634 

i3 

43 

.0034 

2.161 

.005 

0.039 

.0069 

•5499 

.OO9I 

•4474 

17 

44 

.0034 

2.118 

.005 

0.01 

.0069 

.5289 

.0092 

•4315 

16 

45 

1.0034 

12.076 

1.005 

9.9812 

I.OO7 

8.5079 

1.0092 

7-4I56 

15 

46 

•0035 

2.034 

.0051 

•9525 

.007 

.4871 

.0092 

•399 s 

14 

47 

•0035 

1.992 

.0051 

•9239 

.OO7 

.4663 

• 0093 

•384 

!3 

48 

• 0035 

i-95 

.0051 

•8955 

.OO7I 

•4457 

•0093 

•3683 

12 

49 

.0035 

x.909 

.0052 

.8672 

.OO7I 

.4251 

.OO94 

•3527 

II 

5° 

1.0036 

11.868 

1.0052 

9.8391 

I.OO7I 

8.4046 

I.OO94 

7-3372 

IO 

51 

.0036 

1.828 

.0052 

.8112 

.0072 

•3843 

.OO94 

•3217 

9 

52 

.0036 

1.787 

•0053 

• 7834 

.0072 

.3640 

.0095 

•3 o6 3 

8 

53 

.0036 

i-747 

•0053 

•7558 

.0073 

•3439 

.0095 

.29O9 

7 

54 

.0037 

1.707 

•0053 

•7283 

•0073 

•3238 

.0096 

•2757 

6 

55 

1.0037 

11.668 

1.0053 

9.701 

1.0073 

8-3039 

1.0096 

7.2604 

5 

56 

.-0037 

1.628 

.0054 

•6739 

.0074 

.2840 

.OO97 

•2453 

4 

57 

.0037 

1.589 

.0054 

.6469 

.0074 

.2642 

.OO97 

. 2302 

3 

58 

.0038 

I -55 

.0054 

.62 

.0074 

.2446 

.OO97 

.2152 

2 

59 

.0038 

1-512 

•0055 

•5933 

•0075 

•225 

.0098 

.2002 

I 

60 

1.0038 

11.474 

1-0055 

9.5668 

1.0075 

8-2055 

.0098 

co 

10 

00 

O 

/ 

Co-sec’t. 

Secant. 

Co-sec’t . 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

r 


85° 

84° 

83° 

82° 

































NATURAL SECANTS AND CO-SECANTS 


405 



8 ° 

90 

10 ° 

1 u° 


9 

Secant. 

Co-sec’t. 

j Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

* 

O 

1.0098 

7- i8 53 

1.0125 

6.3924 

1-0154 

5-7588 

1.0187 

5.2408 

60 

I 

.0099 

• I 7°4 

.0125 

•3807 

•0155 

•7493 

.0188 

•233 

59 

2 

.0099 

•1557 

.0125 

•369 

•0155 

•7398 

.0188 

.2252 

58 

3 

.0099 

.1409 

.0126 

•3574 

.0156 

•7304 

.0189 

.2174 

57 

4 

.01 

.1263 

.0126 

•3458 

.0156 

. 721 

.0189 

•2097 

56 

5 

1.01 

7.1117 

1.0127 

6-3343 

1-0157 

5 - 7 IX 7 

1.019 

5.2019 

55 

6 

.0101 

.0972 

.0127 

.3228 

•0157 

.7023 

.Q191 

•1942 

54 

7 

.0101 

.0827 

.0128 

•3113 

.0158 

•693 

.0191 

.1865 

53 

8 

.0102 

.0683 

.0128 

.2999 

.0158 

.6838 

.0192 

. 1788 

52 

9 

.0102 

•0539 

.0129 

.2885 

.0159 

•6745 

.0192 

.1712 

5 i 

10 

1.0102 

7.0396 

1.0129 

6.2772 

1.0159 

5.6653 

1-0193 

5.1636 

50 

11 

.0103 

.0254 

.013 

.2659 

.016 

• 6561 

.0193 

.156 

49 

12 

.0103 

.0112 

.013 

.2546 

.016 

.647 

.0194 

•1484 

48 

13 

.0104 

6.9971 

.0131 

•2434 

.0161 

•6379 

•0195 

.1409 

47 

14 

.0104 

•983 

.0131 

.2322 

.0162 

.6288 

•0195 

• I 333 

46 

15 

1.0104 

6.969 

1.0132 

6.2211 

1.0162 

5.6197 

1.0196 

5-1258 

45 

16 

.0105 

•955 

.0132 

.21 

.0163 

.6107 

.0196 

.1183 

44 

17 

.0105 

.9411 

.0133 

.199 

.0163 

.6017 

• OI 97 

.1109 

43 

18 

.0106 

• 9 2 73 

•0133 

.188 

.0164 

•5928 

.0198 

• io 34 

42 

x 9 

.0106 

•9*35 

.0134 

• I 77 

.0164 

.5838 

.0198 

.096 

4 i 

20 

1.0107 

6.8998 

1-0134 

6.1661 

1.0165 

5-5749 

1.0199 

5.0886 

40 

21 

.0107 

.8861 

•0135 

•1552 

.0165 

•566 

.0199 

.0812 

39 

22 

.0107 

.8725 

•0135 

•1443 

.0166 

•5572 

.02 

•0739 

38 

23 

.0108 

.8589 

.0136 

•1335 

.0166 

•5484 

.0201 

.0666 

37 

24 

.0108 

•8454 

.0136 

. 1227 

.0167 

•5396 

.0201 

•°593 

36 

25 

1.0109 

6.832 

1.0136 

6.112 

1.0167 

5 - 5308 

1.0202 

5-052 

35 

26 

.0109 

.8185 

.0137 

.1013 

.0168 

.5221 

.0202 

•0447 

34 

27 

.011 

.8052 

•0137 

.0906 

.0169 

• 5 i 34 

.0203 

•0375 

33 

28 

.011 

.7919 

.0138 

.08 

.0169 

•5047 

.0204 

.0302 

32 

29 

.0111 

.7787 

.0138 

.0694 

.017 

.496 

.0204 

•023 

3 i 

30 

I.OIII 

6 - 7 6 55 

1-0139 

6.0588 

1.017 

5-4874 

1.0205 

5-0158 

30 

31 

.0111 

•7523 

.0139 

.0483 

.0171 

.4788 

.0205 

.O087 

29 

32 

.0112 

• 739 2 

.014 

•0379 

.0171 

.4702 

.0206 

.0015 

28 

33 

.0112 

. 7262 

.014 

.0274 

.0172 

.4617 

.0207 

4-9944 

27 

34 

.0113 

• 7 1 32 

.0141 

.017 

.0172 

•4532 

.0207 

• 9 8 73 

26 

35 

I-OII3 

6.7003 

1.0141 

6.0066 

I - OI 73 

5-4447 

1.0208 

4.9802 

25 

36 

.OII4 

.6874 

.0142 

5-9963 

* OI 74 

.4362 

.0208 

•9732 

24 

37 

.OII4 

■6745 

.0142 

.986 

• OI 74 

.4278 

.0209 

.9661 

23 

38 

.0115 

.6617 

.0143 

•9758 

•0175 

.4194 

.021 

• 959 1 

22 

39 

.0115 

.649 

.0143 

•9655 

•0175 

.411 

.021 

.9521 

21 

40 

I.0115 

6.6363 

1.0144 

5-9554 

1.0176 

5.4026 

1.0211 

4.9452 

20 

4 i 

.OIl6 

.6237 

.0144 

•9452 

.0176 

•3943 

.0211 

.9382 

*9 

42 

.OIl6 

.6lII 

.0145 

• 935 i 

• OI 77 

.386 

.0212 

•9313 

l8 

43 

.OII7 

• 59 8 5 

.0145 

•925 

.0177 

•3777 

.0213 

•9243 

I 7 

44 

.OII7 

.586 

.0146 

•915 

.0178 

•3695 

•0213 

• 9 1 7 5 

16 

45 

I.OIl8 

6-5736 

1.0146 

5.9049 

1.0179 

5.3612 

1.0214 

4.9106 

15 

46 

.OIl8 

.5612 

.0147 

•895 

• OI 79 

•353 

• 0215 

• 9°37 

14 

47 

.OII9 

.5488 

.0147 

.885 

.018 

•3449 

• 0215 

.8969 

13 

48 

.OI19 

• 5365 

.0148 

•8751 

.018 

•3367 

.0216 

.8901 

12 

49 

.OII9 

•5243 

.0148 

.8652 

.0181 

.3286 

.0216 

•8833 

11 

50 

1 . 012 

6.5121 

1.0149 

5 ' 8554 

1.0181 

5- 3205 

1.0217 

4.8765 

10 

5 i 

.012 

•4999 

.015 

.8456 

.0182 

.3124 

.0218 

.8697 

9 

52 

.0121 

.4878 

.015 

•8358 

.0182 

•3044 

.0218 

.863 

8 

53 

.0121 

•4757 

• 0151 

.826l 

.0183 

.2963 

.0219 

•8563 

7 

54 

.0122 

•4637 

.0151 

.8163 

.0184 

.2883 

.022 

.8496 

6 

55 

I 0122 

6.4517 

1.0152 

5.8067 

1.0184 

5.2803 

1.022 

4.8429 

5 

56 

.0123 

•4398 

.0152 

•797 

.0185 

.2724 

.0221 

.8362 

4 

57 

.0123 

4279 

•0153 

•7874 

.0185 

.2645 

.0221 

.8296 

3 

58 

.0124 

.416 

•0153 

•7778 

.0186 

.2566 

.0222 

.8229 

2 

59 

.0124 

.4042 

.0154 

.7683 

.0186 

.2487 

.0223 

.8163 

1 

60 

1.0x25 

6.3924 

1-0154 

5.7588 

1.0187 

5.2408 

1.0223 

4.8097 

0 

»- 

' 

Co-sec’t. 

Secant. 

Co-sec’t. | 

Secant. I 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

9 


8 ] 

O 

80 

0 

790 

78 

























































406 natural secants and co-secants. 



32 ° 

13 ° 

I40 

15 ^ 


/ 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

i 

o 

1.0223 

4.8097 

1.0263 

4 - 4454 

1.0306 

4- i 336 

1-0353 

3-8637 

60 

I 

.0224 

.8032 

.0264 

• 439 8 

.0307 

.1287 

•0353 

•8595 

59 

2 

.0225 

. 7966 

.0264 

•4342 

.0308 

.1239 

•0354 

•8553 

58 

3 

.0225 

.7901 

.0265 

.4287 

.0308 

. II9I 

•0355 

.8512 

57 

4 

.0226 

• 7 8 35 

J0266 

.4231 

.0309 

.1144 

•0356 

.847 

56 

5 

1.0226 

4-777 

1.0266 

4.4x76 

1-031 

4.1096 

1-0357 

3.8428 

55 

6 

.0227 

.7706 

.0267 

.4121 

.0311 

. 1048 

•0358 

•8387 

54 

7 

.0228 

.7641 

.0268 

.4065 

.0311 

. IOOI 

•0358 

• 8346 

53 

8 

.0228 

• 757 6 

.0268 

.4011 

.03x2 

•0953 

•0359 

.8304 

52 

9 

.0229 

• 75 i 2 

.0269 

•3956 

■0313 

.0906 

‘036 

.8263 

5 i 

IO 

1.023 

4.7448 

1.027 

4 - 39 01 

1-0314 

4.0859 

1.0361 

3.8222 

5 o 

II 

.023 

•7384 

.0271 

•3847 

.0314 

,0812 

.0362 

.8181 

49 

12 

.0231 

•732 

.0271 

• 379 2 

•0315 

•0765 

.0362 

.814 

48 

13 

.0232 

• 7 2 57 

.0272 

•3738 

• 0316 

.07x8 

•0363 

.81 

47 

14 

.0232 

• 7 I 93 

.0273 

.3684 

.0317 

.0672 

.0364 

• 8059 

46 

15 

1.0233 

4 - 7 I 3 

1.0273 

4-363 

1-0317 

4.0625 

1-0365 

3.8018 

45 

16 

.0234 

.7067 

.0274 

•3576 

• 0318 

•0579 

.0366 

•7978 

44 

1 7 

•0234 

.7004 

.0275 

•3522 

<0319 

•0532 

.0367 

•7937 

43 

IO 

•0235 

.6942 

.0276 

•3469 

• 032 

.0486 

.0367 

.7897 

42 

19 

•0235 

.6879 

.0276 

• 34 i 5 

•032 

.044 

.0368 

•7857 

4 i 

20 

1.0236 

4.6817 

1.0277 

4-3362 

I.0321 

4.0394 

1.0369 

3.7816 

40 

21 

.0237 

■6754 

.0278 

•3309 

.0322 

.0348 

•037 

•7776 

39 

22 

.0237 

.6692 

.0278 

• 3 2 56 

• 0323 

.0302 

•0371 

•7736 

38 

23 

.0238 

.6631 

.0279 

•3203 

• 0323 

.0256 

•0371 

•7697 

37 

24 

.0239 

.6569 

.028 

• 3 i 5 

.0324 

.0211 

.0372 

•7657 

36 

25 

1.0239 

4.6507 

1.028 

4.3098 

1-0325 

4.0165 

1-0373 

3 - 76 i 7 

35 

26 

.024 

.6446 

.0281 

•3045 

.0326 

.012 

•0374 

•7577 

34 

27 

.0241 

•6385 

.0282 

•2993 

• 0327 

.0074 

•0375 

•7538 

33 

28 

.0241 

.6324 

.0283 

.2941 

•0327 

.OO29 

.0376 

.7498 

32 

29 

.0242 

.6263 

.0283 

.2888 

.0328 

3.9984 

.0376 

•7459 

3 i 

30 

1.0243 

4.6202 

1.0284 

4.2836 

1.0329 

3-9939 

1-0377 

3.742 

30 

31 

.0243 

6142 

0285 

.2785 

•033 

•9894 

•0378 

•738 

29 

32 

.0244 

6081 

.0285 

•2733 

•033 

•985 

•0379 

*7341 

28 

33 

.0245 

.6021 

.0286 

.2681 

•0331 

.9805 

• 038 

•7302 

27 

34 

.0245 

• 59 61 

.0287 

•263 

•0332 

•976 

• 0381 

.7263 

26 

35 

1.0246 

4.5901 

1.0288 

4-2579 

1-0333 

3.9716 

1.0382 

3.7224 

25 

36 

.0247 

.5841 

.0288 

.2527 

•0334 

.9672 

.0382 

.7186 

24 

37 

.0247 

.5782 

,0289 

.2476 

‘0334 

.9627 

•0383 

• 7 I 47 

23 

3 « 

.0248 

.5722 

.029 

.2425 

•0335 

:9583 

.0384 

.7108 

22 

39 

.0249 

•5663 

*0291 

•2375 

•0336 

■'9539 

•0385 

•707 

21 

40 

1.0249 

4.5604 

I.0291 

4.2324 

1-0337 

3-9495 

1.0386 

3 - 703 I 

20 

4 i 

.025 

•5545 

.0292 

.2273 

•0338 

•9451 

.0387 

•6993 

*9 

42 

.0251 

.5486 

.0293 

.2223 

•0338 

.9408 

•0387 

•6955 

18 

43 

.0251 

.5428 

.0293 

•2173 

•0339 

;9364 

.0388 

.69x7 

17 

44 

.0252 

•5369 

.0294 

.2122 

•034 

•932 

•0389 

.6878 

16 

45 

1.0253 

4 - 53 H 

1.0295 

4.2072 

1.0341 

3 - 9 2 77 

1.039 

3.684 

15 

46 

.0253 

•5253 

.0296 

. 2022 

.0341 

•9234 

.0391 

.6802 

14 

47 

.0254 

• 5 i 95 

.0296 

.1972 

.•0342 

.9x9 

• 0392 

• 6765 

13 

48 

•0255 

• 5 i 37 

.0297 

.1923 

•0343 

.9x47 

•0393 

.6727 

12 

49 

•0255 

•5079 

.0298 

•1873 

•0344 

.9104 

■0393 

.6689 

II 

50 

1.0256 

4.5021 

1.0299 

4.1824 

1-0345 

3.9061 

1.0394 

3-6651 

IO 

5 i 

.0257 

.4964 

.0299 

•1774 

•0345 

.9018 

•0395 

.6614 

9 

52 

.0257 

.4907 

•°3 

•1725 

•0346 

.8976 

.0396 

•6576 

8 

53 

.0258 

•485 

.0301 

. 1676 

•0347 

•8933 

•0397 

•6539 

7 

54 

.0259 

•4793 

.0302 

. 1627 

.0348 

.899 

•0398 

.6502 

6 

55 

1.026 

4-4736 

1.0302 

4-1578 

1.0349 

3.8848 

1.0399 

3.6464 

5 

56 

,026 

.4679 

•0303 

.1529 

•0349 

.8805 

•0399 

.6427 

4 

57 

.0261 

.4623 

.0304 

. 1481 

•035 

.8763 

.04 

•639 

3 

58 

.0262 

.4566 

•0305 

.1432 

•0351 

.8721 

.O4OI 

•6353 

2 

59 

.0262 

• 45 i 

•0305 

.1384 

•0352 

.8679 

.0402 

•6316 

I 

60 

1.0263 

4-4454 

1.0306 

4 - i 336 

i -°353 

3.8637 

1.0403 

3.6279 

O 

/ 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 1 

Secant. 

/ 


770 

76 ° 

75 ° 

740 



































NATURAL SECANTS AND CO-SECANTS 


407 



16° 

170 

180 

19° 


/ 

Secant. | 

Co-sec’t. | 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

* 

0 

1.0403 

3.6279 

1-0457 

3-4203 

1-0515 

3.2361 

1,0576 

3-07I5 

60 

I 

.O4O4 

.6243 

.0458 

.417 

• 0516 

• 2332 

•0577 

.069 

59 

2 

.0405 

.6206 

•0459 

.4138 

•0517 

• 2303 

.0578 

.0664 

58 

3 

.0406 

.6169 

.046 

.4106 

• 0518 

.2274 

•0579 

.0638 

57 

4 

.0406 

•6i33 

.0461 

■4073 

.0519 

•2245 

.058 

.0612 

56 

5 

1.0407 

3.6096 

1.0461 

3-4041 

1.052 

3.2216 

1.0581 

3.0586 

55 

6 

.0408 

.606 

.0462 

.4OO9 

• 0521 

.2188 

.0582 

.0561 

54 

7 

.0409 

.6024 

.0463 

•3977 

.0522 

•2159 

.0584 

•0535 

53 

8 

.041 

•59 8 7 

.0464 

•3945 

•0523 

^2131 

•0585 

.0509 

52 

9 

.O4II 

•595i 

• 0465 

•39 r 3 

• 0524 

.2102 

.0586 

.0484 

5i 

IO 

I.0412 

3-59*5 

1.0466 

3-388 i 

1-0525 

3.2074 

1.0587 

3-0458 

50 

II 

.0413 

•5879 

.0467 

•3849 

.0526 

•2045 

.0588 

•0433 

49 

12 

.0413 

•5843 

.0468 

•3817 

.0527 

.2017 

.0589 

.0407 

48 

13 

.0414 

.5807 

.0469 

•3785 

.0528 

.1989 

•059 

.0382 

47 

14 

.0415 

•5772 

•047 

•3754 

.0529 

.196 

.0591 

•0357- 

46 

15 

1.0416 

3-573 6 

1.0471 

3-3722 

1-053 

3-1932 

1.0592 

3-033I 

45 

l6 

.0417 

•57 

.0472 

•369 

•053 1 

.1904 

•0593 

.0306 

44 

17 

.0418 

•5665 

•0473 

•3659 

•0532 

. 1876 

•0594 

.0281 

43 

18 

.0419 

.5629 

•0474 

.3627 

•0533 

.1848 

•0595 

.0256 

42 

*9 

.042 

•5594 

•0475 

•3596 

•0534 

.182 

•0596 

•0231 

4i 

20 

1.042 

3-5559 

1.0476 

3-3565 

i-°535 

3-1792 

1.0598 

3.0206 

40 

21 

.0421 

•5523 

•0477 

•3534 

•0536 

.1764 

•0599 

.0181 

39 

22 

.0422 

.5488 

•0478 

.3502 

•0537 

•1736 

.06 

• 0156 

38 

23 

.0423 

•5453 

.0478 

•3471 

•0538 

. 1708 

.0601 

• 0131 

37 

24 

.O424 

.5418 

•0479 

•344 

•0539 

. 1681 

.0602 

.0106 

36 

25 

1.0425 

3-5383 

1.048 

3- 3409 

1-054 

3-1653 

1.0603 

3.0081 

35 

26 

.0426 

•5348 

.0481 

•3378 

.0541 

.1625 

.0604 

.0056 

34 

27 

.0427 

•5313 

.0482 

•3347 

•0542 

.1598 

.0605 

0031 

33 

28 

.0428 

•5279 

•0483 

• 33 l6 

•0543 

•i57 

.0606 

.0007 

32 

29 

.0428 

•5244 

.0484 

.3286 

■0544 

•1543 

.0607 

2.9982 

3i 

30 

I.O429 

3-5209 

1.0485 

3-3255 

1-0545 

3-i5i5 

1.0608 

2 -9957 

3° 

31 

•°43 

•5175 

.0486 

.3224 

.0546 

. 1488 

.0609 

•9933 

29 

32 

.0431 

•5i4 

.0487 

•3i94 

•0547 

. 1461 

.0611 

.9908 

28 

33 

.0432 

.5106 

.0488 

•3163 

.0548 

•1433 

.0612 

.9884 

27 

34 

•0433 

.5072 

.0489 

•3133 

•0549 

.1406 

•0613 

•9 8 59 

26 

35 

I -°434 

3-5037 

1.049 

3.3102 

1-055 

3-1379 

1.0614 

2-9835 

25 

39 

•0435 

•5003 

.0491 

•3072 

•0551 

•1352 

•0615 

.981 

24 

37 

.0436 

.4969 

.0492 

•3042 

•0552 

•1325 

.0616 

.9786 

23 

3« 

•0437 

•4935 

•0493 

•3 011 

•0553 

.1298 

.0617 

.9762 

22 

39 

.0438 

.4901 

•0494 

.2981 

•0554 

. 1271 

.0618 

•9738 

21 

40 

1.0438 

3.4867 

1.0495 

3-2951 

1-0555 

3.1244 

1.0619 

2.9713 

20 

4i 

•0439 

•4833 

.0496 

.2921 

•0556 

.1217 

.062 

.9689 

19 

42 

.044 

•4799 

.0497 

.2891 

•0557 

.119 

.0622 

.9665 

l8 

43 

.0441 

.4766 

.0498 

. 2861 

•0558 

.1163 

.0623 

.9641 

*7 

44 

.0442 

•4732 

•0499 

•2831 

•0559 

•1137 

.0624 

.9617 

16 

45 

1.0443 

3.4698 

1.05 

3.2801 

1.056 

3. hi 

1.0625 

2-9593 

15 

46 

.0444 

.4665 

.0501 

.2772 

.0561 

.1083 

.0626 

•9569 

14 

47 

•0445 

.4632 

.0502 

.2742 

.0562 

•1057 

.0627 

•9545 

13 

48 

.0446 

•459 3 

•0503 

.2712 

•0363 

.103 

.0628 

.9521 

12 

49 

.0447 

•4565 

.0504 

.2683 

•0565 

. 1004 

.0629 

•9497 

11 

5o 

1.0448 

3-4532 

-1-0505 

3-2653 

1.0566 

3-0977 

1.063 

2.9474 

IO 

5i 

.0448 

•4498 

. -0506 

.2624 

•0567 

.0951 

.0632 

•945 

9 

52 

.0449 

•4465 

•0507 

•2594 

.0568 

■0925 

•0633 

.9426 

8 

53 

•045 

•4432 

.0508 

•2565 

•0569 

.0898 

•0634 

.9402 

7 

54 

.0451 

•4399 

.0509 

•2535 

•057 

.0872 

•0635 

•9379 

6 

55 

1.0452 

3-4366 

1-051 

3.2506 

1-0571 

3.0846 

1.0636 

2-9355 

5 

56 

•0453 

•4334 

• 0511 

•2477 

•0572 

.082 

•0637 

•9332 

4 

57 

•0454 

.4301 

.0512 

.2448 

•0573 

•0793 

.0638 

• 9308 

3 

58 

•0455 

.4268 

•0513 

.2419 

•0574 

.0767 

•0639 

.9285 

2 

59 

.0456 

.4236 

.0514 

•239 

•0575 

.0741 

.0641 

.9261 

I 

60 

1-0457 

3-4203 

1-0515 

3.2361 

1.0576 

3-0715 

1.0642 

2.9238 

O 


Co-sec’t. 

i Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant, j 

Co-sec’t. 

Secant. 

/ 


73° 

72° 1 

71° I 

70° 











































40 8 NATURAL SECANTS AND CO-SECANTS. 



20 ° 

21 ° 

22 ° 

23 ° 


/ 

Secant. 

Co-sec't 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

| Co-sec’t 

/ 

o 

1.0642 

2.9238 

1.0711 

2.7904 

1.0785 

2.6695 

1.0864 

2-5593 

60 

I 

•0643 

.9215 

•0713 

•7883 

.0787 

•6675 

.0865 

•5575 

59 

2 

.0644 

.9191 

.0714 

.7862 

.0788 

.6656 

.0866 

•5558 

58 

3 

•0645 

.9168 

•0715 

.7841 

.0789 

.6637 

.0868 

•554 

57 

4 

.0646 

- 9*45 

.0716 

. 782 

.079 

.6618 

.0869 

•5523 

56 

5 

1.0647 

2.9122 

I-07I7 

2.7799 

1.0792 

2.6599 

1.087 

2.5506 

55 

6 

.0648 

.9098 

•°7 I 9 

•7778 

0793 

•658 

.0872 

.5488 

54 

7 

.065 

•9°75 

.072 

•7757 

.0794 

•6561 

.0873 

•547 1 

53 

8 

.0651 

.9052 

.0721 

•7736 

•0795 

•6542 

.0874 

-5453 

52 

9 

.0652 

.9029 

.0722 

•7715 

1 -0797 

•6523 

.0876 

•5436 

51 

IO 

1-0653 

2.9006 

1.0723 

2.7694 

1.0798 

2.6504 

1.0877 

2.5419 

50 

ii 

.0654 

.8983 

.0725 

.7674 

.0799 

6485 

.0878 

•54° 2 

49 

12 

•0655 

.896 

.0726 

•7653 

.0801 

.6466 

.088 

•5384 

48 

13 

.0656 

•8937 

.0727 

.7632 

.0802 

.6447 

.0881 

•5367 

47 

14 

..0658 

.8915 

.0728 

.7611 

.0803 

.6428 

.0882 

•535 

46 

15 

1.0659 

2.8892 

1.0729 

2-759 1 

1.0804 

2.641 

1.0884 

2-5333 

45 

16 

.066 

. 8869 

•0731 

•757 

.0806 

.6391 

.0885 

•5316 

44 

17 

.0661 

.8846 

.0732 

■*755. 

.0807 

6372 

.0886 

•5299 

43 

18 

.0662 

.8824 

•0733 

•7529 

.0808 

•6353 

.0888 

.5281 

42 

s 9 

.0663 

.8801 

•0734 

•7509 

.081 

■6335 

.0889 

.5264 

41 

20 

1.0664 

00 

00 

c* 

1.0736 

2.7488 

1.0811 

2.6316 

1.0891 

2.5247 

40 

21 

.0 666 

.8756 

•0737 

.7468 

.0812 

.6297 

.0892 

•523 

39 

22 

.0667 

•8733 

.0738 

•7447 

.0813 

.6279 

.0893 

•5213 

38 

23 

.0668 

.8711 

•0739 

.7427 

.0815 

.626 

•0895 

.5196 

37 

24 

.0669 

.8688 

.074 

.7406 

.0816 

.6242 

.0896 

•5179 

36 

25 

1.067 

2.8666 

1.0742 

2.7386 

1.0817 

2.6223 

1.0897 

2-5163 

35 

26 

.0671 

. 8644 

•0743 

.7366 

.0819 

.6205 

.0899 

.5146 

34 

27 

.0673 

. 8621 

.0744 

•7346 

.082 

.6186 

.09 

.5129 

33 

28 

.0674 

■8599 

•0745 

7325 

.0821 

.6168 

.0902 

.5112 

32 

2 9 

.0675 

•8577 

.0747 

•7305 

.0823 

.615 

.0903 

•5095 

3i 

30 

1.0676 

2-8554 

1.0748 

2.7285 

1.0824 

2.6131 

1.0904 

2.5078 

30 

3i 

.0677 

•8532 

.0749 

.7265 

.0825 

.61x3 

.0906 

.5062 

29 

32 

.0678 

.851 

•075 

•7245 

.0826 

•6095 

.0907 

•5045 

28 

33 

.0679 

.8488 

•0751 

.7225 

.0828 

.6076 

.0908 

.5028 

27 

34 

.0681 

.8466 

•°75 3 

.7205 

.0829 

.6058 

.091 

• 5 °ii 

26 

35 

1.0682 

2.8444 

1-0754 

2.7185 

1.083 

2.604 

1.0911 

2-4995 

25 

36 

.0683 

.8422 

•0755 

■7165 

.0832 

.6022 

.0913 

•4978 

24 

37 

.0684 

,8 4 

.0756 

•7i45 

•0833 

.6003 

.0914 

.4961 

23 

3« 

.0685 

.8378 

.0758 

•7125 

.0834 

•5985 

.0915 

•4945 

22 

39 

.0686 

■8356 

•0759 

•7i°5 

.0836 

•59 6 7 

•0917 

.4928 

21 

40 

1.0688 

2-8334 

1.076 

2.7085 

1.0837 

2-5949 

1.0918 

2.4912 

20 

4i 

.0689 

.8312 

.0761 

.7065 

.0838 

•593i 

.092 

•4895 

*9 

42 

.069 

.829 

.0763 

• 7°45 

.084 

•5913 

.0921 

.4879 

18 

43 

.0691 

.8269 

.0764 

. 7026 

.0841 

•5895 

.0922 

.4862 

17 

44 

.0692 

.8247 

.0765 

.7006 

.0842 

■5877 

.0924 

.4346 

16 

45 

1.0694 

2.8225 

1.0766 

2.6986 

1.0844 

2.5859 

1.0925 

2.4829 

15 

46 

•0695 

.8204 

.0768 

.6967 

.0845 

.5841 

.0927 

.4813 

1 4 

47 

.0696 

.8182 

*0769 

.6947 

. 0846 

•5823 

.0928 

•4797 

13 

48 

.0697 

.816 

•0 77 

.6927 

.0847 

.5805 

.0929 

.478 

12 

49 

.0698 

.8139 

.0771 

.6908 

.0849 

•5787 

.0931 

.4764 

11 

50 

1.0699 

2.8117 

I *°773 

2.6888 

1.085 

2-577 . 

1.0932 

2.4748 

10 

5i 

.0701 

.8096 

•°774 

.6869 

.0851 

•5752 

•0934 

•473i 

9 

52 

.0702 

.8074 

•0775 

.6849 

.0853 

•5734 

•0935 

•47i5 

8 

53 

.0703 

•8053 

.0776 

•683 

.0854 

•5716 

.0936 

.4699 

7 

54 

.0704 

•8032 

.0778 

.681 

•0855 

•5699 

.0938 

.4683 

6 

55 

1.0705 

2.801 

1.0779 

2.6791 

1.0857 

2.5681 

I *°939 

2.4666 

5 

56 

.0707 

.7989 

.078 

.6772 

.0858 

•5663 

.0941 

•465 

4 

57 

.0708 

.7968 

.0781 

.6752 

.0859 

.5646 

.0942 

•4^34 

0 

.> 

58 

.0709 

•7947 

.0783 

•6733 

.0861 

.5628 

•°943 

.4618 

2 

59 

.071 

•7925 

.0784 

.6714 

.0862 

.561 

•0945 

.4602 

1 

60 

1.0711 

2.7904 

1.0785 

2.6695 

1.0864 

2-5593 

.0946 

2.4586 

0 


Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant, 

Co-sec’t. 

Secant. 



69 ° 

68 

0 

67 ° l 

66 ° 







































































NATURAL SECANTS AND CO-SECANTS 


409 


24° 

25° 

260 

I 27° 


Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 1 

1 Secant. 

Co-skc’t. 


1.0946 

2.4586 

1.1034 

2.3662 

1.1126 

2.2812 

1.1223 

2.2027 

60 

.0948 

•457 

•1035 

•3647 

.1127 

.2798 

.1225 

.2014 

59 

.0949 

•4554 

• io 37 

•3632 

.1129 

.2784 

. 1226 

.2002 

58 

.0951 

•4538 

.1038 

.3618 

.1131 

.2771 

.1228 

.1989 

57 

.0952 

.4522 

. 104 

•3603 

.1132 

•2757 

.123 

.1977 

56 

1-0953 

2.4506 

1.1041 

2.3588 

I - II 34 

2.2744 

1-1231 

2.1964 

55 

•0955 

•449 

• io 43 

•3574 

•ii35 

•273 

•1233 

.1952 

54 

.0956 

•4474 

.1044 

•3559 

•ii37 

.2717 

•1235 

• J 939 

53 

•0958 

•4458 

.1046 

•3544 

• IJ 39 

.2703 

*1237 

.1927 

52 

•0959 

.4442 

• io 47 

•353 

.114 

.269 

.1238 

• I 9 I 4 

51 

1.0961 

2.4426 

1.1049 

2-3515 

1.1142 

2.2676 

1.124 

2.1902 

50 

.0962 

.4411 

.105 

•35oi 

• II 43 

.2663 

.1242 

.1889 

49 

.0963 

•4395 

.1052 

•3486 

•ii45 

.265 

.1243 

.1877 

48 

.0965 

•4379 

•1053 

•3472 

.1147 

.2636 

.1245 

.1865 

47 

•0966 

•4363 

•1055 

•3457 

.1148 

.2623 

.1247 

.1852 

46 

1.0968 

2.4347 

1.1056 

2-3443 

1.115 

2.261 

1.1248 

2.184 

45 

.0969 

4332 

.1058 

.3428 

.1151 

.2596 

.125 

. 1828 

44 

.0971 

.4316 

.1059 

•34H 

•1153 

.2583 

.1252 

.1815 

43 

.0972 

•43 

. 1061 

•3399 

•ii55 

•257 

.1253 

.1803 

4 2 

•°973 

.4285 

. 1062 

•3385 

.1156 

•2556 

•1255 

.1791 

4i 

1-0975 

2.4269 

1.1064 

2-3371 

1.1158 

2.2543 

1.1257 

2.1778 

40 

.0976 

•4254 

.1065 

3356 

•ii59 

•253 

.1258 

.1766 

39 

.0978 

.4238 

. 1067 

•3342 

.1161 

•2517 

.126 

•1754 

38 

.0979 

4222 

. 1068 

•3328 

.1163 

•2503 

.1262 

.1742 

37 

.0981 

.4207 

. 107 

•33*3 

.1164 

.249 

.1264 

• *73 

36 

n 

00 

On 

0 

M 

2.4191 

1.1072 

2.3299 

1.1166 

2.2477 

1.1265 

2.1717 

35 

.O984 

.4176 

• io 73 

.3285 

. 1167 

. 2464 

.1267 

i7°5 

34 

.0985 

.416 

•1075 

•327 1 

. 1169 

.2451 

.1269 

.1693 

33 

.O986 

•4i45 

. 1076 

.3256 

.1171 

.2438 

.127 

.1681 

32 

.O988 

•4i3 

. 1078 

.3242 

. 1172 

.2425 

. 1272 

. 1669 

3* 

I.O989 

2.4114 

1.1079 

2.3228 

1 ‘ II 74 

2.2411 

1.1274 

2.1657 

3» 

0991 

.4099 

. 1081 

•3214 

.1176 

.2398 

-1275 

.1645 

29 

0992 

.4083 

.1082 

•32 

1177 

-2385 

• I2 77 

•1633 

28 

' 0994 

.4068 

•1084 

• 3 i8 6 

• 11 79 

•2372 

.1279 

. 162 

! 27 

•0995 

•4053 

.1085 

•3172 

.118 

•2359 

.1281 

. 1608 

26 

I O997 

2-4037 

1.1087 

2-3158 

1.1182 

2.2346 

1.1282 

2.1596 

25 

.O998 

.4022 

.1088 

•3143 

.1184 

•2333 

.1284 

•15S4 

24 

* I 

.4007 

. 109 

.3129 

.1185 

.232 

,1286 

•1572 

23 

. IOOI 

•3992 

.1092 

•3115 

.1187 

•2307 

. 1287 

•i5£ 

22 

.1003 

•3976 

.1093 

• 3 101 

.1189 

2294 

. 1289 

.1548. 

21 

1.1004 

2.3961 

1.1095 

2.3087 

I.119 

2.2282 

1,1291 

2.1536 

20 

.1005 

•394 6 

.1096 

•3073 

.1192 

.2269 

.1293 

-1*525. 

J 9 

. 1007 

•3931 

1098 

•3°59 

•ii93 

.2256 

1294 

-1513 

l8 

. 1008 

•39 l6 

.1099 

*3046 

•1195 

•2243 

.1296 

.15P1 

I 7 

. IOI 

•39 31 

. IIOI 

.3032 

.1197 

.223 

. 1298 

.1489 

l6 

I. IOII 

2.3886 

1.1102 

2.30l8 

3.1198 

2.2217 

1.1299 

2-i 477 

*5 

.1013 

•3871 

.1104 

•3004 

.12 

.2204 

- 13o>i 

.1465 


. 1014 

3856 

.1106 

.299 

.1202 

• 2192 

•1303: 

-MSS 


.1016 

.3841 

.1107 

. 2976 

.1203 

•2179 

•13PS 

.1441 

12 

. 1017 

.3826 

.1109 

.2962 

.1205 

.2166 

.1306 

*143 

n 

I.. 1019 

2.3811 

1.hi 

2.2949 

1.1207 

2.2153 

1.13.08 

2.1418 

10 

• 102 

•379 6 

.1112 

•2935 

.1208 

.2141 

; -is* 

.1406' 

9 

. 1022 

•378 i 

.1113 

.2921 

»121 

.2128 


.1394 

8. 

.1023 

.3766 

.1115 

.29O7 

.1212 

.2115 

•1313 

.1382- 

7 

.1025 

•375i 

.1116 

.2894 

.1213 

.2103 

•1315 

•1371 

6> 

1.1026 

2.3736 

1.1118 

2. 288 

1.1215 

2.209. 


2-1359' 

5 

.1028 

•3 7 21 

.112 

.2 866 

.1217 

• 2077 

•1319 

•1347 

4 

.IO29 

.3706 

.1121 

-2853 

.1218 

.2065 

• *3;2 

-1335 

3 

.IO3I 

.3691 

.1123 

.2839 

.122 

.2052 

.1322 

.1324 

2 

.IO32 

•3677 

.1124 

.2825 

. 1222 

•2039 

.1324 

.1313’ 

1 

I. IO34 

2.3662 

1.1126 

2.2812 

1.1223 

2'. 2027 

1.1326 

2.13 

0 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 1 

SECA.NT. 

* 

65 ° 

64 ° 

63 ° 

62 ° 



M Af 






















































410 NATURAL SECANTS AND CO-SECANTS. 



28 ° 

290 

30 ° 

313 

! 

f 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

/ 

o 

1.1326 

2.13 

i -1433 

2.0627 

1-1547 

2 

1.1666 

1.9416 

60 

I 

*1327 

. 1289 

•1435 

.0616 

•1549 

1.999 

. 1668 

9407 

59 

2 

.1329 

•12 77 

•1437 

.0605 

•I 55 i 

.998 

.167 

•9397 

58 

3 

•i 33 i 

. 1266 

•1439 

•0594 

•1553 

•997 

.1672 

.9388 

57 

4 

-1333 

.1254 

• I 44 I 

•0583 

-1555 

.996 

.1674 

•9378 

56 

5 

i -1334 

2.1242 

1443 

2-0573 

i-i 557 

i -995 

1 1676 

19369 

55 

6 

• 1336 

.1231 

•1445 

.0562 

•1559 

•994 

.1678 

•936 

54 

7 

•1338 

. 1219 

.1446 

•0551 

.1561 

•993 

. 1681 

•935 

53 

8 

•134 

. 1208 

.1448 

•054 

.1562 

.992 

.1683 

• 934 1 

52 

9 

•I 34 i 

.1196 

•i 45 

•053 

.1564 

.991 

.1685 

•9332 

5 i 

IO 

I * I 343 

2.1185 

1-1452 

2.0519 

1.1566 

1.99 

1.1687 

1.9322 

50 

ii 

•1345 

1173 

•1454 

.0508 

.1568 

.989 

. 1689 

• 93 i 3 

49 

12 

• I 347 

.1162 

.1456 

.0498 

•157 

.988 

1691 

«9304 

48 

13 

•1349 

•ii 5 

.1458 

.0487 

•1572 

.987 

.1693 

• 9 2 95 

47 

14 

•i 35 

•ii 39 

•1459 

.0476 

•1574 

.986 

.1695 

.9285 

46 

15 

i-i 352 

2.1127 

1.1461 

2.0466 

1.1576 

1.985 

1.1697 

I.9276 

45 

16 

•1354 

.1116 

•h 6 3 

•0455 

•1578 

.984 

- i6 99 

.9267 

44 

17 

•1356 

.1104 

.1465 

•0444 

-158 

•983 

. I7OI 

.9258 

43 

18 

•1357 

.1093 

.1467 

•0434 

.1582 

.982 

• I 7°3 

.9248 

42 

*9 

•1359 

. 1082 

.1469 

.0423 

.1584 

.9811 

• I 7°5 

• 9 2 39 

4 i 

20 

1.1361 

2.107 

1.1471 

2.0413 

1.1586 

1.9801 

1-1707 

1.923 

40 

21 

•1363 

.1059 

•1473 

.0402 

•1588 

• 979 1 

.1709 

.9221 

39 

22 

•1365 

. 1048 

.1474 

-0392 

•159 

.9781 

. 1712 

.9212 

3 8 

23 

.1366 

.1036 

.1476 

.0381 

.1592 

• 977 i 

.1714 

.9203 

37 

24 

.1368 

1025 

.1478 

•037 

•1594 

.9761 

. 1716 

■9 I 93 

36 

25 

I - 1 37 

2.1014 

1.148 

2.036 

i-i 59 6 

1-9752 

1.1718 

I.9184 

35 

26 

•1372 

. 1002 

.1482 

•0349 

.1598 

•9742 

.172 

• 9 I 75 

34 

27 

•1373 

.0991 

.1484 

•0339 

. l6 

•9732 

.1722 

.9166 

33 

28 

•1375 

.098 

.i486 

•0329 

. 1602 

.9722 

.1724 

• 9 I 57 

32 

2 9 

• I 377 

.0969 

. 1488 

.0318 

. 1604 

•9713 

. 1726 

.9148 

3 i 

30 

I * I 379 

2.0957 

1.1489 

2.0308 

I. l6o6 

I * 97°3 

1.1728 

I * 9 I 39 

3 ° 

31 

.1381 

.0946 

.149 1 

•0297 

.l6o8 

•9693 

•i 73 

• 9 I 3 

29 

32 

.1382 

•0935 

•1493 

.0287 

. l6l 

.9683 

•1732 

.9121 

28 

33 

•1384 

.0924 

•H 95 

.0276 

. l6l2 

.9674 

•1734 

.9112 

27 

34 

.1386 

.0912 

.1497 

.0266 

. l6l4 

.9664 

• I 737 

.9102 

26 

35 

1.1388 

2.0901 

1.1499 

2.0256 

I. l6l6 

i- 9 6 54 

I - I 739 

I - 9°93 

25 

36 

•i 39 

.089 

.1501 

.0245 

. l6l8 

•9645 

.1741 

.9084 

24 

37 

• I 39 T 

.0879 

•1503 

•0235 

. l62 

•9635 

•1743 

• 9°75 

23 

3 » 

•1393 

.0868 

•1505 

.0224 

.l622 

.9625 

•1745 

.9066 

22 

39 

•1395 

.0857 

•1507 

. 0214 

. l624 

.9616 

•1747 

• 9°57 

21 

40 

*• I 397 

2.0846 

1.1508 

2.0204 

I. 1626 

I.9606 

1.1749 

I.9O48 

20 

4 i 

• I 399 

•0835 

•151 

• 0194 

, l628 

.9596 

•i 75 i 

• 9°39 

IQ 

42 

. 1401 

.0824 

.1512 

•0183 

.163 

•9587 

• *753 

• 9°3 

18 

43 

. 1402 

.0812 

•1514 

•0173 

.1632 

•9577 

•1756 

.9021 

17 

44 

.1404 

.0801 

.1516 

.0163 

.1634 

.9568 

■1758 

• 9 OI 3 

16 

45 

1.1406 

2.079 

i.i 5 i 8 

2.0152 

I. 1636 

1-9558 

1.176 

1.9004 

15 

46 

.1408 

•°779 

.152 

.0142 

.1638 

•9549 

. 1762 

•8995 

14 

47 

.141 

.0768 

.1522 

•0132 

.164 

•9539 

.1764 

.8986 

13 

48 

. 1411 

•°757 

•1524 

.0122 

. 1642 

•953 

.1766 

.8977 

12 

49 

•1413 

.0746 

.1526 

.0111 

.1644 

•952 

. 1768 

.8968 

II 

50 

I -i 4 I 5 

2-0735 

1.1528 

2.0101 

I. 1646 

I- 95 I 

I - I 77 

1.8959 

IO 

5 i 

.1417 

.0725 

•i 53 

.0091 

. 1648 

.9501 

.1772 

.895 

9 

52 

.1419 

.0714 

•1531 

.0081 

.165 

• 949 1 

-1775 

.8941 

8 

53 

. 1421 

.0703 

•1533 

.0071 

.1652 

.9482 

.1777 

.8932 

7 

54 

.1422 

.0692 

•1535 

.0061 

.1654 

•9473 

.1779 

.8924 

6 

55 

1.1424 

2.0681 

I-I 537 

2.005 

1.1656 

1-9463 

1.1781 

1.8915 

5 

56 

. 1426 

.067 

•1539 

.004 

.3:658 

•9454 

•1783 

.8906 

4 

57 

.1428 

•0659 

•i 54 i 

•003 

. l66 

•9444 

•1785 

.8897 

3 

58 

•143 

.0648 

•1543 

.002 

. l662 

•9435 

•1787 

.8888 

2 

59 

•i 43 2 

.0637 

•1545 

.001 

. 1664 

•9425 

• I 79 

. 8879 

1 

60 

i -1433 

2.0627 

I - I 547 

2 

I. l666 

1.9416 

1.1792 

1.8871 

0 

/ 

Co-sec't. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

/ 


61° 

60 ° 

59 ° 

68° 













































NATURAL SECANTS AND COSECANTS 


41 I 


I 

32 0 

33 ° 

34 ° 

35 ° 

t 

SECANT. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 1 

Secant. 

Co-sec’t. 

° \ 

I. I 792 

1.S871 

1.1924 

1.8361 

1.2062 

1. 7883 

1.2208 

1-7434 

I 

.1794 

.8862 

■ 1926 

•8352 

.2064 

•7875 

.221 

.7427 

o 

.1796 

•8853 

. 1928 

•8344 

.2067 

.7867 

.2213 

.742 

3 

.1798 

. 8844 

• i 93 

833 6 

.2069 

.786 

2215 

•7413 

4 

.l8 

.8836 

•1933 

.8328 

.2072 

.7852 

.2218 

•7405 

5 

I. l8o2 

1.8827 

i-i 935 

1.832 

1.2074 

1.7844 

1.222 

i -7398 

6 

.1805 

.8818 

• r 937 

.8311 

2076 

7837 

.2223 

■ 739 1 

7 

. 1807 

.8809 

■1939 

•8303 

.2079 

.7829 

.2225 

•7384 

8 

. 1809 

.8801 

.1942 

.8295 

.2081 

.7821 

.2228 

•7377 

9 

. l8ll 

.8792 

.1944 

. 8287 

.2083 

.7814 

223 

•7369 

IO 

I.1813 

1.8783 

1.1946 

1.8279 

1.2086 

1.7806 

1.2233 

1.7362 

II 

.1815 

.8785 

.1948 

.8271 

.2088 

.7798 

• 2235 

•7355 

12 

.l8l8 

.8766 

•i 95 i 

8263 

.2091 

• 779 1 

• 2238 

•7348 

13 

. 182 

•8757 

•1953 

•8255 

.2093 

•7783 

.224 

• 734 i 

14 

.l822 

18749 

•1955 

.8246 

.2095 

.7776 

.2243 

•7334 

15 

I. 1824 

1.874 

1.1958 

1.8238 

1.2098 

1.7768 

1-2245 

I -7327 

l6 

. 1826 

•8731 

. 196 

•823 

• 21 

.776 

.2248 

• 73 X 9 

17 

. 1828 

•8723 

. 1962 

.8222 

.2103 

•7753 

■ 225 

• 73 X 2 

18 

.1831 

.8714 

.1964 

• 8214 

.2105 

7745 

• 2253 

•7305 

*9 

•1833 

.8706 

.1967 

• 8206 

.2107 

•7738 

•2255 

•7298 

20 

I'l 835 

1.8697 

1.1969 

1.8198 

I. 2 II 

1-773 

1.2258 

I. 7291 

21 

•1837 

.8688 

.1971 

.819 

.2112 

•7723 

.226 

.7284 

22 

.1839 

.868 

.1974 

.8182 

.2115 

•7715 

.2263 

•7277 

23 

.1841 

.8671 

.1976 

.8174 

.2117 

.7708 

.2265 

•727 

24 

.1844 

.8663 

.1978 

.8166 

.2119 

•77 

.2268 

.7263 

25 

I. 1846 

1.8654 

1.198 

1.8158 

I.2122 

i- 7 6 93 

I.227 

1.7256 

26 

.1848 

.8646 

.1983 

.815 

.2124 

.7685 

•2273 

•7249 

27 

.185 

.8637 

.1985 

.8142 

.2127 

.7678 

.2276 

•7242 

28 

.1852 

.8629 

.1987 

.8x34 

. 2129 

.767 

. 2278 

•7234 

29 

•1855 

. 862 

•*99 

.8126 

.2132 

.7663 

. 2281 

.7227 

30 

I.1857 

1.8611 

I. 1992 

1.8118 

1-2134 

1-7655 

1.2283 

1.722 

3 1 

.1859 

.8603 

.1994 

• 811. 

.2136 

.7648 

.2286 

•7213 

32 

.l86l 

•8595 

• 1 997 

.8102 

.2139 

.764 

.2288 

.7206 

33 

. 1863 

.8586 

.1999 

.8094 

.2141 

•7633 

. 2291 

.7199 

34 

. l866 

.8578 

.2001 

.8086 

.2144 

.7625 

.2293 

.7192 

35 

I. l868 

1.8569 

I.2OO4 

1.8078 

1.2146 

1.7618 

1.2296 

1.7185 

3 & 

. 187 

.8561 

. 2006 

.807 

.2149 

.761 

.2298 

.7178 

37 

. 1872 

■8552 

. 2008 

. 8062 

.2151 

.7603 

.2301 

.7171 

3 » 

.1874 

•8544 

. 201 

.8054 

• 2153 

■7596 

• 2304 

.7164 

39 

.1877 

•8535 

.2013 

.8047 

.2156 

•7588 

.2306 

• 7 X 57 

40 

I. 1879 

1.8527 

1.2015 

1.8039 

1.2158 

1.7581 

1.2309 

1-7x51 

4 i 

• l88l 

.8519 

.2OI7 

.8031 

.2161 

•7573 

.2311 

• 7 X 44 

42 

.1883 

.851 

.202 

.8023 

.2163 

.7566 

•2314 

•7137 

43 

.1886 

.8502 

.2022 

.8015 

.2166 

•7559 

.2316 

•713 

44 

.1888 

•8493 

.2024 

.8007 

.2168 

•7551 

.2319 

•7x23 

45 

I. 189 

1.8485 

1.2027 

x.7999 

I.2171 

x -7544 

I.2322 

1.7116 

46 

. 1892 

.8477 

.2029 

.7992 

• 2173 

•7537 

.2324 

•7x09 

47 

.1894 

.8468 

.2031 

.7984 

• 2x75 

•7529 

.2327 

.7102 

48 

.1897 

.846 

.2034 

.7976 

.2178 

.7522 

•2329 

•7095 

49 

.1899 

.8452 

.2036 

. 7968 

.218 

•7514 

• 2332 

.7088 

50 

I. 1901 

1.8443 

1.2039 

1.796 

1.2183 

I -7507 

x -2335 

1.7081 

5 i 

• I 9°3 

•8435 

.2041 

•7953 

.2185 

•75 

■2337 

•7075 

52 

.1906 

.8427 

.2043 

•7945 

.2188 

•7493 

•234 

. 7068 

53 

. 1908 

.8418 

.2046 

•7937 

. 219 

- 7485 

.2342 

.7061 

54 

.191 

. 841 

. 2048 

.7929 

.2193 

• -7478 

•2345 

•7054 

55 

I. 1912 

1.8402 

1.205 

I. 7921 

1-2195 

1.7471 

1.2348 

1.7047 

56 

• 1915 

•8394 

•2053 

.7914 

.2198 

•7463 

•235 

.704 

57 

.1917 

•8385 

.2055 

.7906 

.22 

•7456 

•2353 

• 7°33 

58 

.1919 

•8377 

.2057 

.7898 

.2203 

•7449 

•2355 

. 7027 

59 

. 1921 

.8369 

. 206 

.7891 

. 2205 

.7442 

•2358 

.702 

60 

I. 1922 

1.8361 

1.2062 

1.7883 

1.2208 

x -7434 

1.2361 

1.7013 

t 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 


57 ° 

56 ° 

550 

54 ° 


I 

o 


w m m h m h m m w m to (0 to tO 10 10 10 10 10 10 W lu U U U Ol W W U U uiuounounuiounui O' 

to OJ -P- On CT\^J 00VO 0 M 10 OJ -f. <Jl 0^0 ccvo O m WW+W O'-! OCvO 0 w 10 OJ 4*. Cn Ov'O 00^0 0 H 10 OJ 4^ Ul CT\'4 OCO 0 H 10 OO -fc- On O'J 00^0 0 































412 NATUKAL SECANTS AND CO-SECANTS. 



36 ° 

37 ° 

38 ° 

390 


/ 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

| Co-sec’t. 

/ 

o 

1.2361 

I -7 OI 3 

1.2521 

1.6616 

1.269 

1.6243 

1.2867 

1.589 

60 

I 

•2363 

.7006 

.2524 

.661 

.2693 

.6237 

.2871 

.5884 

59 

2 

.2366 

.6999 

.2527 

.6603 

.2696 

.6231 

.2874 

•5879 

58 

3 

.2368 

•6993 

•253 

•6597 

.2699 

.6224 

.2877 

•5873 

57 

4 

•2371 

.6986 

.2532 

.659 1 

.2702 

.6218 

.288 

■5867 

56 

5 

1-2374 

1.6979 

1-2535 

1.6584 

1.2705 

1.6212 

1.2883 

1.5862 

55 

6 

.2376 

.6972 

.2538 

.6578 

.2707 

.6206 

.2886 

•5856 

54 

7 

•2379 

.6965 

•254 1 

.6572 

.271 

.62 

.2889 

•585 

53 

8 

.2382 

•6959 

•2543 

•6565 

•2713 

.6194 

.2892 

•5845 

52 

9 

.2384 

.6952 

.2546 

•6559 

.2716 

.6188 

.2895 

•5839 

5i 

IO 

1.2387 

1.6945 

1.2549 

1-6552 

1.2719 

1.6182 

1.2898 

1-5833 

50 

ii 

.2389 

.6938 

*2552 

.6546 

.2722 

.6176 

.2901 

.5828 

49 

12 

.2392 

.6932 

•2554 

•654 

.2725 

.617 

.2904 

.5822 

48 

13 

•2395 

.6925 

•2557 

•6533 

.2728 

.6164 

.2907 

.5816 

47 

14 

•2397 

.6918 

.256 

.6527 

•2731 

.6159 

.291 

.5811 

46 

15 

1.24 

1.6912 

1-2563 

1.6521 

1-2734 

1.6153 

1-2913 

1-5805 

45 

16 

.2403 

.6905 

•2565 

.6514 

•2737 

.6147 

. 2916 

•5799 

44 

i7 

.2405 

.6898 

.2568 

.6508 

•2739 

.6141 

.2919 

■5794 

43 

18 

.2408 

.6891 

•2571 

.6502 

.2742 

•6135 

.2922 

.5788 

42 

19 

.2411 

.6885 

•2574 

.6496 

•2745 

.6129 

.2926 

•5783 

4i 

20 

1*2413 

1.6878 

1-2577 

1.6489 

1.2748 

1.6123 

1.2929 

1-5777 

40 

21 

.2416 

.6871 

• 2 579 

.6483 

•2751 

.6117 

.2932 

• 577 1 

39 

22 

.2419 

.6865 

.2582 

.6477 

•2754 

.6111 

•2035 

•5766 

38 

23 

.2421 

.6858 

•2585 

.647 

•2757 

.6105 

.2938 

•576 

37 

24 

.2424 

•6851 

.2588 

.6464 

.276 

.6099 

.2941 

•5755 

36 

25 

1.2427 

1.6845 

1.2591 

1.6458 

1.2763 

1.6093 

1.2944 

1-5749 

35 

26 

.2429 

.6838 

•2593 

.6452 

.2766 

.6087 

.2947 

•5743 

34 

27 

•2432 

.6831 

.2596 

.6445 

.2769 

.6081 

•295 

-5738 

33 

28 

•2435 

.6825 

•2599 

•6439 

.2772 

.6077 

•2953 

•5732 

32 

29 

•2437 

.6818 

. 2602 

•6433 

•2775 

.607 

.2956 

•5727 

3i 

30 

1.244 

1.6812 

1.2605 

1.6427 

1.2778 

1.6064 

1.296 

I.57 21 

3° 

3 1 

•2443 

.6805 

.2607 

.642 

.2781 

.6058 

.2963 

•57 l6 

29 

32 

•2445 

.6798 

.26l. 

.6414 

.2784 

.6052 

.2966 

•571 

28 

33 

. 2448 

.6792 

.2613 

.6408 

.2787 

.6046 

.2969 

•57<>5 

27 

34 

•2451 

.6785 

.26l6 

.6402 

.279 

.604 

.2972 

•5699 

26 

35 

1-2453 

i- 6 779 

I.26l9 

1.6396 

1-2793 

1.6034 

1-2975 

1.5694 

25 

36 

.2456 

.6772 

. 2622 

.6389 

•2795 

.6029 

.2978 

.5688 

24 

37 

•2459 

.6766 

.2624 

•6383 

. 2798 

.6023 

.2981 

•5683 

23 

33 

.2461 

•6759 

.2627 

•6377 

. 2801 

.6017 

.2985 

•5677 

22 

39 

.2464 

.6752 

.263 

•6371 

.2804 

.6011 

.2988 

.5672 

21 

40 

1.2467 

I.6746 

1.2633 

1-6365 

1.2807 

1.6005 

1.2991 

I. 5666 

20 

4i 

.247 

• 6 739 

.2636 

• 6359 

.281 

.6 

.2994 

■ 5661 

x 9 

42 

>2472 

•6733 

.2639 

• 6352 

.2813 

•5994 

.2997 

•5655 

18 

43 

•2475 

.6726 

.264I 

.6346 

.2816 

-59 88 

•3 

•565 

1 7 

44 

.2478 

.672 

.2644 

•634 

.2819 

.5982 

•3 00 3 

•5644 

16 

45 

1.248 

1.6713 

I.2647 

1-6334 

1.2822 

I-5976 

1.3006 

1-5639 

i5 

46 

.2483 

.67O7 

.265 

.6328 

.2825 

•5971 

•3 01 

•5633 

I 4 

47 

, 2486 

.67 

.2653 

.6322 

.2828 

•5965 

•3013 

.5628 

x 3 

48 

.2488 

.6694 

.2656 

.6316 

.2831 

•5959 

• 3° l6 

.5622 

12 

49 

.249 

.6687 

• 2659 

.6309 

.2834 

•5953 

.3019 

•5617 

11 

50 

1.2494 

I.668l 

I. 266l 

1.6303 

1.2837 

I -5947 

1.3022 

1.5611 

10 

5i 

.2497 

.6674 

.2664 

.6297 

.284 

•5942 

.3025 

.5606 

9 

52 

.2499 

.6668 

.2667 

.6291 

.2843 

•5936 

.3C)29 

•56 

8 

53 

.2502 

.6661 

.267 

.6285 

.2846 

•593 

•3°3 2 

•5595 

7 

54 

•2505 

•6655 

•2673! 

.6279 

.2849 

*59 2 4 

•3035 

•559 

6 

55 

1.2508 

1.6648 

I.2676 

1.6273 

1.2852 

I -59 I 9 

i-3°3 8 

1-5584 

5 

56 

.251 

.6642 

.2679 

.6267 

■2855 

• 59 I 3 

.3041 

•5579 

4 

57 

•2513 

.6636 

.268l 

.6261 

.2858 

•59°7 

•3044 

•5573 

3 

58 

.2516 

.6629 

.2684 

•6255 

.2861 

•59 01 

.3048 

.5568 

2 

59 

.2519 

.6623 

.2687 

.6249 

.2864 

.5896 

•3051 

•5563 

1 

60 

1.2521 

1.6616 

I.269 

1.6243 

1.2867 

1.589 

I -3°54 

1-5557 

0 

e 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

r 


53° 

520 1 

510 

500 









































NATURAL SECANTS AND CO-SECANTS 


413 



40 ° I 

410 

42 ° I 

43 ° 



Secant. 

Co-SKC’T.j 

Secant. 

Co-sbc’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

r 

0 

*- 3°54 

1-5557 I 

I -325 

1.5242 

I -3456 

*■4945 

*•3673 

1.4663 

60 

I 

•3057 

•5552 

•3253 

•5237 

•346 

•494 

•3677 

.4658 

59 

2 

.306 

•5546 

•3257 

•5232 

•3463 

•4935 

.3681 

•4654 

58 

3 

.3064 

•5541 

.326 

•5227 

•3467 

•493 

•3684 

•4649 

57 

4 

.3067 

•5536 

•3263 

.5222 

•347 

•4925 

.3688 

•4644 

56 

5 

*- 3°7 

i -553 

1.3267 

*•5217 

*•3474 

I.492I 

1.3692 

1.464 

55 

6 

• 3°73 

•5525 

•327 

.5212 

•3477 

.4916 

■3695 

•4635 

54 

7 

.3076 

•552 

•3274 

•5207 

•3481 

.4911 

•3699 

.4631 

53 

8 

• 3°8 

• 55 i 4 

•3277 

.5202 

•3485 

. 4906 

•3703 

.4626 

52 

9 

•3083 

•5509 

.328 

• 5*97 

.3488 

.4901 

•3707 

.4622 

5 * 

IO 

1.3086 

I -5503 

1.3284 

1.5192 

1.3492 

1.4897 

*- 37 * 

1.4617 

50 

II 

.3089 

•5498 

.3287 

•5*87 

•3495 

.4892 

• 37*4 

.4613 

49 

12 

.3092 

•5493 

•329 

• 5*82 

•3499 

.4887 

• 37*8 

.4608 

48 

13 

.3096 

•5487 

•3294 

• 5*77 

.3502 

. 4882 

.3722 

.4604 

47 

14 

•3099 

•5482 

•3297 

• 5 * 7 * 

• 35 o 6 

•4877 

•3725 

•4599 

46 

15 

I. 3102 

1-5477 

*- 33 °* 

1.5166 

*•3509 

*•4873 

*•3729 

*•4595 

45 

16 

• 3 i °5 

• 547 1 

•3304 

• 5 * 6 i 

• 35*3 

. 4868 

•3733 

•459 

44 

17 

.3109 

.5466 

•3307 

•5*56 

• 35*7 

•4863 

•3737 

.4586 

43 

18 

.3112 

.5461 

• 33 ** 

• 5 * 5 * 

•352 

.4858 

•374 

.4581 

42 

*9 

• 3 ii 5 

•5456 

• 33*4 

•5*46 

•3524 

•4854 

•3744 

•4577 

4 * 

20 

i- 3**8 

i -545 

*• 33*8 

*■ 5 * 4 * 

*•3527 

1.4849 

*•3748 

1-4572 

40 

21 

.3121 

•5445 

• 332 i 

•5*36 

• 353 * 

.4844 

•3752 

.4568 

39 

22 

•3125 

•544 

•3324 

• 5 * 3 * 

•3534 

•4839 

•3756 

•4563 

38 

23 

• 3 I2 8 

•5434 

•3328 

• 5*26 

■3538 

•4835 

•3759 

•4559 

37 

24 

• 3 * 3 * 

•5429 

• 333 * 

• 5*21 

•3542 

•483 

•3763 

•4554 

36 

25 

i- 3 i 34 

1.5424 

*•3335 

1.5116 

*•3545 

1.4825 

*•3767 

1-455 

35 

26 

• 3 I 3 8 

• 54 i 9 

•3338 

• 5**1 

•3549 

.4821 

• 377 * 

•4545 

34 

27 

• 3 * 4 * 

• 54 i 3 

•3342 

.5106 

•3552 

.4816 

•3774 

• 454 * 

33 

28 

•3144 

.5408 

•3345 

.5101 

•3556 

.4811 

•3778 

•4536 

32 

29 

.3148 

•5403 

•3348 

.5096 

•356 

.4806 

.3782 

4532 

3 * 

30 

i- 3 i 5 i 

i -5398 

1-3352 

1.5092 

*■3563 

1.4802 

1.3786 

* 4527 

30 

31 

• 3 I 54 

•5392 

•3355 

.5087 

•3567 

•4797 

•379 

•4523 

29 

32 

•3157 

•5387 

•3359 

.5082 

• 357 * 

•4792 

•3794 

• 45*8 

28 

33 

.3161 

•5382 

•3362 

•5077 

•3574 

.4788 

•3797 

• 45*4 

27 

34 

.3164 

•5377 

•3366 

.5072 

•3578 

•4783 

.3801 

• 45 * 

26 

35 

1.3167 

1 - 537 * 

*•3369 

1.5067 

i- 358 i 

*■4778 

*■3805 

1-4505 

25 

3 6 

• 3 i 7 

•5366 

•3372 

.5062 

•3585 

•4774 

.3809 

.4501 

24 

37 

• 3 U 4 

536 i 

•3376 

•5057 

•3589 

•4769 

•3813 

•4496 

23 

38 

•3177 

•5356 

•3379 

•5052 

•3592 

•4764 

.3816 

•4492 

22 

39 

• 3 l8 

• 535 * 

•3383 

.5047 

•3596 

•476 

.382 

.4487 

21 

40 

1.3184 

*•5345 

1.3386 

1.5042 

1.36 

*•4755 

1.3824 

1.4483 

20 

4 1 

•3187 

•534 

•339 

■5037 

•3603 

•475 

.3828 

•4479 

*9 

42 

• 3 I 9 

•5335 

•3393 

.5032 

.3607 

.4746 

•3832 

•4474 

18 

43 

• 3*93 

•533 

•3397 

.5027 

.3611 

.4741 

•3836 

•447 

*7 

44 

• 3 I 97 

•5325 

•34 

.5022 

.3614 

•4736 

•3839 

•4465 

l6 

45 

1.32 

1 - 53*9 

1.3404 

1.5018 

1.3618 

1-4732 

*■3843 

1.4461 

*5 

46 

• 3203 

• 53*4 

•3407 

•5013 

. 3622 

.4727 

•3847 

•4457 

*4 

47 

.3207 

•5309 

• 34 ** 

.5008 

•3625 

•4723 

•3851 

•4452 

*3 

48 

.321 

•5304 

• 34*4 

•5003 

.3629 

• 47*8 

•3855 

•4448 

12 

49 

•3213 

■5299 

■ 34*8 

•4998 

•3633 

• 47*3 

•3859 

•4443 

II 

50 

1-3217 

I -5294 

1.3421 

*•4993 

1.3636 

1.4709 

1-3863 

*•4439 

IO 

51 

• 322 

.5289 

•3425 

.4988 

•364 

.4704 

.3867 

•4435 

9 

S 2 

.3223 

•5283 

•3428 

•4983 

•3644 

•4699 

■387 

•443 

8 

53 

• 3227 

•5278 

•3432 

•4979 

•3647 

•4695 

•3874 

.4425 

7 

54 

•323 

•5273 

•3435 

•4974 

•3651 

•469 

•3878 

.4422 

6 


I -3233 

1.5268 

*■3439 

1.4969 

*•3655 

1.4686 

1.3882 

*■ 44*7 

5 

56 

•3237 

•5263 

•3442 

• 49 6 4 

•3658 

.4681 

.3886 

• 44*3 

4 

57 

•324 

•5258 

•3446 

•4959 

. 3662 

.4676 

•389 

. 4408 

3 

58 

•3243 

•5253 

•3449 

•4954 

.3666 

.4672 

•3894 

.4404 

2 

59 

•3247 

•5248 

•3453 

•4949 

.3669 

•3667 

.3898 

•44 

1 

60 

1-325 

1.5242 

I -3456 

*■4945 

*•3673 

1.4663 

1.3902 

*■4395 

0 


Co-8 EC’T. 

Secant. 

1 Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

r 


490 

I 48° 

470 

460 

\ - 


M M* 



























































4*4 


NATURAL SECANTS AND CO-SECANTS. 



440 



440 



440 


/ 

Secant. 

Co-sec’t. 

t 

i 

Secant. 

Co-sec’t. 

t 

t 

Secant. 

Co-sec’t. 

/ 

0 

1.3902 

1-4395 

60 

21 

i- 39 8 4 

i- 43°5 

39 

4 i 

i. 4 o6 5 

I. 4221 

19 

I 

• 39°5 

• 439 1 

59 

22 

• 39 88 

.4301 

3 8 

42 

.4069 

.4217 

l8 

2 

■ 39°9 

•4387 

5 8 

23 

•3992 

.4297 

37 

43 

• 4°73 

.4212 

17 

3 

• 39 J 3 

.4382 

57 

24 

• 399 6 

.4292 

39 

44 

.4077 

.4208 

l6 

4 

• 39*7 

• 437 8 

59 

25 

1.4 

1.4288 

35 

45 

1.4081 

1.4204 

15 

5 

1.3921 

1-4374 

55 

26 

.4OO4 

.4284 

34 

46 

.4085 

.42 

14 

6 

• 39 2 5 

•437 

54 

27 

.4008 

.428 

33 

47 

.4089 

.4196 

13 

7 

39 2 9 

■4365 

53 

28 

.4012 

.4276 

32 

48 

• 4°93 

.4192 

12 

8 

•3933 

.4361 

5 2 

29 

.4016 

.4271 

3 i 

49 

• 4°97 

.4188 

II 

9 

•3937 

•4357 

5 i 

3 ° 

1.402 

1.4267 

3 ° 

50 

1.410X 

i- 4 i8 3 

IO 

IO 

i- 394 i 

i- 435 2 

5 ° 

3 i 

.4024 

.4263 

29 

5 i 

.4105 

.4179 

9 

II 

•3945 

• 434 8 

49 

3 2 

.4028 

•4259 

28 

52 

.4109 

•4175 

8 

12 

•3949 

•4344 

48 

33 

.4032 

• 4 2 54 

2 7 

53 

•4113 

.4171 

7 

13 

•3953 

•4339 

47 

34 

.4036 

• 4 2 5 

26 

54 

.4117 

.4167 

6 

14 

•3957 

•4335 

46 

35 

1.404 

1.4246 

25 

55 

I. 4122 

1.4163 

5 

15 

1.396 

i- 433 i 

45 

3 6 

.4044 

.4242 

24 

59 

.4126 

•4159 

4 

16 

•3964 

• 43 2 7 

44 

37 

.4048 

.4238 

23 

57 

• 4 i 3 

•4154 

3 

17 

.3968 

• 43 22 

43 

3 8 

.4052 

•4333 

22 

58 

• 4 i 34 

• 4 i 5 

2 

18 

• 397 2 

.4318 

4 2 

39 

.4056 

.4229 

21 

59 

.4138 

.4146 

1 

19 

•3976 

• 43 H 

4 i 

40 

1.406 

1.4225 

20 

60 

I.4142 

1.4142 

0 

20 

i- 39 8 

I - 43 I 

40 









t 

Co-sec’t. 

Secant. 

t 

f 

Co-sec’t. 

Secant. 

! 

I / 

Co-sec’t. 

Secant. 

r 


45 ° 



450 


1 

45 ° 



Preceding Table contains Natural Secants and Co-secants for every 
minute of the Quadrant to Radius i. 

If Degrees are taken at head of column, Minutes, Secant, and Co-secant 
must be taken from head also; and if they are taken at foot of column, 
Minutes, etc., must be taken from foot also. 

Illustration. —1.05 is secant of 17 0 45' and co secant of 72 0 15'. 

To Compute Secant or Co-secant of any Angle. 
Rule. —Divide 1 by Cosine of angle for Secant, and by Sine for Co-secant. 
Example i.—W hat is secant of 25 0 25'? 

Cosine of angle = .903 21. Then 1 — .903 21 = 1.1072, Secant. 

2.—What is co secant of 64° 35'? 

Sine of angle = .903 21. Then 1-7-.903 21 = 1.1072, Co-secant. 

To Compute Degrees, Minutes, and Seconds of a Secant 

or Co-secant. 

When Secant is given, 

Proceed as by Rule, page 402, for Sines, substituting Secants for Sines. 
Example. —What is secant for 1.1607? 

The next less secant is 1.1606, arc for which == 30 0 30'. 

Next greater secant is 1.1608, difference between which and next less is 1.1608 — 
1.1606 = .0002. 

■ Difference between less tab. secant and one given is 1.1607 — 1.1606 = .0001. 

Then .0002 : .0001 :: 60 : 30, which, added to 30 0 3o' = 3o° 30' 30". 

When Co-secant is given , 

Proceed as by Rule, page 402, substituting Co-secants for Cosines. 




































NATURAL TANGENTS AND CO-TANGENTS. 415 

IN'atnral Tangents and. Co-tangents. 



0 

D 

1 

0 

2 

0 

3 

0 


/ 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

/ 

0 

.000 00 

Infinite. 

.017 46 

57-29 

.03492 

28.6363 

.05241 

I9.08H 

60 

I 

.000 29 

3437-75 

•017 75 

6.3506 

•03521 

8.3994 

.0527 

8-9755 

59 

2 

.000 58 

I7l8. 87 

.018 04 

5-4415 

•0355 

8.1664 

•05299 

8.87II 

58 

3 

.000 87 

I 45 - 9 2 

•01833 

4-5613 

•03579 

7-9372 

•053 28 

8.7678 

57 

4 

.001 16 

859 - 43 6 

.018 62 

3.7086 

.036 09 

7 - 7 TI 7 

•05357 

8.6656 

56 

5 

.001 45 

687.549 

.018 91 

52.8821 

036 38 

27.4899 

•05387 

18.5645 

55 

6 

.001 75 

572-957 

.019 2 

2.0807 

•036 67 

7-2715 

•05416 

8.4645 

54 

7 

. 002 04 

491. 106 

.01949 

1.3032 

.036 96 

7.0566 

•054 45 

8-3655 

53 

8 

.002 33 

29. 7l8 

.019 78 

0.5485 

03725 

6.845 

•054 74 

8.2677 

52 

9 

.002 62 

381.971 

.020 07 

49.8157 

•037 54 

6.6367 

•05503 

8.1708 

5 i 

10 

.002 91 

343-774 

.020 36 

49. IO39 

•03783 

26.4316 

•05533 

18.075 

5 ° 

11 

.003 2 

12.521 

.020 66 

8.4121 

.038 12 

6.2296 

•05562 

7.9802 

49 

12 

.003 49 

286.478 

.02095 

7-7395 

.038 42 

6.0307 

•0559 1 

7.8863 

48 

13 

.003 78 

64.441 

.021 24 

7-0853 

•03871 

5.8348 

.0562 

7-7934 

47 

14 

. 004 07 

45-552 

.021 53 

6.4489 

•039 

5.6418 

.05649 

7 - 7 oi 5 

46 

15 

.004 36 

229.182 

.021 82 

45.8294 

.039 29 

25 - 45 I 7 , 

•05678 

17.6106 

45 

16 

.00465 

14.858 

.02211 

5.226l 

•039 58 

5.2644 

•05708 

7-5205 

44 

17 

.00495 

02.219 

.022 4 

4.6386 

•039 87 

5.0798 

•05737 

7 - 43 I 4 

43 

18 

.005 24 

190.984 

.022 69 

4.066l 

.040 16 

4.8978 

.057 66 

7-3432 

42 

19 

•005 53 

80.932 

.022 98 

3 - 5 o 8 i 

. 040 46 

4 - 7 i8 5 

•057 95 

7-2558 

41 

20 

.005 82 

171.885 

.023 28 

42.964I 

.040 75 

24.54x8 

•05824 

i 7- i6 93 

40 

21 

.006 11 

63-7 

•023 57 

2-4335 

.041 04 

4-3675 

•05854 

7.0837 

39 

22 

.006 4 

56.259 

.023 86 

1.9158 

•041 33 

4-1957 

.058 83 

6.999 

38 

23 

006 69 

49.465 

.024 15 

I.4IO6 

,041 62 

4.0263 

.059 12 

6.915 

37 

24 

.006 98 

43-237 

.02444 

°- 9 I 74 

041 91 

3-8593 

.05941 

6.8319 

36 

25 

.007 27 

137.507 

.02473 

40.4358 

042 2 

23.6945 

•0597 

16.7496 

35 

26 

.007 56 

32.219 

.025 02 

39-9655 

0425 

3-5321 

•059 99 

6.6681 

34 

27 

.007 85 

27.321 

• 025 31 

9 - 5059 

•042 79 

3 - 37 l8 

! .060 29 

6.5874 

33 

28 

.008 14 

22.774 

1 -025 6 

9.0568 

.04308 

3.2137 

.060 58 

6-5075 

32 

29 

. 008 44 

18.54 

• 025 89 

8.6177 

•043 37 

3-0577 

.060 87 

6.4283 

3 i 

30 

.00873 

114.589 

.026 I9 

38.1885 

.043 66 

22.9038 

.06l l6 

16.3499 

30 

31 

. 009 02 

10.892 

.02648 

7.7686 

•043 95 

2.7519 

.06145 

6.2722 

29 

32 

.00931 

07.426 

.026 77 

7-3579 

.044 24 

2.602 

.061 75 

6.1952 

28 

33 

.009 6 

04.171 

.027 06 

6.956 

.044 54 

2-4541 

.062 04 

6.119 

27 

34 

. 009 89 

01.107 

•027 35 

6.5627 

•044 83 

2.3081 

.06233 

6.0435 

26 

35 

.010 18 

98.2179 

.027 64 

36.1776 

.045 12 

22.164 

.062 62 

15.9687 

25 

36 

.010 47 

5-4895 

.027 93 

5.8006 

•04541 

2.0217 

.062 9I 

5-8945 

24 

37 

.01076 

2.9085 

.028 22 

5-4313 

0457 

1.8813 

.063 21 

5.8211 

23 

38 

.011 05 

0.4633 

.028 51 

5-0695 

.04599 

1.7426 

•0635 

5-7483 

22 

39 

•on 35 

88.1436 

.028 8l 

4 - 7 I 5 I 

.046 28 

1.6056 

•063 79 

5.6762 

21 

40 

.011 64 

85-9398 

.O29 I 

34.3678 

.046 58 

21.4704 

.06408 

15.6048 

20 

4 i 

.011 93 

3-8435 

.02939 

4.0273 

.046 87 

1-3369 

.064 37 

5-534 

*9 

42 

.012 22 

1.847 

.029 68 

3 -6935 

.047 16 

1.2049 

.064 67 

5.4638 

18 

43 

.012 51 

79-9434 

.029 97 

3.3662 

•04745 

I -°747 

.064 96 

5-3943 

J 7 

44 

.012 8 

8.1263 

.030 26 

3-0452 

•047 74 

0.946 

•065 25 

5-3254 

16 

45 

.01309 

76.39 

•03055 

32.7303 

.04803 

20.8188 

•065 54 

15-2571 

15 

46 

• 01338 

4.7292 

.03084 

2.42I3 

.048 32 

0.6932 

.065 84 

5-1893 

1 4 

47 

.01367 

3- 1 39 

.03114 

2 . Il8l 

.048 62 

0.5691 

.0 66 13 

5. 1222 

13 

48 

.0x396 

1.6151 

•03143 

I.8205 

.048 91 

0.4465 

.066 42 

5-0557 

12 

49 

• 014 25 

0-1533 

.031 72 

1.5284 

• 049 2 

0.3253 

.066 71 

4.9898 

11 

50 

■01455 

68.75OI 

.032 01 

31.2416 

-049 49 

20.2056 

.067 

14.9244 

10 

5 i 

.014 84 

7.4019 

•0323 

0.9599 

.04978 

0.0872 

.0673 

4.8596 

9 

52 

.01513 

6.1055 

.032 59 

0.6833 

.05007 

19.9702 

•067 59 

4-7954 

8 

53 

.015 42 

4.858 

.032 88 

O.4I16 

•05037 

9.8546 

.067 88 

4-7317 

7 

54 

.01571 

3-6567 

•°33 17 

O. 1446 

.050 66 

9-7403 

.068 17 

4.6685 

6 

55 

.016 

62.4992 

•03346 

29. 8823 

•05095 

19.6273 

.068 47 

14.6059 

5 

56 

.016 29 

1.3829 

•03376 

9.6245 

.051 24 

9.5156 

.068 76 

4-5438 

4 

57 

.016 58 

0.3058 

•03405 

9 - 37 11 

•051 53 

9.4051 

.069 05 

4.4823 

3 

58 

.016 87 

59-2659 

•034 34 

9.122 

.051 82 

9.2959 

•069 34 

4.4212 

2 

59 

.017 16 

8.26i2 

•034 63 

8.8771 

.052 12 

9.1879 

.069 63 

4.3607 

1 

60 

.017 46 

57-29 

.034 92 

28.6363 

.05241 

19.0811 

•06993 

14.3007 

0 

/ 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

/ 


89 ° 

88° 

87 ° 

86° 





























































41 6 NATURAL TANGENTS AND CO-TANGENTS. 



40 

50 

(50 

70 


r 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

f 

O 

.069 93 

I4.3OO7 

.08749 

II.43O I 

• 105 1 

9 - 5 I 4 36 

. 122 78 

8-144 35 

60 

I 

.070 22 

4. 2411 

.087 78 

I - 39 I 9 

• 105 4 

.487 8l 

. 123 08 

.124 8l 

59 

2 

.07051 

4. l82I 

.088 07 

1-354 

• 105 69 

.461 41 

• 12338 

•105 36 

58 

3 

.070 8 

4- 12 35 

.088 37 

i- 3 i6 3 

.105 99 

• 435 i 5 

.123 67 

.086 

57 

4 

.O7I I 

4' o6 55 

.088 66 

1.278 9 

. 106 28 

. 4O9 04 

.12397 

.066 74 

56 

5 

•071 39 

I4.OO79 

.08895 

11.241 7 

.10657 

9-383 07 

. 124 26 

8.O47 56 

55 

6 

.071 68 

3-9507 

.089 25 

1.204 8 

. 106 87 

•357 24 

.124 56 

.028 48 

54 

7 

.07197 

3-894 

.08954 

1.168 i 

.10716 

• 33 1 54 

.124 85 

8.OO9 48 

53 

8 

.O72 27 

3-8378 

.08983 

1.13 1 6 

. 107 46 

•305 99 

•125 15 

7.99058 

52 

9 

• 072 56 

3.7821 

•090 13 

1.0954 

•10775 

. 280 58 

.125 44 

.971 76 

5 i 

IO 

.072 85 

13.7267 

.O9O 42 

11.0594 

. 108 05 

9-255 3 

•125 74 

7-953 02 

50 

II 

■07314 

3-6719 

.O9O 71 

1.0237 

.108 34 

.230 16 

. 126 03 

■934 38 

49 

12 

•073 44 

3 - 6 i 74 

.O9I OI 

0.988 2 

.108 63 

.205 16 

.12633 

.91582 

48 

13 

•073 73 

3-5634 

•09 1 3 

0.9529 

.10893 

. 180 28 

. 126 62 

•897 34 

47 

m 

.074 02 

3-5098 

.091 59 

0.917 8 

. IO9 22 

•i 55 54 

.126 92 

.87895 

46 

15 

•07431 

13.4566 

.091 89 

10.882 9 

.10952 

9 - x 3 ° 93 

. I27 22 

7.86064 

45 

l6 

.074 61 

3-4039 

.09218 

0.8483 

. 109 81 

. 106 46 

.127 51 

.842 42 

44 

17 

.0749 

3 - 35 I 5 

.09247 

0.813 9 

.HO II 

.082 11 

. 127 81 

. 824 28 

43 

l8 

•075*9 

3.2996 

.092 77 

0.7797 

.IIO 4 

•05789 

. 1281 

.806 22 

42 

l 9 

.07548 

3.248 

.093 06 

o- 745 7 

• IIO 7 

•033 79 

. 128 4 

.788 25 

4 i 

20 

.07578 

13.1969 

•09335 

10.711 9 

.IIO99 

9.009 83 

.128 69 

7-770 35 

40 

21 

.076 07 

3.1461 

.09365 

0.6783 

.hi 28 

8.98598 

.128 99 

•752 54 

39 

22 

.076 36 

3-0958 

.09394 

0.645 

.in 58 

.962 27 

. 129 29 

•734 8 

38 

23 

.07665 

3-0458 

-09423 

0.611 8 

.hi 87 

.93867 

.12958 

• 7 I 7 *5 

37 

24 

.07695 

2.9962 

•09453 

0.5789 

.11217 

.9152 

. 129 88 

•699 57 

36 

25 

.07724 

12.9469 

.094 82 

10.546 2 

. 112 46 

8.891 85 

■ 130 1 7 

7.68208 

35 

26 

•07753 

2.8981 

•095 n 

0-5136 

. 112 76 

.86862 

•13047 

.66466 

34 

27 

.077 82 

2.8496 

.09541 

0.481 3 

•11305 

■845 5i 

.13076 

•64732 

33 

28 

.078 12 

2.8014 

•0957 

0.449 1 

•11335 

.822 52 

. 131 06 

•63005 

32 

29 

.078 41 

2.7536 

.096 

O. 417 2 

.11364 

•799 64 

•13136 

.612 87 

3 i 

30 

.0787 

12.7062 

.096 29 

10.3854 

■ XI 3 94 

8.776 89 

.131 65 

7-595 75 

30 

31 

.078 99 

2.6591 

.096 58 

0-3538 

.11423 

•754 25 

•i 3 i 95 

.57872 

29 

32 

.O79 29 

2.6124 

.096 88 

0.3224 

.11452 

• 73 i 72 

. 132 24 

.561 76 

28 

33 

.07958 

2.566 

•097 17 

0.2913 

. 114 82 

.70931 

•13254 

•544 87 

27 

34 

.079 87 

2.5199 

.097 46 

0.260 2 

.11511 

.687 01 

.13284 

.52806 

26 

35 

.080 17 

12.4742 

.097 76 

IO.229 4 

.11541 

8.664 82 

•i 33 1 3 

7.511 32 

25 

36 

.080 46 

2.4288 

.09805 

0.198 8 

•115 7 

.642 75 

•1 33 43 

.49465 

24 

37 

• 08075 

2-3838 

•09834 

0.168 3 

.Il6 

.620 78 

•133 72 

.478 06 

23 

38 

.08 r 04 

2-339 

.098 64 

0.138 1 

. 116 29 

•598 93 

.13402 

.461 54 

22 

39 

.081 34 

2.2946 

.098 93 

0.108 

.11659 

•577 i8 

•134 32 

•445 09 

21 

40 

.081 63 

12.2505 

•09923 

10.078 

.11688 

8-555 55 

.13461 

7.428 71 

20 

41 

.081 92 

2.2067 

.09952 

0.048 3 

.11718 

•534 02 

• J 34 9 1 

.4124 

19 

42 

.082 21 

2.1632 

.099 81 

0.018 7 

■ II 7 47 

•51259 

•13521 

•396 16 

18 

43 

.082 51 

2. 1201 

. IOO II 

9.9893 

.11777 

.49128 

•1355 

■379 99 

17 

44 

.082 8 

2.O772 

. 100 4 

.960 07 

. 118 06 

•47007 

•1358 

•36389 

l6 

45 

.08309 

12.0346 

. 100 69 

9.931 01 

. 118 36 

8.448 96 

.13609 

7.347 86 

15 

46 

.083 39 

1.9923 

. 100 99 

. 902 I I 

.11865 

•427 95 

•136 39 

• 33 i 9 

h 

47 

083 68 

1.9504 

. 101 28 

•873 3 8 

.11895 

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• 3 l6 

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48 

.08397 

r.9087 

.10158 

.844 82 

. 119 24 

•38625 

.13698 

. 300 18 

12 

49 

.084 27 

1 8673 

.10187 

. 816 41 

.119 54 

•365 55 

.13728 

.284 42 

11 

50 

.084 56 

n.8262 

. 102 16 

9.788 I? 

.11983 

8.344 96 

•137 58 

7.26873 

IO 

5 i 

.084 85 

1-7853 

. 102 46 

.760 09 

. 120 13 

• 32446 

•13787 

•253 1 

9 

52 

.085 14 

1.7448 

.IO275 

• 732 17 

. 120 42 

. 304 06 

.13817 

•237 54 

8 

53 

.08544 

1-7045 

• 103 05 

.70441 

.120 72 

.283 76 

.13846 

.222 04 

7 

54 

•085 73 

1.6645 

■IO334 

.676 8 

.121 OI 

•263 55 

.13876 

.206 61 

6 

55 

.086 02 

n.6248 

.103 63 

9-649 35 

.12131 

8-24345 

.13906 

7 - I 9 I 25 

5 

56 

.086 32 

i- 5853 

• 103 93 

.622 05 

.121 6 

.223 44 

•139 35 

•175 94 

4 

57 

.086 61 

1.5461 

. IO4 22 

•594 9 

. 121 9 

• 203 52 

•13965 

. 160 71 

3 

58 

.086 9 

1-5072 

.10452 

•567 9 1 

.122 19 

•1837 

•13995 

•145 53 

2 

59 

.087 2 

1.4685 

.10481 

.541 06 

. 122 49 

.16398 

. I4O 24 

. 130 42 

I 

60 

.08749 

n- 43 °i 

.1051 

9 - 5 J 4 36 

. 122 78 

8 -144 35 

.140 54 

7 -H 537 

O 

/ 

Co-TANG. 

Tang. 

Co-TANG. 

Tang. 

Co-TANG. 

Tang. 

Co-TANG. 

Tang, i 

/ 


85° 

84o 

830 

82° i 







































NATURAL TANGENTS AND CO-TANGENTS. 



8° 

90 

/ 

Tang. | 

CO-TANG. 

Tang. | 

CO-TANG. 

o 

.140 54 

7 -115 37 j 

.15838 

6-31375 ; 

I 

. 140 84 

.10038 ! 

.15868 

.30189 j 

2 

.14113 

.08546 

.15898 

.29007 1 

3 

•141 43 

.07059 

.15928 

. 278 29 | 

4 

•Hi 73 

•05579 

•159 58 

.266551 

5 

. I42 02 

7.04105 

.15988 

6.254 86 

6 

.142 32 

•02637 

.160 17 

• 24321 

7 

.142 62 

.011 74 

. 160 47 

.231 6 

8 

.142 91 

6.997 18 

. 160 77 

.220 03 

9 

.14321 

.982 68 

.161 07 

.208 51 

IO 

143 51 

6.968 23 

•i6i 37 

6.19703 

II 

.14381 

•953 85 

. 161 67 

•18559 

12 

.1441 

•939 52 

.161 96 

•i 74 19 

13 

.1444 

•92525 

.162 26 

. 162 83 

14 

1447 

.91 I O4 

.162 56 

•151 5 i 

15 

.144 99 

6.89688 

.162 86 

6.14023 

l6 

• 145 29 

.882 78 

.163 16 

. 128 99 

17 

• 14559 

.868 74 

.163 46 

.117 79 

18 

.145 88 

•854 75 

.163 76 

. 106 64 

19 

.146 18 

. 840 82 

.16405 

•095 52 

20 

.146 48 

6.826 94 

•16435 

6.084 44 

21 

. 146 78 

.81312 

•164 65 

•0734 

22 

• i 47 07 

•799 36 

.16495 

.062 4 

23 

• J 47 37 

.78564 

.16525 

•051 43 

24 

.14767 

•77199 

•16555 

.04051 

25 

.147 96 

6.75838 

.16585 

6.029 62 

26 

. 148 26 

•744 83 

.166 15 

.018 78 

27 

.148 56 

• 73 i 33 

.16645 

.007 97 

28 

. 148 86 

.71789 

. 166 74 

5-997 2 

29 

•149 i 5 

•704 5 

. 167 04 

.986 46 

3 ° 

• I 49 45 

6.691 16 

•16734 

5-975 76 

3 i 

• J 49 75 

.677 87 

. 167 64 

.965 1 

32 

.15005 

.66463 

.167 94 

•954 48 

33 

•15034 

.651 44 

.168 24 

•943 9 

34 

.15064 

•63831 

.16854 

•933 35 

35 

•15094 

6.625 23 

. 168 84 

5.92283 

36 

.151 24 

.612 19 

. 169 14 

• 9 12 35 

37 

•i 5 iS 3 

•599 21 

.169 44 

.9OI 91 

3 * 

.15183 

.58627 

.169 74 

.891 51 

39 

.15213 

•573 39 

. 170 04 

.881 14 

40 

•152 43 

6.56055 

•17033 

5.870 8 

4 i 

.152 72 

•547 77 

.17063 

. 860 51 

42 

•15302 

•535 03 

.17093 

.85024 

43 

•153 32 

•522 34 

. 17123 

.84001 

44 

• 15362 

•509 7 

•171 53 

.829 82 

45 

•i 53 9 1 

6.497 1 

•17183 

5.819 66 

46 

.15421 

.48456 

.172 13 

•80953 

47 

•i 545 i 

. 472 06 

•17243 

■ 799 44 

48 

.15481 

.45961 

•17273 

.789 38 

49 

•i 55 11 

.4472 

•17303 

•779 36 

50 

•1554 

6.434 84 

•173 33 

5-769 37 

5 i 

■1557 

•42253 

•17363 

•759 41 

52 

.156 

.410 26 

•173 93 

•749 49 

53 

•1563 

.39804 

•174 23 

•739 6 

54 

.1566 

•385 87 

•174 53 

•729 74 

55 

.156 89 

6-373 74 

•174 83 

5.71992 

5 <> 

•i 57 19 

•361 65 

•I 75 I 3 

.71013 

57 

•157 49 

.34961 

•17543 

.70037 

58 

•i 57 79 

• 3376 i 

•175 73 

.69064 

59 

.15809 

.325 66 

.17603 

.68094 

60 

•158 38 

6-31375 

•U 6 33 

5.671 28 

/ 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 


[ 81 ° 

80 ° 


IO® 

11° 


Tang. 

CO-TANG. 

Tang. 

CO-TANG. 


.17633 

5.671 28 

.19438 

5-144 55 

60 

•17663 

• 66l 65 

. 194 68 

•13658 

59 

•17693 

.65205 

.19498 

. 128 62 

58 

•177 23 

.642 48 

•19529 

. 120 69 

57 

•177 53 

•632 95 

•195 59 

.112 79 

56 

•177 83 

5-623 44 

•i 95 89 

5.1049 

55 

•17813 

.61397 

.19619 

.09704 

54 

•17843 

.604 52 

.196 49 

.089 21 

53 

•17873 

• 595 11 

. 196 8 

•081 39 

52 

•i 79°3 

•585 73 

•i 97 1 

.0736 

5 i 

•179 33 

5-57638 

•1974 

5.065 84 

50 

•17963 

• 567 °6 

•197 7 

.05809 

49 

•17993 

•557 77 

. 198 01 

•05037 

48 

. 180 23 

•54851 

.19831 

.042 67 

47 

.18053 

•539 27 

. 198 61 

.03499 

46 

. 180 83 

5-53007 

. 198 91 

5.02734 

45 

.181 13 

.5209 

. I99 21 

.OI9 7I 

44 

.18143 

• 5 ii76 

.19952 

.012 I 

43 

•181 73 

. 502 64 

. 199 82 

•004 51 

42 

.18203 

•493 56 

.200 12 

4.99695 

4 i 

•18233 

5-484 5 I 

. 200 42 

4.9894 

40 

. 182 63 

•475 48 

• 20073 

.981 88 

39 

•182 93 

.466 48 

.201 03 

■974 38 

38 

•18323 

• 4575 i 

•20133 

.966 9 

37 

•18353 

■448 57 

. 20164 

•959 45 

36 

•183 83 

5-439 66 

.201 94 

4.95201 

35 

.18414 

•430 77 

. 202 24 

.9446 

34 

.18444 

.421 92 

• 202 54 

•937 2 i 

33 

•18474 

.41309 

.202 85 

.929 84 

32 

• 18504 

. 4O4 29 

•20315 

.92249 

3 i 

•i 85 34 

5-395 52 

•20345 

4.915 16 

30 

• 18564 

•38677 

• 203 76 

.90785 

29 

•185 94 

•37805 

.204 06 

. 900 56 

28 

.186 24 

•369 36 

• 204 36 

•893 3 

27 

.18654 

.3607 

.204 66 

. 886 05 

26 

. 186 84 

5 - 352 o 6 

.20497 

4.878 82 

25 

.18714 

•343 45 

• 205 27 

.871 62 

24 

•18745 

•334 87 

•205 57 

.864 44 

23 

•187 75 

.32631 

.205 88 

.857 27 

22 

.18805 

•31778 

.206 18 

■850 13 

21 

.18835 

. 5-30928 

.206 48 

4-843 

20 

.18865 

.3008 

. 206 79 

•8359 

19 

• 18895 

•29235 

. 207 09 

. 828 82 

18 

.18925 

•28393 

•207 39 

•82175 

17 

■18955 

•275 53 

.207 7 

.81471 

16 

. 189 86 

5-267 15 

.208 

4.807 69 

15 

. 190 16 

.258 8 

.208 3 

. 800 68 

14 

. 19046 

.25048 

.208 61 

•793 7 

13 

. 190 76 

.242 18 

.208 91 

.786 73 

12 

. 191 06 

•23391 

.209 21 

.77978 

II 

.191 36 

5.225 66 

• 20952 

4.772 86 

IO 

. 191 66 

•21744 

.209 82 

• 7 6 5 95 

9 

.19197 

.209 25 

.210 13 

.75906 

8 

. I92 27 

.201 07 

• 21043 

•752 19 

7 

•19257 

.19293 

• 21073 

•745 34 

6 

.192 87 

5.1848 

. 211 04 

4-73851 

5 

•i 93 17 

.176 71 

•211 34 

• 73 i 7 

4 

•193 47 

.16863 

.21164 

.7249 

j 

•19378 

.16058 

.21195 

. 718 13 

2 

. 194 08 

• 15256 

212 25 

•711 37 

1 

■194 38 

5-144 55 

.212 56 

4.704 63 

0 

CO-TANG. 

Tang, i 

CO-TANG. 

Tang. 

/ 

79° 

1 780 










































4 i 8 


NATUKAL TANGENTS AND CO-TANGENTS. 



12° 

13 ° 

14 ? 

1 

/ 

Tang. 

CO-TANG. 

Tang, 

CO-TANG. 

Tang. 

I CO-TANG. 

Tang. | 

0 

.212 56 

4.704 63 

.230 87 

4 - 33 i 48 

•249 33 

4.OIO 78 

.26795 

I 

.212 86 

•697 9 1 

.23117 

•325 73 

.249 64 

.005 82 

.268 26 

2 

.213 16 

.691 21 

.23148 

.32001 

■24995 

.000 86 

.268 57 

3 

•21347 

•684 52 

•231 79 

• 3 H 3 

.25026 

3-995 9 2 

.268 88 

4 

•21377 

.677 86 

.232 09 

.3086 

.25056 

. 99 0 99 

.269 2 

5 

.214 08 

4.671 21 

.2324 

4.302 9 1 

.25087 

3.98607 

.269 51 

6 

.214 38 

.664 58 

.23271 

•297 24 

.25118 

.981 17 

.269 82 

7 

.21469 

•65797 

.23301 

.29159 

• 25 M 9 

.976 27 

.27013 

8 

.21499 

■65138 

• 233 32 

.285 95 

• 251 8 

• 97 1 39 

.27044 

9 

.215 29 

.644 8 

•23363 

.280 32 

. 252 II 

.96651 

.270 76 

10 

.2156 

4.638 25 

•233 93 

4.27471 

.252 42 

3.96165 

.271 07 

11 

•2159 

.631 7 i 

.234 24 

.269 11 

•252 73 

.9568 

.27138 

12 

.216 21 

.625 18 

•234 55 

.26352 

•253 04 

.95196 

.271 69 

13 

.216 51 

.61868 

•23485 

•25795 

•253 35 

•94713 

.272 01 

14 

.216 82 

.612 19 

.23516 

•25239 

• 25366 

•94232 

.27232 

15 

.217 12 

4.605 72 

•23547 

4.24685 

•253 97 

3-937 51 

.272 63 

16 

•21743 

•59927 

•23578 

.241 32 

.254 28 

•932 7 1 

.27294 

*7 

•21773 

.592 83 

.236 08 

•2358 

•254 59 

•927 93 

.27326 

18 

.218 04 

.58641 

.236 39 

•2303 

•2549 

.92316 

•27357 

*9 

.21834 

.58001 

•2367 

.224 81 

.25521 

• 9 l8 39 

.273 88 

20 

.218 64 

4-57363 

•237 

4-219 33 

•255 52 

3 - 9 r 3 64 

.27419 

21 

.21895 

.567 26 

•23731 

.21387 

•23583 

.908 9 

•27451 

22 

.21925 

.56091 

.237 62 

. 208 42 

•25614 

.904 17 

.274 82 

23 

.219 56 

•55458 

•237 93 

.202 98 

•25645 

•89945 

•27513 

24 

.219 86 

.54826 

•238 23 

•197 56 

.25676 

•89474 

•275 45 

25 

.220 17 

4.54196 

•23854 

4.19215 

•25707 

3.890 04 

•27576 

26 

.220 47 

•53568 

•23885 

.186 75 

•25738 

.88536 

.276 O7 

27 

.220 78 

.52941 

■23916 

.18137 

•257 69 

.880 68 

• 27638 

28 

.22108 

.52316 

.239 46 

. 176 

•258 

.876 OI 

. 276 7 

29 

.221 39 

•51693 

•239 77 

. 17064 

•25831 

•871 36 

.277 OI 

30 

.221 69 

4-51071 

.240 08 

4-1653 

.258 62 

3.866 71 

•277 32 

31 

.222 

•50451 

.24039 

•i 59 97 

.25893 

.862 08 

•27764 

32 

.222 31 

.49832 

.240 69 

•15465 

.259 24 

•857 45 

•27795 

33 

. 222 61 

.49215 

.241 

•149 34 

•259 55 

.852 84 

.278 26 

34 

.222 92 

.486 

.241 31 4 

.144 05 

.259 86 

.848 24 

.278 58 

35 

.223 22 

4.47986 

.241 62 

4-13877 

.260 17 

3-84364 

.278 89 

36 

•22353 

•473 74 

.24193 

133 5 

.200 48 

.83906 

279 2 

37 

.22383 

.467 64 

.24223 

. 128 25 

. 260 79 

•834 49 

.27952 

38 

.224 14 

•46155 

.24254 

. 123 01 

.26l I 

.829 92 

• 27983 

39 

.224 44 

•45548 

.242 85 

.117 78 

.26l 4I 

•82537 

.280 15 

40 

•22475 

4.44942 

.24316 

4.11256 

.26l 72 

3.820 83 

.28046 

4 i 

•22505 

•443 38 

•243 47 

• 107 36 

.262 03 

.8l6 3 

.280 77 

42 

.225 36 

•437 35 

•243 77 

.102 16 

.26235 

.8ll 77 

.281 09 

43 

.22567 

• 43 i 34 

.244 08 

. 096 99 

.262 66 

. 807 26 

.281 4 

44 

•22597 

•425 34 

•244 39 

.091 82 

.262 97 

.802 76 

.281 72 

45 

.226 28 

4.41936 

.2447 

4.086 66 

.263 28 

3.798 27 

.282 03 

46 

.226 58 

• 4 i 34 

.24501 

.081 52 

•263 59 

•793 78 

• 28234 

47 

. 226 89 

•407 45 

•245 32 

•076 39 

.2639 

.78931 

.282,66 

48 

.227 19 

• 4 QI 52 

.24562 

.071 27 

.26421 

.78485 

.282 97 

49 

•2275 

•3956 

•245 93 

.066 16 

.264 52 

.7804 

• 283 29 

50 

.227 81 

4.38969 

.246 24 

4.06107 

.264 83 

3-775 95 

.2836 

5 i 

.228 11 

•38381 

.24655 

•055 99 

•26515 

• 77 i 52 

.28391 

52 

.228 42 

•377 93 

. 246 86 

.05092 

.265 46 

.767 09 

.284 23 

53 

.228 72 

•37207 

.24717 

.045 86 

•26577 

.762 68 

.28454 

54 

.229 03 

• 366 23 

•247 47 

. 040 81 

.26608 

.75828 

.284 86 

55 

•229 34 

4.360 4 

.24778 

4-03578 

.266 39 

3-75388 

.28517 

56 

.229 64 

•354 59 

. 248 09 

•03075 

.266 7 

•7495 

.28549 

57 

.22995 

•34879 

. 248 4 

•02574 

.267 01 

•745 12 

.285 8 

58 

.230 26 

•343 

.248 71 

.020 74 

•267 33 

•74075 

.286 12 

59 

.23056 

•337 23 

.249 02 

.015 76 

.267 64 

•7364 

.286 43 

60 

.23087 

4-33148 

•249 33 

4.010 78 

.267 95 

3-73205 

.28675 

f 

Co-tang. 

Tang. 

I Co-TANG. 

Tang. 

| CO-TANG. 

Tang. 

CO-TANG. 


770 

760 

750 

74 


Co-TANG. 

3 * 73 2 °5 

.72771 

• 72338 

.71907 
.71476 
3 . 7 io 46 
.706 16 
.701 88 
.697 61 

•69335 

*3.680 OQ 

.68485 
.680 6l 
.676 38 
.67217 
3.667 go 
.663 76 
•659 57 
•65538 

• 651 21 

3-64705 

.642 89 
.63874 

• 63461 
.63048 

3.626 36 
.622 24 
.61814 
.614 05 
. 609 96 
3.605 88 
.601 81 
•597 75 
•593 7 
.589 66 
3-58562 
.581 6 

•57758 

•57357 
•569 57 
3-565 57 
•56159 
•55761 
•55364 

.54968 
3-545 73 
• 54 i 79 
•537 85 
•533 93 

• 53 ° 01 

3.52609 

.522 IQ 

.518 29 
• 5 I 44 I 

•510 53 
3.506 66 
.502 79 
.49894 
.49509 

• 49 1 2 5 

3.48741 

Tang. 













































NATURAL TANGENTS AND CO-TANGENTS. 



16 ° 

170 

180 

190 , 

/ 

Tang. 

COTANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

Co-TANG, | 

o 

.28675 

3.48741 

•30573 

3.27085 

.32492 

3.077 68 

•344 33 

2.9O4 21 

I 

. 287 06 

• 4 8 3 59 

• 306 05 

.26745 

•32524 

.07464 

•34465 

.90* 47 

2 

.287 38 

•479 77 

• 306 37 

.264 06 

•32556 

.071 6 

.34498 

.89873 

o 

O 

.287 69 

•475 9 6 

. 306 69 

260 67 

.325 88 

.068 57 

•345 3 

.896 

4 

.288 

.472 16 

•307 

•257 29 

. 326 21 

•065 54 

•34563 

.89327 

5 

.288 32 

3-46837 

• 307 32 

3-253 92 

•32653 

3.062 52 

•345 96 

2.89055 

6 

.288 64 

.46458 

• 307 64 

•25055 

.326 85 

•059 5 

.34628 

.887 83 

7 

.28895 

.460 8 

.307 96 

•247 *9 

•327 *7 

.05649 

.34661 

.88511 

8 

.289 27 

•45703 

.308 28 

•24383 

• 327^9 

•053 49 

•346 93 

.882 4 

9 

.289 58 

•453 27 

.3086 

.240 49 

.327 82 

.05049 

•34726 

.8797 

IO 

.289 9 

3-449 51 

•30891 

3-237 *4 

.32814 

3-047 49 

•34758 

2.877 

II 

.290 21 

• 4457 6 

.30923 

•23381 

.32846 

•0445 

•347 9 * 

■8743 

12 

.29053 

. 442 02 

•309 55 

.23048 

.32878 

.041 52 

.34824 

.871 6l 

13 

. 290 84 

.43829 

.309 87 

.227 15 

• 329 n 

•038 54 

•348 56 

.868 92 

14 

.291 16 

•434 56 

.310 19 

.223 84 

•329 43 

•035 56 

.34889 

.866 24 

15 

.29147 

3 - 43 ° 84 

•31051 

3.22053 

•32975 

3.032 6 

.34922 

2.863 56 

l6 

.29179 

•427 13 

•31083 

.217 22 

• 33 ° 07 

.029 63 

•349 54 

.860 89 

17 

.292 1 

•423 43 

• 3 ii 15 

-21392 

•330 4 

.026 67 

•349 87 

.858 22 

l8 

.292 42 

•41973 

• 3 IM 7 

.210 63 

•33072 

.023 72 

•35019 

•85555 

*9 

.292 74 

.416 04 

.31178 

.207 34 

• 33*04 

.020 77 

•35052 

.852 89 

20 

•293 05 

3.41236 

.312 1 

3.204 06 

•33136 

3017 83 

•35085 

2.85° 23 

21 

•293 37 

. 408 69 

.31242 

.200 79 

•33169 

.014 89 

•35117 

.84758 

22 

.29368 

.405 02 

•31274 

•197 52 

.33201 

.011 96 

• 35 * 5 

.844 94 

23 

.294 

.40136 

.3x306 

.194 26 

•332 33 

.009 03 

•35183 

.842 29 

24 

.294 32 

•397 7 1 

•31338 

.191 

.332 66 

.006 11 

.35216 

•83965 

25 

.29463 

3-39406 

•3137 

3-18775 

•33298 

3.00319 

.35248 

2.837 02 

26 

.29495 

.39042 

• 3 I 4 02 

.18451 

•333 3 

.000 28 

•35281 

•83439 

27 

.295 26 

.386 79 

• 3*4 34 

. 181 27 

•33363 

2.99738 

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.83176 

28 

•295 58 

•38317 

.31466 

. 178 04 

•333 95 

•994 47 

•35346 

.829 14 

29 

•295 9 

•379 55 

.31498 

.17481 

•334 27 

.99158 

•353 79 

.82653 

30 

.296 21 

3-375 94 

• 3*53 

3-17159 

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2.988 68 

•35412 

2.823 91 

31 

•29653 

•372 34 

•31562 

.16838 

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.9858 

•354 45 

.8213 

32 

•29685 

•36875 

•31594 

.16517 

•335 24 

.982 92 

•354 77 

.818 7 

33 

.297 16 

•365 16 

.316 26 

. 161 97 

•335 57 

.980 04 

•355 * 

.816 1 

34 

.29748 

•36158 

•31658 

•15877 

•33589 

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•355 43 

•813 5 

35 

. 297 8 

3-358 

.3169 

3-155 58 

•33621 

2-974 3 

•35576 

2.81091 

36 

.298 11 

•354 43 

• 3*722 

.1524 

•336 54 

.97144 

.35608 

•808 33 

37 

.298 43 

.35087 

• 3*754 

. 149 22 

• 336 86 

.96858 

•35641 

.805 74 

38 

.29875 

•347 32 

.317 86 

.146 05 

•337 18 

•96573 

■35674 

.803 16 

39 

. 299 06 

•343 77 

.31818 

. 142 88 

•337 51 

.962 88 

•357 07 

. 800 59 

40 

.299 38 

3-34023 

• 3 i8 5 

3 -I 39 72 

•33783 

>2.960 04 

•3574 

2.798 02 

4 i 

.2997 

•3367 

.3l8 82 

.136 56 

• 338 16 

•957 2 i 

•357 72 

•79545 

42 

.30001 

•333 17 

• 3 * 9*4 

•*33 4 * 

•33848 

•95437 

•35805 

.79289 

43 

• 3 °° 33 

• 329 65 

.31946 

.13027 

•33881 

• 95 i 55 

•35838 

•790 33 

44 

• 3 oo6 5 

.32614 

•31978 

.127 13 

•339 13 

.94872 

•35871 

.787 78 

45 

. 3 00 97 

3.32264 

.3201 

3.124 

•339 45 

2.9459 

•359 °4 

2.785 23 

46 

. 3OI 28 

• 3 i 9 *4 

•320 42 

. 120 87 

•33978 

•943 09 

•359 37 

.782 69 

47 

• 3 01 6 

•31565 

•32074 

•117 75 

• 34 °i 

.940 28 

•359 69 

.780 14 

48 

• 3 01 92 

.312 16 

.321 06 

. 114 64 

•340 43 

•93748 

.36002 

•77761 

49 

.302 24 

.30868 

•32139 

•in 53 

•340 75 

.93468 

•36035 

•77507 

50 

•302 55 

3.30521 

.32171 

3.108 42 

•34 1 08 

2.93189 

.360 68 

2.77254 

5 i 

.302 87 

•3 01 74 

.32203 

•105 32 

• 34 M 

.9291 

.361 01 

.77002 

52 

•303 19 

.298 29 

•32235 

. 102 23 

•34173 

.926 32 

•36134 

•7675 

53 

•30351 

.29483 

.32267 

.09914 

.342 05 

•923 54 

.36167 

.764 98 

54 

.30382 

.29139 

.32299 

. 096 06 

•34238 

.920 76 

.36199 

.76247 

55 

.30414 

3-28795 

•323 3 * 

3.09298 

•342 7 

2.9x799 

.36232 

2.75996 

56 

.304 46 

.28452 

•32363 

.089 91 

•343 03 

• 9*523 

.36265 

•757 46 

57 

• 304 78 

. 281 09 

.32396 

.086 85 

•343 35 

.912 46 

.36298 

•75496 

58 

•305 09 

. 277 67 

.324 28 

.08379 

•34368 

.90971 

•36331 

.75246 

59 

• 3°5 4 1 

.274 26 

.3246 

.08073 

•344 

.906 96 

•36364 

•74997 

60 

•305 73 

3.27085 

.32492 

3.07768 

•344 33 

2.904 21 

•36397 

2.747 48 

/ 

Co-tang. 

1 Tang. 

CO-TANG. 

Tang. 

Co-tang. 

Tang. 

CO-TANG. 

Tang. 


730 

720 

710 

70 ° 


O ON 00 lO-trON m o ON 00 t^.'O LD T^- ro CS h o OnOO t^vO m -tfOO m O 0\CO iO t*- ro Cl m 0 OnCO NVO ir> rj- rO Cl m 0 CL 00 C>sV3 iO CO Cl 

O L/i lo uo lo lo to lo Ln lo cocorocororococoforo Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl M M M H H M M H H H 





















































NATURAL TANGENTS AND CO-TANGENTS 



20 ° 

21 ° 

220 

230 


/ 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

/ 

o 

•36397 

2.747 48 

.38386 

2.605 09 

. 404 03 

2.47509 

•424 47 

2-355 85 

60 

I 

•3643 

•744 99 

.3842 

.602 83 

• 4°4 36 

•47302 

.424 82 

•353 95 

59 

2 

•36463 

■ 74 2 5 i 

•38453 

• 600 57 

.4O47 

■ 47 ° 95 

.42516 

.352 05 

58 

3 

•364 96 

.740 04 

.38487 

• 59 8 3 i 

.405 04 

.46888 

•42551 

•350 15 

57 

4 

•36529 

•737 56 

•3852 

.59606 

•405 38 

. 466 82 

•42585 

•34825 

56 

5 

.36562 

2.735 09 

•38553 

2.59381 

. 4O5 72 

2.464 76 

.426 19 

2.34636 

55 

6 

• 365 95 

■73263 

•38587 

• 59 1 56 

. 406 c6 

.4627 

•42654 

• 344 47 

54 

7 

.366 28 

• 73 ° 1 7 

.3862 

•58932 

. 406 4 

.46065 

.426 88 

■34258 

53 

8 

.366 61 

• 7 2 7 7 i 

.38654 

.58708 

.406 74 

.4586 

.427 22 

. 340 69 

52 

9 

.36694 

.725 26 

.386 87 

• 584 84 

. 407 07 

•456 55 

•427 57 

•33881 

5 i 

IO 

.36727 

2.722 81 

.38721 

2.582 6l 

.40741 

2.45451 

.42791 

2-336 93 

50 

ii 

•3676 

.720 36 

•38754 

.58038 

•407 75 

.45246 

.428 26 

•335 05 

49 

12 

.36793 

• 7 J 7 9 2 

.387 87 

•57815 

. 408 09 

•450 43 

.428 6 

•333 17 

48 

13 

.368 26 

•71548 

.388 21 

•57593 

.408 43 

•448 39 

. 428 94 

• 33 i 3 

47 

i 4 

•36859 

• 7 I 305 

• 38854 

•573 71 

• 4°8 77 

.44636 

.42929 

•32943 

46 

i 5 

.36892 

2.710 62 

.38888 

2 - 57 1 5 

.4O9 II 

2-444 33 

.42963 

2.327 56 

45 

16 

• 369 25 

.708 19 

.389 21 

.56928 

•409 45 

•4423 

.429 98 

•3257 

44 


•36958 

.70577 

•38955 

.56707 

.40979 

.44027 

• 43o 32 

•323 83 

43 

18 

.36991 

• 7°3 35 

.38988 

.56487 

.41013 

•43825 

.43067 

•321 97 

42 

*9 

• 37 ° 24 

.70094 

.390 22 

.56266 

4IO47 

•43623 

• 43 i 01 

. 320 12 

4 i 

20 

• 37 ° 57 

2.698 53 

• 39 ° 55 

2. 560 46 

.41081 

2.43422 

•43136 

2.318 26 

40 

21 

• 37 ° 9 

.696 12 

• 39 ° 8 9 

•55827 

• 4 11 15 

.4322 

•431 7 

.31641 

39 

22 

• 37 1 24 

•693 7 1 

• 39 1 22 

.55608 

• 4 11 49 

.43019 

•43205 

•31456 

38 

23 

• 37 i 57 

.691 31 

• 39 1 56 

•55389 

.41183 

.428 19 

•432 39 

.31271 

37 

24 

• 37 i 9 

.688 92 

• 39 1 9 

•5517 

.41217 

.426 18 

•43274 

.310 86 

36 

25 

.37223 

2.68653 

• 39 22 3 

2.54952 

• 4 12 51 

2.424 18 

■43308 

2.309 02 

35 

26 

•37256 

.684 14 

• 39 2 57 

•54734 

.41285 

.422 18 1 

■43343 

.30718 

34 

27 

.37289 

.68175 

• 39 2 9 

•54516 

• 4 r 3 J 9 

.420 19 

•43378 

■30534 

33 

28 

•373 22 

•67937 

•39324 

•542 99 

•41353 

.418 19 

•434 12 

•30351 

32 

29 

•373 55 

.677 

•393 57 

.54082 

•41387 

.416 2 

•434 47 

.30167 

3 i 

30 

•37388 

2.674 62 

*393 9 1 

2.53865 

.414 21 

2.414 21 

.43481 

2.299 84 

30 

31 

•374 22 

.67225 

• 394 2 5 

•53648 

•414 55 

.41223 

•43516 

.298 01 

29 

32 

•374 55 

. 669 89 

•39458 

•534 32 

.4149 

• 4 IQ 25 

! -435 5 

.296 19 

28 

33 

.37488 

.667 52 

•394 9 2 

•53217 

•41524 

.408 27 

I -43585 

•294 37 

27 

34 

•375 21 

.665 16 

•39526 

.53001 

■41558 

. 406 29 

.4362 

.292 54 

26 

35 

•375 54 

2.662 81 

•395 59 

2.527 86 

.41592 

2.404 32 

•430 54 

2.29073 

25 

36 

•37588 

.660 46 

•395 93 

•525 7 * 

.416 26 

•402 35 

•436 89 

.288 91 

24 

37 

.376 21 

.65811 

. 396 26 

•52357 

.416 6 

. 400 38 

•437 24 

. 287 1 

23 

3B 

•376 54 

•65576 

.3966 

.52142 

.41694 

.39841 

•43758 

.285 28 

22 

39 

•37687 

•65342 

•396 94 

.51929 

.417 28 

•39645 

•437 93 

.28348 

21 

40 

•377 2 

2.651 09 

•397 27 

2-517 15 

•41763 

2.39449 

.43828 

2.281 67 

20 

4 i 

•377 54 

.64875 

■39761 

• 51502 

•41797 

•392 53 

.43862 

.27987 

19 

42 

•377 87 

.646 42 

•397 95 

• 5 12 89 

.41831 

•39058 

•43897 

.278 06 

18 

43 

.3782 

.6441 

.39829 

.51076 

.41865 

.388 62 

•439 32 

.276 26 

17 

44 

•37853 

.64177 

.39862 

. 508 64 

.41899 

.38668 

•439 66 

•274 47 

16 

45 

•37887 

2-63945 

.398 96 

2.50652 

•419 33 

2-38473 

.440 01 

2.272 67 

15 

46 

•379 2 

•63714 

•399 3 

•504 4 

.419 68 

.382 79 

.44036 

.270 88 

14 

47 

•379 53 

.634 83 

•399 63 

. 502 29 

.420 02 

.38084 

.44O7I 

.269 09 

13 

48 

.37986 

•632 52 

•39997 

. 500 l8 

.42036 

•37891 

•44105 

.267 3 

12 

49 

.3802 

.630 21 

.40031 

.49807 

.4207 

•37697 

• 44 i 4 

•265 52 

11 

50 

•38053 

2.627 9 1 

.40065 

2 .495 97 

.42105 

2.37504 

• 44 i 75 

2.263 74 

10 

5 i 

.380 86 

.625 61 

. 400 98 

.493 86 

•42139 

•373 n 

.442 1 

. 261 96 

9 

52 

.3812 

.62332 

• 4 QI 32 

• 49 1 77 

■421 73 

.37118 

.44244 

.260 18 

8 

53 

•38153 

.621 03 

.401 66 

.489 67 

.422 07 

•36925 

•442 79 

.2584 

7 

54 

. 381 86 

.618 74 

.402 

.487 58 

.422 42 

•367 33 

•443 14 

•25663 

6 

55 

.382 2 

2.61646 

.402 34 

2.485 49 

. 422 76 

2.36541 

•443 49 

2.254 86 

5 

56 

•38253 

.61418 

.402 67 

•4834 

•4231 

•363 49 

•44384 

.25309 

4 

57 

.382 86 

.611 9 

• 4°3 01 

.48132 

•42345 

.36158 

.44418 

.251 32 

3 

53 

•3832 

. 609 63 

• 4°3 35 

.47924 

•423 79 

•35967 

•444 53 

• 249 56 

2 

59 

•38353 

.607 36 

.403 69 

•47716 

.42413 

•35776 

.444 88 

.2478 

1 

60 

.383 86 

2.605 °9 

.404 03 

2.47509 

.42447 

2-35585 

•445 23 

2.246 04 

0 

/ 

CO-TANG. 

Tang. 

Co-tang. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

/ 


690 

680 | 

67 ° 

66° 




























































NATURAL TANGENTS AND CO-TANGENTS 


421 



24 ° 

25 

O 

26 ° 

27 ° 


/ 

Tang. 

CO-TANG. | 

Tang. 

CO-TANG. 

Tang. | 

CO-TANG. 

Tang. I 

Co-tang. 

/. 

0 

•44523 

2. 246 04 

.46631 

2.144 51 

■48773 

2.050 3 

•50953 

1.962 61 

60 

I 

•44558 

.244 28 

.466 66 

. 142 88 

. 488 09 

.048 79 

• 509 89 

.961 2 

59 

2 

•445 93 

.242 52 

.467 02 

.141 25 

.48845 

.O47 28 

.510 26 

•95979 

58 

3 

.44627 

•240 77 

•46737 

•13963 

. 488 81 

•045 77 

.51063 

.958 38 

57 

4 

.446 62 

.239 02 

.46772 

.13801 

.48917 

.04426 

.51099 

.95698 

56 

5 I 

.44697 

2.23727 j 

.46808 

2.136 39 

•48953 

2.042 76 

.51136 

*•955 57 

55 

6 

•447 32 

•235 53 

•468 43 

• I 34 77 

.489 89 

.04125 

•5*i 73 

•95417 

54 

7 

.44767 

■ 23378 

.468 79 

•133 16 

.490 26 

•039 75 

.51209 

•95277 

53 

8 

.44802 

.23204 

.46914 

•* 3 * 54 

.49062 

.03825 

.51246 

• 95 i 37 

52 

9 

•44837 

•230 3 

•4695 

.129 93 

• 4QO q8 

■03675 

.51283 

•94997 

5 i 

10 

.44872 

2. 228 57 

•46985 

2. 128 32 

• 49 1 34 

2.03526 

•513 19 

1.948 58 

50 

11 

•449 °7 

.226 83 

.470 21 

. 126 71 

• 49 1 7 

•03376 

• 51356 

•947 *8 

49 

12 

•44942 

.225 I 

.470 56 

.125 II 

.492 06 

•032 27 

•51393 

•945 79 

48 

13 

•449 77 

•223 37 

.47092 

•1235 

.49242 

•03078 

•5143 

•944 4 

47 

14 

‘450 12 

.221 64 

.471 28 

. 121 9 

•49278 

•O29 29 

•51467 

.94301 

46 

i 5 

■45047 

2. 219 92 

•47*63 

2. 120 3 

•493 *5 

2.027 8 

•51503 

1.941 62 

45 

16 

.45082 

.21819 

• 47 1 99 

.I187I 

•493 5 * 

.026 31 

•5154 

.94023 

44 

*7 

• 45 i 17 

.2l6 47 I 

•472 34 

•11711 

•493 87 

.024 83 

•5*577 

•93885 

43 

18 

• 45 i 52 

•21475 

•4727 

•115 52 

•494 23 

•° 2 3 35 

.516x4 

•93746 

42 

19 

•45187 

.21304 

•473 05 

.11392 

•494 59 

.021 87 

•5*651 

.93608 

4 i 

20 

.45222 

2. 211 32 

• 4734 i 

2.11233 

•49495 

2.020 39 

.516 88 

1-934 7 

40 

21 

•45257 

.209 6l 

•473 77 

.IIO75 

•495 32 

.018 91 

•51724 

•933 32 

39 

22 

.45292 

.207 9 

•474 12 

. IO9 l6 

•49568 

.01743 

.51761 

• 93*95 

38 

23 

•453 27 

.206 19 

•47448 

.107 58 

. 496 04. 

.01596 

•5*798 

•930 57 

37 

24 

•45362 

.204 49 

•47483 

.106 

•4964 

.01449 

•5*835 

.929 2 

36 

25 

•45397 

2. 202 78 

•475 *9 

2. IO4 42 

•49677 

2.013 02 

.5*8 72 

I.927 82 

35 

26 

•454 32 

.201 08 

•475 55 

. 102 84 

•49713 

• OII 55 

.51909 

• 92645 

34 

27 

•45467 

.19938 j 

•4759 

. 101 26 

•497 49 

.OIO08 

.51946 

.92508 

33 

28 

•455 02 

. I97 69 1 

.476 26 

. O99 69 

.497 86 

.008 62 

•5*983 

•923 7 1 

32 

29 

•455 37 

•195 99 

.476 62 

.098 11 

.498 22 

•00715 

.5202 

• 9 22 35 

3 i 

3 ° 

•45573 

2.1943 

.47698 

2.09654 

.49858 

2.005 69 

•52057 

1.92098 

30 

3 i 

.45608 

.192 6l 1 

•477 33 

.O94 98 

.49894 

.004 23 

.52094 

.919 62 

29 

32 

•45643 

. I9092 

.47769 

.O934I 

•499 31 

.002 77 

.52131 

.918 26 

28 

33 

.45678 

.189 23 

•47805 

.C9184 

•49967 

031 3 1 

.521 68 

•9*69 

2 7 

34 

•457 13 

■187 55 

.4784 

.C90 28 

. 500 04 

i -999 86 

.52205 

• 9 I 5 54 

26 

35 

1 -45748 

2. 185 87 

.47876 

2.088 72 

.5OO4 

1-99841 

.52242 

1.914 18 

25 

36 

! -45784 

.184 19 

•479 12 

.087 16 

.50076 

•99695 

.52279 

.912 82 

24 

37 

.45819 

.182 51 

.479 48 

.0856 

•501 13 

•995 5 

•523*6 

• 9 11 47 

23 

38 

■45854 

. l8o 84 

.47984 

.08405 

.5OI49 

•994 06 

•52353 

.91012 

22 

39 

.45889 

.17916 

.480 19 

■082 5 

.50185 

.99261 

•5239 

.908 76 

21 

40 

•459 24 

2.17749 

•48055 

2.080 94 

.502 22 

vi.99 1 x 6 

•52427 

1.907 41 

20 

4 1 

•4596 

.17582 

.48091 

•07939 

.502 58 

• 98972 

.524 64 

.906 07 

J 9 

42 

•45995 

.17416 

.481 27 

.077 85 

•50295 

.98828 

‘ 5 2 5 oi 

.90472 

18 

43 

.4603 

.17249 

.48163 

•0763 

•503 31 

.986 84 

•52538 

•903 37 

17 

44 

.46065 

.17083 

.481 98 

.07476 

.50368 

• 9854 

•52575 

.902 03 

16 

45 

.461 01 

2. 169 17 

.48234 

2.O73 21 

.50404 

i- 9 8 3 q 6 

.52613 

1.900 69 

i 5 

46 

.46136 

.167 51 

.4827 

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.50441 

.98253 

•5265 

•89935 

14 

47 

.46171 

• 16585 

. 483 06 

.070 14 

•504 77 

.981 x 

.52687 

.89801 

13 

48 

.462 06 

. 164 2 

.48342 

.0686 

•505 14 

•979 66 

.52724 

.896 67 

12 

49 

.462 42 

• 16255 

.48378 

.067 06 

•505 5 

• 97823 

.52761 

•89533 

11 

5 ° 

.46277 

2.1609 

.484i 4 

2.065 53 

• 505 87 

I.9768 

•527 9 s 

1.894 

10 

51 

•46312 

•159 25 

•4845 

.064 

• 50623 

•975 38 

.528 36 

.892 66 

9 

52 

.46348 

• i 57 6 

. 484 86 

.062 47 

.5066 

•973 95 

•52873 

•89133 

8 

53 

•46383 

•15596 

.485 21 

. 060 94 

. 506 96 

•972 53 

• 529 1 

.89 

7 

54 

.464 18 

•154 32 

•48557 

•059 42 

•507 33 

• 97 111 

•529 47 

.888 67 

6 

55 

•464 54 

2.152 68 

•48593 

2.057 9 

.50769 

1.969 69 

•529 84 

1.887 34 

5 

56 

.464 8o 

.151 04 

.486 29 

•05637 

. 508 06 

.968 27 

•53022 

.886 02 

4 

57 

■46525 

.1494 

.486 65 

• 05485 

• 508 43 

.966 85 

•530 59 

.884 69 

3 

58 

.4656 

•i 47 77 

.487 01 

•053 33 

.508 79 

•965 44 

•53096 

•88337 

2 

59 

•46595 

. 146 14 

•48737 

.051 82 

.50916 

.964 02 

- 53 1 34 

. 882 05 

1 

60 

.46631 

2.144 51 

•48773 

2.O503 

•509 53 

1.962 61 

' - 53 i 7 i 

1.880 73 

0 

/ 

CO-TANG 

Tang. 

Co-tang. 

Tang. 

CO-TANG 

Tang. 

| CO-TANG 

1 Tang. 

/ 


65 0 

64 ° . 

63 ° 

II 62 ° 



N N 





















































































ON O H M fT) irjvo t^OO On 0 h CM CO rt* mvO t^OO Os 0 h N fO -t i/NVO t^.00 ON 0 H CM CO rh LO\0 t^CO On O h CM CO ^ ionO t^OO On O 

m hmhhhhhmmcm N « Cl N N O N W N CO ncncOfOrOrOMfOrO-t lOiOiOiOiommiO U-J'O 


NATURAL TANGENTS AND CO-TANGENTS. 


28° 

290 

300 

31° 


Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

t 

•53i 7 1 

I.88073 

•55431 

I.80405 

•57735 

1.73205 

. 600 86 

I.664 28 

j 60 

.53208 

.87941 

•55469 

.802 8l 

•577 74 

• 73089 

.601 26 

.66 3 18 

59 

•53246 

.878 09 

•555 07 

.801 58 

•57813 

•729 73 

•60165 

. 662 09 

58 

•53283 

.876 77 

•555 45 

.80034 

•57851 

• 72857 

. 602 05 

.660 99 

57 

•5332 

.87546 

•55583 

.799 11 

•5789 

.7274I 

.60245 

•6599 

56 

•53358 

1.87415 

•55621 

1.797 88 

•579 29 

I. 726 25 

. 602 84 

I.658 8l 

55 

•53395 

• 87283 

•55659 

■79665 

.57968 

.72509 

. 603 24 

.65772 

54 

•534 3 2 

•87152 

•55697 

•79542 

•58007 

•723 93 

.60364 

• 65663 

53 

•5347 

.870 21 

•55736 

.79419 

.58046 

. 722 78 

.604 03 

•65554 

5? 

•535 07 

.868 91 

•55774 

.792 96 

•58085 

• 72163 

• 604 43 

•65445 

5i 

•53545 

1.867 6 

•55812 

I -79 I 74 

•58124 

I. 720 47 

.60483 

i -653 37 

50 

•53582 

.8663 

•5585 

•79051 

.58162 

• 7 I 9 32 

.605 22 

.652 28 

49 

•5362 

.864 99 

.55888 

.789 29 

.58201 

.718 17 

. 605 62 

• 651 2 

48 

•53657 

.86369 

•559 26 

.788 07 

.5824 

.717 02 

. 606 02 

.65011 

47 

•53694 

.862 39 

•55964 

.786 85 

.58279 

.71588 

.606 42 

•649 03 

46 

•537 32 

1.861 09 

•560 03 

x.78563 

•58318 

1-71473 

.606 81 

1.64795 

45 

•53769 

•859 79 

.56041 

.78441 

•583 57 

■71358 

.607 21 

.646 87 

44 

•53807 

.8585 

•560 79 

•78319 

.58396 

• 7 12 44 

.607 61 

•645 79 

43 

•53844 

•8572 

.56117 

.781 98 

•58435 

.711 29 

.608 OI 

.64471 

42 

•53882 

•85591 

.56156 

. 780 77 

•584 74 

• 7 IQ 15 

.608 41 

•64363 

4i 

.5392 

1.854 62 

.56194 

J - 779 55 

•58513 

1.709 OI 

.608 81 

I.642 56 

4° 

•539 57 

•85333 

.56232 

•778 34 

•58552 

.70787 

.609 21 

.64I 48 

39 

•53995 

.85204 

.5627 

•77713 

■5859 1 

•70673 

.609 6 

.64O4I 

38 

•54 0 32 

•85075 

• 563 09 

•775 92 

•58631 

.7056 

.61 

•639 34 

37 

•5407 

.849 46 

•56347 

•77471 

.5867 

.704 46 

.6104 

.638 26 

36 

•54107 

1.848 18 

•56385 

i-773 5i 

.58709 

1.70332 

.610 8 

1.637 19 

35 

•54145 

.846 89 

.56424 

•77 23 

.58748 

• 70219 

.611 2 

. 636 12 

34 

•54183 

•84561 

.56462 

.771 1 

.58787 

.701 06 

.61x6 

•63505 

33 

.5422 

•84433 

.565 

•7699 

.588 26 

.6gg Q2 

.612 

•6339 s 

32 

•542 58 

■S43 °5 

•565 39 

.768 69 

•58865 

.698 79 

.612 4 

.632 92 

3i 

•54296 

1.84177 

•56577 

1.76749 

•589 04 

1.697 66 

.612 8 

1.63185 

30 

•543 33 

.84049 

. 566 16 

•7663 

.58944 

•696 53 

•6132 

.630 79 

29 

•543 7i 

.83922 

■56654 

•7651 

.58983 

•60541 

.6136 

.629 72 

28 

•54409 

•83794 

•566 93 

•7639 

.59022 

.694 28 

. 614 

.62866 

27 

•54446 

.83667 

•56731 

.762 71 

.59061 

•69316 

.6144 

.627 6 

26 

•544 84 

1-8354 

.56769 

1.761 5 i 

•59 1 OI 

1.692 03 

.614 8 

1.626 54 

25 

•545 22 

•83413 

.568 08 

. 760 32 

•59 1 4 

.69091 

.6152 

.62548 

24 

•5456 

.832 86 

.56846 

•759 13 

•59 1 79 

.689 79 

.61561 

.624 42 

23 

•54597 

•83159 

•56885 

•757 94 

•59218 

.688 66 

.616 OI 

.623 36 

22 

•54635 

•83033 

• 569 23 

•75675 

•59258 

•68754 

.61641 

.622 3 

21 

•54673 

1.829 06 

.569 62 

1-75556 

•59297 

I.68643 

.616 81 

1.621 25 

20 

•54711 

.827 8 

•57 

•754 37 

•593 36 

•68531 

.617 21 

.62019 

19 

•54748 

.826 54 

•570 39 

•753 19 

•593 76 

.684 I9 

.617 61 

.619 14 

18 

.54786 

.825 28 

.57078 

•752 

•59415 

.68308 

.618 OI 

. 618 08 

*7 

.548 24 

.824 02 

.57116 

75082 

•594 54 

.68l 96 

.618 42 

.61703 

16 

.54862 

1.822 76 

•57i 55 

1.74964 

•59494 

1.680 85 

.61882 

1.61598 

15 

•549 

.8215 

•57193 

.74846 

•59533 

•679 74 

.619 22 

•61493 

14 

•54938 

.820 25 

•57232 

.74728 

•59573 

•67863 

.619 62 

.613 88 

13 

•54975 

.818 99 

•57271 

•7461 

•59612 

•677 52 

.62003 

.612 83 

12 

•55013 

•81774 

• 57309 

.•744 92 

•59651 

.676 4I 

• 62043 

.611 79 

11 

•55051 

1.816 49 

•57348 

*•743 75 

•5969 1 

I-675 3 

.620 83 

1.610 74 

10 

•55089 

.815 24 

•573 86 

•742 57 

•5973 

.67419 

.621 24 

.609 7 

9 

•55127 

•81399 

•574 25 

.7414 

•5977 

.67309 

621 64 

.608 65 

8 

•55165 

.812 74 

•57464 

.740 22 

•598 09 

.67I 98 

.622 04 

.607 61 

7 

•55203 

•8n 5 

•575 03 

•739 05 

.59849 

.670 88 

.62245 

.606 57 

6 

•55241 

1.810 25 

•57541 

1.73788 

.59888 

1.669 78 

.622 85 

1-60553 

5 

•552 79 

. 809 01 

•575 8 

•73671 

•599 28 

.668 67 

.62325 

.604 49 

4 

•553 17 

•80777 

•57619 

•735 55 

• 599 67 

•667 57 

.623 66 

•603 45 

3 

•553 55 

.80653 

•576 57 

•73438 

. 600 07 

.666 47 

.624 06 

.602 41 

2 

•553 93 

•80529 

•57696 

•733 21 

. 600 46 

•665 38 

• 624 46 

•60137 

1 

•5543i 

1.804 °5 

•577 35 

1-73205 

.600 86 

1.664 28 

.624 87 

1.60033 

0 

CO-TANG. 

Tang. 

Co-tang. 

Tang. 

CO-TANG. 

Tang. 

COTANG. 

Tang. 

/ 

610 

60 ° 

590 

58 ° 















































NATURAL TANGENTS AND CO-TANGENTS. 


1 

32 ° | 

330 

340 

35 ° 


/ 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

/ 

o 

.624 87 

1.60033 

.64941 

1.539 86 

•67451 

I.482 56 

.700 21 

1.42815 

60 

I 

.625 27 

•599 3 

.649 82 

.53888 

•674 93 

.48163 

.700 64 

.427 26 

59 

2 

.625 68 

.59826 

•65023 

• 5379 i 

•67536 

.4807 

.701 07 

.426 38 

58 

3 

.626 08 

• 597 2 3 

•65065 

•53693 

.67578 

•479 77 

• 7 01 5 i 

•425 5 

57 

4 

.626 49 

.5962 

.651 06 

•535 95 

.676 2 

.47885 

.701 94 

.424 62 

56 

5 

.626 89 

i -595 17 

.65148 

1-534 97 

.676 63 

I- 4779 2 

.70238 

I.42374 

35 

6 

.6273 

•59414 

.651 89 

*534 

.67705 

.47699 

.702 81 

.422 86 

54 

7 

.6277 

•593 11 

•65231 

•533 02 

.67748 

.47607 

70325 

.421 98 

53 

8 

.628 11 

.59208 

.652 72 

•53205 

•6779 

•47514 

• 7°3 68 

.421 1 

52 

9 

.628 52 

.59105 

•65314 

•53107 

.67832 

.47422 

.704 12 

.420 22 

5 i 

IO 

.628 92 

I.59O 02 

•65355 

1.530 1 

.67875 

1-473 3 

•70455 

I - 4 I 9 34 

5 o 

II 

.62933 

• 589 

•65397 

•52913 

•679 J 7 

.47238 

.704 99 

.41847 

49 

12 

.62973 

•58797 

.654 38 

.528 16 

.679 6 

.47146 

•705 42 

•417 59 

48 

13 

.63014 

.58695 

.6548 

•52719 

.680 02 

•470 53 

• 7°5 86 

.416 72 

47 

14 

•63055 

• 58593 

•65521 

.526 22 

.68045 

. 469 62 

.706 29 

.41584 

46 

15 

.63095 

1.5849 

•65563 

1-52525 

.68088 

1.468 7 

.70673 

1.41497 

45 

l6 

.63136 

.58388 

.65604 

.524 29 

•6813 

.46778 

• 7°7 x 7 

.41409 

44 

17 

•63177 

.582 86 

.65646 

•52332 

.68173 

.466 86 

.707 6 

.41322 

43 

l8 

.63217 

.58184 

.65688 

•52235 

.682 15 

•465 95 

. 708 04 

•41235 

42 

*9 

.632 58 

.58083 

.65729 

•521 39 

.682 58. 

•46503 

. 708 48 

.411 48 

4 i 

20 

.63299 

1.57981 

•65771 

1.52043 

.683 01 

I.464 II 

. 708 91 

1.410 61 

40 

21 

•6334 

•57879 

.65813 

.51946 

.68343 

.4632 

■ 709 35 

.409 74 

39 

22 

•6338 

•57778 

•65854 

•5185 

.683 86 

.462 29 

•709 79 

.408 87 

38 

23 

.63421 

.576 76 

.65896 

•51754 

.684 29 

•461 37 

• 7 IQ 23 

.408 

37 

24 

.63462 

•57575 

•659 38 

.516 58 

.684 71 

. 460 46 

.710 66 

.407 14 

36 

25 

•63503 

i -574 74 

•6598 

1-515 62 

•68514 

1-459 55 

.711 1 

1.406 27 

35 

26 

•63544 

•573 72 

.660 21 

.51466 

•685 57 

.45864 

•71154 

•405 4 

34 

27 

•63584 

• 57 2 7 1 

.660 63 

•5137 

.686 

■457 73 

.711 98 

.404 54 

33 

28 

.63625 

•5717 

.661 05 

•51275 

.686 42 

• 456 82 

.712 42 

.40367 

32 

29 

.636 66 

•570 69 

.661 47 

• 5 ii 79 

.686 85 

•455 9 2 

.71285 

.402 81 

3 i 

30 

•63707 

1.56969 

.661 89 

1.51084 

.687 28 

455 01 

• 7 1 3 29 

1.4°i 95 

30 

31 

•637 48 

.56868 

.662 3 

.50988 

.687 71 

• 454 i 

•71373 

.401 09 

29 

32 

.63789 

.56767 

.662 72 

•50893 

.68814 

•453 2 

.71417 

. 400 22 

28 

33 

.6383 

.56667 

•66314 

•507 97 

•68857 

.45229 

.71461 

•399 36 

27 

34 

.638 71 

.565 66 

■663 56 

• 507 02 

.689 

•45139 

•71505 

•3985 

26 

35 

•63912 

1.564 66 

.663 98 

1.506 07 

.689 42 

1.450 49 

•71549 

1.397 64 

25 

36 

•639 53 

• 563 66 

.66 4 4 

•505 12 

.68985 

•449 58 

•71593 

•396 79 

24 

37 

.63994 

.56265 

.664 82 

•50417 

.690 28 

.448 68 

.71637 

•395 93 

23 

38 

•64035 

.561 65 

.665 24 

.50322 

.690 71 

.44778 

716 81 

•395 07 

22 

39 

. 640 76 

•56065 

.665 66 

.502 28 

.691 14 

.446 88 

•71725 

•394 2 i 

21 

40 

.641 17 

1.559 66 

. 666 08 

1*501 33 

•69157 

,1.445 98 

.71769 

i -393 36 

20 

4 i 

.64158 

.55866 

.6665 

• 5 °o 38 

692 

.44508 

.71813 

•392 5 

19 

42 

.64199 

•55766 

.666 92 

• 499 44 

. 692 43 

.44418 

• 7 i8 57 

• 39 i 65 

18 

43 

.642 4 

• 556 66 

.66734 

.49849 

.692 86 

•443 29 

.719 OI 

.390 79 

!7 

44 

.642 81 

•55567 

.667 76 

■49755 

.693 29 

.442 39 

.71946 

.38994 

16 

45 

.643 22 

1-55467 

.66818 

1.496 61 

.69372 

I - 44 I 49 

.7199 

1.389 09 

15 

46 

.643 63 

•55368 

.668 6 

•49566 

.694 16 

.440 6 

.720 34 

.388 24 

14 

47 

. 644 04 

•55269 

. 669 02 

•494 72 

694 59 

•439 7 

.72078 

•387 38 

13 

48 

.644 46 

• 55 i 7 

. 669 44 

•49378 

.69502 

.43881 

.721 22 

•38653 

12 

49 

.64487 

•55071 

. 669 86 

.492 84 

•69545 

•437 9 2 

.721 66 

.385 68 

11 

5 o 

.645 28 

I - 549 7 2 

.670 28 

I - 49 I 9 

.695 88 

1-437 03 

.722II 

1.384 84 

10 

SI 

.64569 

•54873 

.670 71 

.49097 

.696 31 

•43614 

•72255 

•383 99 

9 

52 

.646 1 

•54774 

.67113 

.490 03 

.69675 

•43525 

.722 QQ 

•383 14 

8 

53 

.646 52 

•54675 

•67155 

.489 09 

697 l8 

•434 36 

•723 44 

.382 29 

7 

54 

.64693 

54576 

•67197 

.488 16 

•697 6l 

•433 47 

.723 88 

•381 45 

6 

55 

•647 34 

1.54478 

•67239 

1.487 22 

.698 04 

1.432 58 

•724 32 

1.380 6 

5 

56 

•647 75 

•543 79 

.672 82 

.486 29 

.69847 

.43169 

.72477 

•37976 

4 

57 

.648 17 

.542 81 

.67324 

•48536 

.698 9I 

.4308 

.725 21 

•37891 

3 

58 

.64858 

•54183 

■ 673 66 

.484 42 

.699 34 

.429 92 

•72565 

.37807 

2 

59 

.648 99 

• 540 85 

.67409 

.483 49 

•699 77 

.42903 

. 726 I 

.37722 

1 

60 

.64941 

1.539 86 

•67451 

1.482 56 

. 7OO 21 

1.42815 

.726 54 

1.37638 

0 

/• 

CO-TANG. 

Tang. 

Co-tang. 

Tang. 

! CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

/ 


57 ° 

56 0 

j 55 ° 

54 ° 






















































424 NATURAL TANGENTS AND COTANGENTS. 



36 ° 

I 37 ° 

38 ° 

39 ° 


/ 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

| CO-TANG. 

/ 

0 

.72654 

1.37638 

•753 55 

I.327O4 

.781 29 

I.27994 

.809 78 

I.2349 

60 

I 

.726 99 

•375 54 

• 754 oi 

.326 24 

•78175 

.279 17 

. 810 27 

.23416 

59 

2 

•727 43 

•3747 

•754 47 

• 3 2 5 44 

.782 22 

.278 41 

.81075 

•233 43 

58 

3 

.727 88 

•37386 

•754 9 2 

.32464 

.782 69 

.277 64 

.8u 23 

.2327 

57 

4 

.72832 

•373 02 

•75538 

•32384 

.78316 

.276 88 

.811 71 

.23196 

56 

5 

.728 77 

1.37218 

•75584 

I.323O4 

•78363 

1.276 11 

.812 2 

I‘23I 23 

55 

6 

. 729 21 

•37134 

.75629 

.32224 

.7841 

•27535 

.812 68 

• 230S 

54 

7 

.729 66 

•370 5 

•75675 

•321 44 

•784 57 

.27458 

.813 16 

. 229 77 

53 

8 

• 73 ° 1 

.36967 

•757 2 i 

.32064 

.78504 

.27382 

.81364 

.229 04 

52 

9 

• 73 ° 55 

•368 83 

•75767 

.31984 

•7855* 

.273 06 

.81413 

.228 31 

5 i 

10 

• 73 i 

1.368 

.75812 

1.31904 

.78598 

1.272 3 

. 814 61 

1.227 58 

50 

11 

• 73 M 4 

•36716 

•75858 

•3*825 

.78645 

•271 53 

.8151 

. 226 85 

49 

12 

•73189 

•36633 

•75904 

•31745 

.786 92 

.270 77 

.81558 

.226 12 

48 

13 

•73234 

•36549 

•7595 

.316 66 

•78739 

.27001 

.816 06 

•225 39 

47 

14 

.732 78 

1.364 66 

•759 96 

.31586 

. 787 86 

.269 25 

•81655 

.224 67 

46 

i 5 

•733 23 

■363 83 

.760 42 

1-31507 

.78834 

1.268 49 

.81703 

I.22394 

45 

16 

•73368 

•363 

.760 88 

•31427 

.788 81 

.267 74 

.81752 

.223 21 

44 

17 

•734 13 

.36217 

•76134 

•3*348 

. 789 28 

.266 98 

.818 

.222 49 

43 

18 

•734 57 

•36133 

.761 8 

.312 69 

•78975 

.266 22 

.818 49 

.221 76 

42 

*9 

•735 02 

•36051 

.762 26 

•3119 

.790 22 

.26546 

.81898 

.221 04 

4 i 

20 

•73547 

1.35968 

.762 72 

1.3H 1 

• 79 ° 7 

1.264 71 

.819 46 

1.220 31 

40 

21 

•73592 

•35885 

.76318 

• 3 I ° 3 I 

.791 17 

•263 95 

.81995 

.21959 

39 

22 

•736 37 

.35802 

.763 64 

•30952 

.79164 

•26319 

. 820 44 

.218 86 

38 

23 

•736 81 

•357 19 

.7641 

•30873 

.79212 

.262 44 

.82092 

.21814 

37 

24 

•737 26 

•35637 

.764 56 

•307 95 

• 79 2 59 

.261 69 

.821 41 

.217 42 

36 

25 

• 7377 i 

1-355 54 

.76502 

1.30716 

.79306 

1.260 93 

.821 9 

1.216 7 

35 

26 

•73816 

•354 72 

.76548 

•30637 

•79354 

.260 18 

.822 38 

.215 98 

34 

27 

.73861 

•35389 

•76594 

■30558 

.79401 

•259 43 

.822 87 

.215 26 

33 

28 

• 739 °6 

•35307 

.7664 

.3048 

•794 49 

.25867 

.823 36 

.21454 

32 

29 

•739 51 

•35224 

.766 86 

.30401 

.79496 

.25792 

.823 85 

.213 82 

3 i 

30 

•739 96 

I- 35 I 42 

•767 33 

1.303 23 

•795 44 

1.257 *7 

.82434 

1-213 1 

30 

3 i 

740 41 

■3506 

.76779 

.302 44 

•795 9 1 

.25642 

.82483 

.21238 

29 

32 

.740 86 

•34978 

.76825 

. 301 66 

.79639 

•25567 

.825 31 

.21166 

28 

33 

• 74 i 31 

.34896 

.768 71 

• 3 °° 87 

.796 86 

.25492 

.8258 

.210 94 

27 

34 

.74176 

•34814 

.769 18 

.300 O9 

•797 34 

•254 *7 

.826 29 

.210 23 

26 

35 

.742 21 

1-347 32 

.76964 

1.299 31 

.79781 

1-253 43 

. 826 78 

1.209 51 

25 

36 

.74267 

•3465 

• 77 oi 

•29853 

.798 29 

.252 68 

. 827 27 

.208 79 

24 

37 

•74312 

•34568 

•77057 

•297 75 

.79877 

•25193 

. 827 76 

.208 08 

23 

38 

•743 57 

•34487 

.77103 

. 296 96 

.79924 

.25118 

.828 25 

.207 36 

22 

39 

•744 02 

•344 05 

• 77 M 9 

.296 l8 

•799 72 

.25044 

.828 74 

.206 65 

21 

40 

•74447 

I -343 23 

.77196 

I .29541 

. 800 2 

1.249 69 

.82923 

1.205 93 

20 

4 i 

.74492 

.34242 

.77242 

.29463 

.80067 

•2489s 

.829 72 

.205 22 

1 9 

42 

•74538 

.3416 

.772 89 

•29385 

.801 15 

.248 2 

. 830 22 

.20451 

18 

43 

•74583 

•340 79 

•773 35 

•293 07 

.801 63 

.24746 

.830 71 

.20379 

17 

44 

.746 28 

•339 98 

.77382 

.292 29 

. 802 11 

.246 72 

.8312 

.203 08 

16 

45 

.746 74 

1.339 16 

.77428 

I- 29 I 52 

.802 58 

1.245 97 

.83169 

1.202 37 

15 

46 

•747 19 

•33835 

•77475 

.290 74 

. 803 06 

•24523 

.832 18 

. 201 66 

14 

47 

•747 64 

•337 54 

•77521 

. 289 97 

• 80354 

•244 49 

.832 68 

.20095 

13 

48 

.7481 

•33673 

.77568 

.289 I9 

.80402 

•243 75 

•833 *7 

. 200 24 

12 

49 

•74855 

•335 92 

•77615 

. 288 42 

• 8045 

.24301 

.83366 

•199 53 

11 

50 

•749 

i -335 11 

.776 61 

I. 287 64 

. 804 98 

1.24227 

•834*5 

1.198 82 

10 

5 i 

.74946 

•334 3 

.77708 

.286 87 

. 805 46 

•241 53 

■83465 

w 

VO 

OO 

M 

H 

9 

52 

•749 9 1 

•333 49 

•777 54 

.286 I 

• 805 94 

.240 79 

•835 *4 

.197 4 

8 

53 

•750 37 

.33268 

.778 01 

•285 33 

. 806 42 

.24005 

•83564 

.I96 69 

7 

54 

.75082 

•33187 

.77848 

.284 56 

.806 9 

•23931 

•83613 

•i 95 99 

6 

55 

•75 1 28 

1-33107 

•77895 

*•383 79 

. 807 38 

1.23858 

.836 62 

1.195 28 

5 

56 

•75173 

.33026 

•77941 

.283 02 

. 807 86 

.23784 

.83712 

•*94 57 

4 

57 

•75219 

.32946 

.779 88 

.282 25 

.80834 

•237 1 

. 837 6l 

.19387 

0 

0 

58 

.75264 

.32865 

•78035 

.281 48 

.808 82 

•23637 

.838 11 

.193 16 

2 

59 

•753 1 

•32785 

.780 82 

. 280 71 

.8093 

•23563 

.8386 

. I92 46 

1 

60 

•753 55 

1.32704 

.781 29 

I - 2 79 94 

.809 78 

I * 2 34 9 

•839 1 

I * I 9 I 75 

0 


CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

/ 


53 ° 

52 ° 

51 ° 

50 ° 



























































NATURAL TANGENTS AND CO-TANGENTS. 425 



40 ° 

41 ° 

420 

430 


/ 

Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

f 

0 

•839 1 

x - I 9 I 75 

.869 29 

x - 1 5 ° 37 

.900 4 

I. IIO 61 

•932 52 

I.O72 37 

60 

I 

.8396 

.19105 

.8698 

.149 69 

• 9 °° 93 

. IO9 96 

•933 06 

. 07 x 74 

59 

2 

. 840 09 

• 19035 

.87031 

. I49 02 

.901 46 

.IO93I 

•933 6 

.071 12 

58 

0 

D 

.840 59 

. 189 64 

. 870 82 

.148 34 

.901 99 

. I08 67 

• 934 x 5 

.07049 

57 

4 

.841 08 

. 188 94 

•871 33 

.14767 

.902 51 

. I08 02 

•934 69 

.069 87 

56 

5 

.841 58 

1.188 24 

. 871 84 

1.146 99 

• 9°3 04 

I.IO7 37 

•935 24 

1.069 25 

55 

6 

. 842 08 

•18754 

. 872 36 

.146 32 

• 9°3 57 

. 106 72 

•935 78 

.068 62 

54 

7 

.842 58 

. 186 84 

. 872 87 

•14565 

.904 1 

. 106 07 

•936 33 

.068 

53 

8 

.84307 

.18614 

•87338 

.14498 

.90463 

•105 43 

.93688 

.06738 

52 

9 

•843 57 

.18544 

•87389 

• x 44 3 

• 905 x6 

. 104 78 

■937 42 

.066 76 

5 1 

IO 

.844 07 

1.184 74 

.87441 

x - x 43 6 3 

.905 69 

I. IO4 14 

•937 97 

1.06613 

50 

II 

•844 57 

.18404 

.87492 

. 142 96 

.906 21 

.10349 

•93852 

•06551 

49 

12 

.84507 

• x8 3 34 

•875 43 

. 142 29 

.90674 

. 102 85 

.93906 

.064 89 

48 

13 

.84556 

.182 64 

•875 95 

. 141 62 

.9O7 27 

. 102 2 

.93961 

.064 27 

47 

14 

. 846 06 

. 181 94 

. 876 46 

.14095 

.907 81 

. 101 56 

.94016 

• 06365 

46 

15 

.846 56 

1.18125 

. 876 98 

1.140 28 

•90834 

1.100 91 

94071 

1.06303 

45 

l6 

. 847 06 

•18055 

.87749 

.13961 

.908 87 

. 100 27 

.941 25 

.062 41 

44 

17 

.847 56 

. 179 86 

.878 01 

.13894 

•909 4 

.099 63 

.941 8 

.06179 

43 

18 

.848 06 

. 179 16 

.87852 

. 138 28 

•909 93 

.098 99 

•942 35 

.06117 

42 

x 9 

.848 56 

. 178 46 

.87904 

.137 61 

.910 46 

•09834 

•942 9 

.060 56 

4 1 

20 

.849 06 

x -177 77 

•879 55 

1.13694 

.91099 

1.0977 

•943 45 

1.05994 

40 

21 

.84956 

. 177 08 

.880 07 

.136 27 

• 9 XX 53 

.097 06 

•944 

•05932 

39 

22 

.85006 

.17638 

.88059 

• x 356 i 

.912 06 

.096 42 

•944 55 

•0587 

38 

23 

•85057 

•17569 

.881 1 

• x 34 94 

.91259 

.09578 

•945 i 

.05809 

37 

24 

.851 07 

• x 75 

.881 62 

.13428 

• 9 X 3 x 3 

•095 x 4 

•94565 

•05747 

36 

25 

•851 57 

i - x 74 3 

.88214 

1.13361 

.913 66 

1.0945 

.946 2 

1.056 85 

35 

26 

.85207 

.17361 

.882 65 

• x 32 95 

.91419 

,093 86 

.946 76 

.056 24 

34 

27 

•852 57 

. 172 92 

.88317 

.132 28 

• 9 X 4 73 

•093 22 

•947 3 X 

.055 62 

33 

28 

•85307 

.172 23 

.88369 

.131 62 

• 9 X 5 26 

.09258 

.947 86 

•05501 

32 

29 

•85358 

• I 7 I 54 

.884 21 

.130 96 

• 9 X 5 8 

.09195 

•948 41 

•05439 

3 X 

30 

.85408 

1-170 85 

•88473 

I. I3O 29 

•9 x6 33 

1.091 31 

.948 96 

1.05378 

30 

31 

•854 58 

.170 16 

.885 24 

.129 63 

.916 87 

.09067 

.94952 

•053 x 7 

29 

32 

•85509 

.169 47 

•88576 

. 128 97 

.9174 

.O9O O3 

•950 07 

•05255 

28 

33 

•855 59 

. 168 78 

.886 28 

.128 31 

• 9 X 7 94 

.089 4 

.95062 

.051 94 

27 

34 

.85609 

. 168 09 

.886 8 

• x2 7 65 

.91847 

.088 76 

• 95 x 18 

•051 33 

26 

35 

.8566 

1.167 41 

.887 32 

1.126 99 

.91901 

1.088 13 

• 95 x 73 

1.050 72 

25 

36 

•857 1 

. 166 72 

. 887 84 

.126 33 

• 9 X 9 55 

.08749 

.95229 

.050 1 

24 

37 

.85761 

. 166 03 

.88836 

.12567 

.92008 

.086 86 

.95284 

.049 49 

23 

38 

.858 11 

•16535 

.888 88 

.12501 

.920 62 

.086 22 

•9534 

.048 88 

22 

39 

.858 62 

. 164 66 

.8894 

•124 35 

.921 16 

•08559 

•95395 

.048 27 

21 

40 

.859 12 

1.163 98 

.889 92 

1.123 69 

.921 7 

..1.084 96 

•954 5 X 

1.047 66 

20 

41 

•85963 

.16329 

.89045 

.12303 

. Q 22 23 

.08432 

•955 o6 

•047 05 

x 9 

42 

.860 14 

. 162 61 

.89097 

. 122 38 

.922 77 

.08369 

•955 62 

.046 44 

18 

43 

.860 64 

. 161 92 

.891 49 

. 121 72 

• 9 2 3 3 X 

.083 06 

.95618 

•04583 

x 7 

44 

.86115 

.161 24 

.892 01 

.12106 

• 9 2 3 85 

.082 43 

•956 73 

• 045 22 

l6 

45 

.86166 

1.160 56 

.89253 

I. 120 41 

•924 39 

1.081 79 

•957 29 

1.044 61 

x 5 

46 

.862 16 

•159 8 7 

.89306 

• xx 9 75 

• 9 2 4 93 

.08116 

•957 85 

.044 01 

x 4 

47 

. 862 67 

• x 59 x 9 

.893 58 

. 119 O9 

•925 47 

.08053 

•95841 

•043 4 

x 3 

48 

.863 18 

•158 5 1 

.8941 

.11844 

.92601 

•079 9 

•95897 

.042 79 

12 

49 

.86368 

•15783 

.89463 

.11778 

•92655 

.O79 27 

•959 52 

.04218 

II 

5 ° 

.864 19 

i-i 57 x 5 

•89515 

1.117 13 

.927 O9 

1.078 64 

. 960 08 

1.041 58 

IO 

51 

.8647 

•15647 

.89567 

. 116 48 

•92763 

.078 01 

.960 64 

.04097 

9 

52 

.865 21 

•15579 

.896 2 

.115 82 

.92817 

•07738 

.961 2 

.04036 

8 

53 

.865 72 

•i 55 ii 

. 896 72 

• xx 5 x 7 

.928 72 

.076 76 

.961 76 

.03976 

7 

54 

. 866 23 

•15443 

•897 25 

.11452 

.929 26 

■07613 

.962 32 

•039 x 5 

6 

55 

. 866 74 

x - 1 53 75 

•89777 

1.113 87 

.929 8 

x -o 75 5 

. 962 88 

1-03855 

5 

56 

.867 25 

• 15308 

.8983 

.11321 

•930 34 

•074 87 

.96344 

•037 94 

4 

57 

. 867 76 

.1524 

.89883 

.11256 

.93088 

•07425 

.964 

•037 34 

3 

58 

.868 27 

.15172 

•89935 

. Ill 91 

• 93 x 43 

•07362 

•964 57 

•03674 

2 

59 

.86878 

.15104 

. 899 88 

.111 26 

• 93 x 97 

.072 99 

•965 x 3 

■036 13 

1 

60 

.869 29 

1-15037 

.900 4 

1. no 61 

.93252 

1.072 37 

.965 69 

x -°35 53 

0 

/ 

Co-tang, i Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

/ 


49° 

48° 

47° 

460 



N N* 















































426 


NATURAL TANGENTS AND CO-TANGENTS. 



440 



440 



440 


f 

Tang. 

CO-TANG. 

/ 

f 

Tang. 

CO-TANG. 


/ 

Tang. 

CO-TANG. 

/ 

0 

.96569 

1-035 53 

60 

21 

•977 56 

1.022 95 

39 

41 

.989 01 

I.OII 12 

D 

I 

.96625 

1-034 93 

59 

22 

•978 13 

1.022 36 

38 

42 

.98958 

1.01053 

18 

2 

.966 81 

1-034 33 

58 

23 

.9787 

1.02176 

37 

43 

. 990 16 

I.OO9 94 

17 

3 

.96738 

1.033 72 

57 

24 

.97927 

1.021 17 

36 

44 

• 99 ° 73 

1.00935 

16 

4 

.967 94 

I.O33 12 

56 

25 

.97984 

1.020 57 

35 

45 

• 99 1 3 i 

1.008 76 

15 

5 

.9685 

I.O32 52 

55 

26 

.980 41 

1.01998 

34 

46 

.991 89 

1.00818 

14 

6 

.969 07 

I 03I 92 

54 

27 

.980 98 

1.01939 

33 

47 

.99247 

1.007 59 

13 

7 

.96963 

I- 03 I 32 

53 

28 

• 9 Sl 55 

1.018 79 

32 

48 

•993 04 

I.OO7 OI 

12 

8 

.97O 2 

I.O3O72 

52 

29 

.98213 

1.018 2 

3 i 

49 

.99362 

1.006 42 

II 

9 

.970 76 

I.O3O 12 

5 i 

30 

.982 7 

1.017 61 

30 

50 

•994 2 

1.005 83 

10 

IO 

• 97 1 33 

1.029 52 

50 

31 

.98327 

I.OI7 02 

29 

51 

.99478 

1.005 25 

9 

II 

.97189 

1.028 92 

49 

32 

.983 84 

1.016 42 

28 

52 

•995 36 

1.004 67 

8 

12 

.97246 

1.028 32 

48 

33 

.98441 

1.01583 

27 

53 

•995 94 

1.004 08 

7 

13 

.97302 

I.O27 72 

47 

34 

.98499 

1.015 24 

26 

54 

.99652 

1-003 5 

6 

14 

•973 59 

1-02713 

46 

35 

•985 56 

1.01465 

25 

55 

•997 1 

1.002 91 

5 

15 

.97416 

1.02653 

45 

36 

.986 13 

1.01406 

24 

56 

.997 68 

1.00233 

4 

16 

.97472 

1.02593 

44 

37 

.986 71 

1.01347 

23 

57 

.998 26 

1.00175 

3 

17 

•975 29 

1.02533 

43 

38 

.987 28 

1.012 88 

22 

58 

. 998 84 

1.00116 

2 

l8 

.97586 

I.O24 74 

42 

39 

.987 86 

I.OI2 29 

21 

59 

•999 42 

1.000 58 

1 

D 

•976 43 

I.O24 14 

4 i 

40 

.98843 

i.oii 7 

20 

60 

I 

I 

0 

20 

•977 

1-02355 

40 









' 

CO-TANG. 

Tang. 

/ 

f 

CO-TANG. 

Tang. 

1 

r 

CO-TANG. 

Tang. 

/ 


1 45 ° 



45 ° 



450 



Preceding Table contains Natural Tangents and Co-tangents for every 
minute of the quadrant, to the radius of i. 

If Degrees are taken at head of columns, Minutes, Tangents, and Co-tan¬ 
gents must be taken from head also; and if they are taken at foot of col¬ 
umns, Minutes, etc., must be taken from foot also. 

Illustration.—. 1974 is tangent for n° 10', and co tangent for 78° 50'. 


To Compute Tangents and. Co-tangents for Seconds. 
Ascertain tangent or co-tangent of angle for degrees and minutes from 
Table; take difference between it and tangent or co-tangent next below it. 

Then as 60 seconds is to difference, so are seconds given to result required, 
which is to be added to tangent and subtracted from co-tangent. 
Illustration.— What is the tangent and co-tangent of 54 0 40' 40"? 

Tangent of 50 0 40', per Table = 1.41061) 0 

Tangent of 54° 4 D “ = 1.4x148 } - 000 8 ? 

Then 60 : .00087 4° : 00058, which, added to 1.41061 = 1.411x9 tangent. 

Co-tangent of 54° 40' per Table = . 708 91 ) difference 

Co-tangent of 54 0 41', “ =.70848! ■ 000 43 a W ei ence - 

Then 6o° : .00043 :: 40 : 29, which, subtracted from .70891 = .70862 co-tangent. 


To Compute Tangent, or Co-tangent of any- Angle in 
Degrees, NEinntes, and Seconds. 

Divide Sine by Cosine for Tangent, and Cosine by Sine for Co-tangent. 
Example. —What is tangent of 25 0 18'? 

Sine = .427 36; cosine = .904 08. Then ’ 4 2 7 36 __ tangent. 

. 904 08 

To Compute Number of Degrees, Minutes, and Seconds 
of a given Tangent or Co-tangent. 

When Tangent is given .—Proceed as by Rule, page 402, for Sines, substi¬ 
tuting Tangents for Sines. 

Example. —What is tangent for 1.41119? 

Next less tangent is 1.41061, arc for which is 54 0 40'. Next greatest tangent is 
x.411 48, difference betiveen which and next less is .00087. 

Difference between less tabular tangent and one given is 1.410 61 —1.41119 = .00058. 
Then .00087 : .00058 :: 60 : 40, which, added to 54040'= 54° 40' 40". 

When Co-tangent, is given .—Proceed as by Rule, page 402, for Cosines, 
substituting Co-tangents for Cosines. 































AEROSTATICS. 


42; 


AEROSTATICS. 

Atmospheric Air consists, by volume, of Oxygen 21, and Nitrogen 79 
parts; and in 10000 parts there are 4.9 parts of Carbonic acid gas. 
By weight, it consists of 77 parts of Oxygen, and 23 of Nitrogen. 

One cube foot of Atmospheric Air at surface of Earth, when barome¬ 
ter is at 30 ins., and at a temperature of 32 0 , weighs 565.0964 grains = 
.080 728 lbs. avoirdupois, being 773.19 times lighter than water. 

Specific gravity compared with water, at 62.418 = .001 293 345. 

Mean weight of a column of air a foot square, and of an altitude 
equal to height of atmosphere (barometer 30 ins.), is 2124.6875 lbs. = 
14.7548 lbs. per sq. inch = support of 34.0393 feet of water. 

Standard pound is computed with a mercurial barometer at 30 ins.; hence, 
as a cube inch of mercury at 6o° weighs .490 776 9 lbs., pressure of atmos¬ 
phere at 6o° = 14.723307 lbs. per square inch. 

12.3873 cube feet of air weigh a pound, and its weight varies about 
1 gr. for each degree of heat. 

Extreme height of barometer in latitude 30° to 35 0 N. = 30.21 ins. 

Rate of expansion of Air, and all other Elastic Fluids for all temperatures, 
is essentially uniform. From 32° to 212 0 they expand from 1000 to 1376 
volumes = .002 088 or 9th part of their bulk for every degree of heat. 
From 212 0 to 68o° they expand from 1376 to 2322=^.002021 for each de¬ 
gree of heat. 

Thus, if volume of air at 132 0 is required. 132 0 — 32 0 = 100, and 1000 
+ too X .002088= 1209 volumes. 

Height, at Equator is estimated at 300 feet greater than at Poles, its 
mean height at 45 0 latitude. 

In like latitudes, air loses about i° for every foot in height above level 
of sea. 

Below surface of Earth, temperature increases. 

Elasticity of air is inversely as space it occupies, and directly as its density. 

When altitude of air is taken in arithmetical proportion, its Rarity will be 
in geometric proportion. Thus, at 7 miles above surface of Earth, air is 4 
times rarer or lighter than at Earth’s surface; at 14 miles, 16 times; at 21 
miles, 64 times, and so on. 

Density of an aeriform fluid mass at 32 0 and at t° will be to each other 
as 1 + .002 088 ( t° — 32 0 ) is to 1. 

For Volume, Pressure, and Density of Air, see Heat, page 521. 

Altitude of Atmosphere at ordinary density is = a column of mercury 30 
ins. in height, divided by specific gravity of air compared with mercury. 

Hence 30 ins. = 2.5 feet, which, divided by .000094987, specific gravity 
of air compared with mercury, = 26319 feet=. 4.985 miles. 

Gay Lussac, Humboldt, and Boussingault estimated it at a minimum of 
30 miles, Sir John Herschell 83, Bravais 66 to 100, Dalton 102, and Liais at 
180 or 204 miles. 

The aqueous vapor always existing in air, in a greater or less quantity, 
being lighter than air, diminishes its weight in mixing with it; and as, other 
things equal, its quantity is greater the higher the temperature of the air, its 
effect is to be considered by increasing the multiplier of t by raising it to 
.002 22. 

Glaisher and Coxwell, in 1862, ascended in a balloon to a height of 37 000 
feet. 



428 


AEROSTATICS. 


At temperature of 32 0 , mean velocity of sound is 1089 feet per second. It 
is increased or diminished about one foot for each degree of temperature 
above or below 32 0 . 

Velocity of sound in water is estimated at 4750 feet per second. 


Velocity of Sound at Various Temperatures. 


0 

Per Second. 

O 

Per Second. 

O 

Per Second. 

O 

Per Second. 


Fe.et. 


Feet. 


Feet. 


Feet. 

5 

1056 

32 

1089 

68 

1122 

95 

1152 

14 

1070 

50 

1102 

77 

1132 

IO4 

1161 

23 

1079 

59 

1112 

86 

II42 

113 

II7I 


Motions of air and all gases , by force of gravity , arc precisely alike to 
those of fluids. 

Sensation of hearing, or sound, cannot exist in an absolute vacuum. The 
human voice can be heard a distance of 3300 feet. 

Echo. —At a less distance than 100 feet there is not a sufficient interval 
between the delivery of a sound and its reflection to render one perceptible. 


To Compute Velocity of Sound, through. ,A.ir. 

1089 X 13V1 + [-002 088 (t — 32J] = v in feet per second, t representing temperature 
of air. 

Illustration. —Flash of a cannon from a vessel was observed 13 seconds before 
report was heard; temperature of air 6o°; what was distance to vessel? 

1089 X 13V1 + [.002 088'(6o° — 32)] rz: 1089 X 13 X 1.029 == I 4 567.55 feet = 2.76 miles. 

Theoretical velocity with which air will flow into a vacuum, if wholly un¬ 
obstructed, is V2, g h = 1347.4 feet per second. In operation, however, it is 
1 347-4 X .707 = 952.61 feet. 


To Compute "Velocity- of -Adr Flowing into a "Vacuum. 


V2 g h X c = v in feet per second, c representing coefficient of efflux. 
Coefficients for openings are as follows: 

Circular aperture in a thin plate... .65 to .7 

Cylindrical adjutage .92 j Conical adjutage .93 


Lead. 
Gold. 


"Velocity- of Sound in Several Solids. 
Velocity in Air = 1. 

3-9 ' 


I Zinc... 

... 9.8 

Pine. 

. 12.5 I Glass .. 

.. 11. 9 I Steel... 


j Oak.... 

... 9.9 

Copper.. 

. 11.2 | Pine_ 

.. 12.5 | Iron ... 

.. 15.1 


To Compute Elevations Toy a Barometer. 
Approximately * 60000 (log. B — log. b) C = height in feet; B and b representing 
heights of barometer at lower and upper stations, and C correction due to T -f- t or 
temperatures of lower and upper stations. 


"Values of C or T-j-t. 


O 

C 

O 

C 

O 

C 

O 

c 

O 

c 

O 

C 

0 

c 

40 

•973 

60 

.996 

80 

1.018 

IOO 

I.04 

120 

1.062 

140 

1.084 

160 

1.106 

42 

.976 

62 

.998 

82 

1.02 

102 

I.O42 

122 

1.064 

I42 

1.087 

162 

1.108 

44 

.978 

64 

I 

84 

1.022 

IO4 

1.044 

124 

1.067 

x 44 

1.089 

164 

I. Ill 

46 

.98 

66 

1.002 

86 

I.O24 

106 

1.047 

126 

1.069 

146 

I.O9I 

166 

I. 113 

48 

.982 

68 

I.OO4 

88 

I.O27 

108 

1.049 

128 

I.O71 

148 

1.093 

168 

i.ii5 

50 

.984 

70 

I.OO7 

90 

I.O29 

no 

1-051 

130 

1.073 

150 

1.096 

17O 

I. I 17 

52 

.987 

72 

I.OO9 

92 

1-031 

112 

1.053 

132 

1.076 

152 

1.098 

172 

1.12 

54 

.989 

74 

I.011 

94 

1.033 

114 

1.056 

134 

1.078 

154 

I. I 

174 

I. 122 

5b 

•99 1 

76 

1.013 

96 

1.036 

Il6 

1.058 

136 

1.08 

156 

1.102 

176 

1.124 

58 

•993 

78 

1.016 

98 

1.038 

Il8 

1.06 

138 

1.082 

158 

I. IO4 

178 

1.126 


* For more exact formulas, see Tables and Formulas, by Capt. T. S. Lee, U. S. Top. Eng., 1853 .. 






























































AEROSTATICS. 


429 


Their values vary approximately .0011 per degree. 

Upper Station. Lower Station. 

Illustration. —Thermometer 70.4 77.6 

Barometer 23.66 3005 

0 = 77.6 + 70.4 = 1.093, log. B = 1.4778, log. 6 = 1.374. 

Then 60000 X (1-4778 —1-374) X 1.093 = 6807.2 feet. 

To Compute Elevations L>y a Thermometer. 

520 B + B 2 X C = height in feet. B representing temperature of water boiling at 
elevated station deducted from 212 0 . 

Correction for temperatures of air at lower and upper stations, or T + 1, to be taken 
from table, page 428, as before. 

Illustration.— Temperature of water boiling at upper station 192°; temperature 
of air 50 0 and 32 0 . C = 1.02. 

__2 

Then 520 x 212 —192 + 212 —192 X 1-02 = n 010 feet. 

To Compute Capacity of a Balloon , etc., see page 218. 

Barometer. 

Elevations lay Barometer Readings- (Astronomer Royal.) 

Mean Temperature of Air 50 0 . 

For correction for temperature, see note at foot. 


Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

. Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

O 

31 

600 

30-325 

1500 

29-34 

4000 

26.769 

7 000 

23-979 

50 

3°-943 

650 

30.269 

1600 

29.233 

4250 

26.524 

7500 

23-543 

loo 

30.886 

700 

30.214 

1750 

29.072 

4500 

26. 282 

8 000 

23 -ii 5 

150 

3°- 8 3 

750 

30 -I 59 

1800 

29.019 

4750 

26 042 

8 500 

22.695 

200 

3°-773 

800 

30.103 

2000 

28.807 

5000 

25.804 

9 000 

22.282 

250 

30.717 

850 

30.048 

2250 

28.544 

5250 

25-569 

9500 

21.877 

300 

30.661 

900 

29.993 

2500 

28.283 

5500 

25-335 

10 000 

21.479 

350 

30.604 

IOOO 

29.883 

2750 

28.025 

5750 

25.104 

10 500 

21.089 

400 

30.548 

I IOO 

29.774 

3000 

27.769 

6000 

24.875 

11 000 

20.706 

450 

30.492 

1200 

29.665 

3250 

27 - 5 I 5 

6250 

24.648 

XI 500 

20.329 

500 

30-436 

1300 

29-556 

3500 

27.264 

6500 

24.423 

12 OOO 

19-959 

550 

30-381 

I4OO 

29.448 

375 ° 

27.015 

6750 

24.2 

12 500 

I 9-952 


Barometer. 

Correction for Capillary Attraction to be added in Inches. 


Diameter of tube. 1 .6 

•55 

•5 

•45 

•4 

•55 

•3 

•25 

.2 

Correction, unboiled. .004 

005 

.007 

.OI 

v ; 014 

.02 

025 

04 

■059 

Correction, boiled.| .002 

.003 

OO4 

005 

.007 

01 

.014 

.02 

029 


To Compute Height. 

Rule. —Subtract reading at lower station from reading at upper station, difference 
is height in feet. 

Table assumes mean temperature of atmosphere to be 50 0 F. or xo° C. For other 
temperatures following correction must be applied. 

Add together temperatures at upper and lower station. If this sum, in degrees 
in F., is greater than ioo°, increase height by part for every degree of excess 
above ioo° ; if sum is less than ioo°, diminish height by -j(hid P art f° r ever y degree 
of defect from ioo°. Or if sum in C° is greater than 20 0 , increase height by 
part for every degree of excess above 20 0 ; if sum is less than 20 0 , diminish height 
by -g-R. part for every degree of defect from 20 0 . 

Barometer Indications. 

Increasing storm. —If mercury falls during a high wind from S. W., S. S. W., W., 
or S. 

Violent but short.—If fall be rapid. 

Less violent but of longer continuance.—If hill be slow. 

Snow.—If mercury falls when thermometer is low. 

Improved weather.—When a gradual continuous rise of mercury occurs with a 
falling thermometer. 







































430 


AEROSTATICS. 


Heavy gales from N.— Soon after first rise of mercury from a very low point. 

Unsettled weather.—With a rapid rise of mercury. 

Settled weather.—With & slow rise of mercury. 

Very fine weather.—With a continued steadiness of mercury with dry air. 

Stormy weather with rain (or snow).—With a rapid and considerable fall of mer¬ 
cury. 

Threatening, unsettled weather.—With an alternate rising and falling of mercury. 

Lightning only. —When mercury is low, storm being beyond horizon. 

Fine weather.—With a rosy sky at sunset. 

Wind and rain.—When sky has a sickly greenish hue. 

Rain.—When clouds are of a dark Indian red. 

Foul weather or much wind.—When sky is red in morning. 

"WeatHer Grlasses. 

Explanatory Card. Vice-Admiral Fitzroy , F. R. S. 

Barometer Rises for Northerly wind (including from N. W. by N, to E.), for dry, 
or less wet weather, for less wind, or for more than one of these changes— 

Except on a few occasions when rain, hail, or snow comes from N. with strong wind. 

Barometer Falls for Southerly wind (including from S. E. by S. to W.), for wet 
weather, for stronger wind, or for more than one of these changes— 

Except on a few occasions when moderate wind with rain (or snow) comes from N. 

For change of wind toward Northerly directions, a Thermometer falls. 

For change of wind toward Southerly directions, a Thermometer rises . 

Moisture or dampness in air (shown by a Hygrometer) increases before rain, fog, 
or dew. 

Add one tenth of an inch to observed height for each hundred feet Barometer is 
above half-tide level. 

Average height of Barometer, in England, at sea-level, is about 29.94 inches; and 
average temperature of air is nearly 50 degrees (latitude of London). 

Thermometer falls about one degree for each 300 feet of elevation from ground, 
but varies with wind. 

“ When the wind shifts against the sun, 

Trust it not, for back it will run.” 

First rise after very low I Long foretold—long last, 

Indicates a stronger blow. | Short notice—soon past. 

Rarefaction of Air. 

In consequence of rarefaction of air, gas loses of its illuminating power 1 cube 
inch for each 2.69 feet of elevation above the sea. (M. Bremond.) 

Clcmds. 

Classification. —1. Cirrus —Like to a feather, commonly termed Mare’s 
tails. 2. Cirro-cumulus — Small round clouds, termed mackerel shy. 
3. Cirro-stratus —Concave or undulated stratus. 4. Cumulus —Conical, 
round clusters, termed wool-packs and cotton balls. 5. Gumulo-stratus — 
Two latter mixed. 6. Nimbus —A cumulus spreading out in arms, and 
precipitating rain beneath it. 7. Stratus —A level sheet. 

Note.— Cirrus is most elevated. 

Height .—Clouds have been seen at a greater height than 37000 feet. 

Velocity .—At an apparent moderate speed, they attain a velocity of 80 
miles per hour. 

E igli t it ing. 

Classification. —1 Striped or Zigzag —Developed with great rapidity. 
2. Sheet —Covering a large surface. 3. Globular —When the electric 
fluid appears condensed, and it is developed at a comparatively lower 
velocity. 4. Phosphoric — When the flash appears to rest upon the 
edges of the clouds. 


AEROSTATICS.-ATMOSPHERIC AIR. 


431 


Weather. 
Fine and 
Fair. 

Wind. 


Wind only. 

Rain. 
Wind and 
Rain. 


WEATHER INDICATIONS. 


Clouds. 

Soft or delicate-looking and in¬ 
definite outlines. 

Hard - edged, oily - looking, and 
tawny or copper-colored, and the 
more hard, “greasy,” and ragged, 
the more wind. 

Light scud alone. 

Small and inky. 

Light scud driving across heavy 
masses. 


Sky. 

Gray in morning and light, 
delicate tints and low dawn. 

High dawn, and sunset of a 
bright yellow. 


Sunset of a pale yellow. 
Orange or copper color. 


Rain and 
Wind. 


Change of 
Wind. 


Hard defined outlines. 

High upper, cross lower in a di¬ 
rection different to their course or 
that of wind. 


Gaudy unusual hues. 


General. 

Fair.—When sea-birds fly early and far out, when dew is deposited, and when’a 
leech, confined in a bottle of water, will curl up at the bottom. 

.Rain.—Clear atmosphere near to horizon and light atmospheric pressure, or a 
good “hearing day,” as it is termed. 

Storm.—When sea-birds remain near to shore or fly inland. 

Rain, Snow, or Wind.—When a leech, confined iu a bottle of water, will rise ex¬ 
citedly to the surface. 

Thunder.—When a leech, confined as above, will be much excited and leave the 
water. 

Value of Indications of Fair Weatlaer, in Days, Com¬ 
pared to one of ITain. 


Front an extended series of observations. {Lowe.) 


Profuse Dew.. 

White Stratus in a valley.., 
Colored Clouds at sunset.. 

Solar Halo. 

Sun red and rayless. 

Sun pale and sparkling.... 

White Frost. 

Lunar Halo. 

Lunar burr, or rough-edged 

Moon dim. 

Moon rising red. 


4-5 

7.2 
2.9 

1 * 9 
10.3 
1 

4.2 

1 

2.8 

2 

7 


Mock Sun or Moon..,.. 
Stars falling abundant.. 

Stars bright. 

Stars dim. 

Stars scintillated. 

Aurora borealis. 

Toads in evening. 

Landrails noisy. 

Ducks and Geese noisy. 

Fish rising. 

Smoke rising vertically. 


For weather-foretelling plants, see page 185. 


3-3 

3-2 

3-4 

1- 5 

6 

1.8 

2.4 

13 

2- 3 
i-5 
5 


ATMOSPHERIC AIR. 

Very pure air contains Oxygen 20.96, Nitrogen 79, and Carbonic Acid .04. 

Air respired by a human being in one hour is about 15 cube feet, produc¬ 
ing 500 grains of carbonic acid, corresponding to 137 grains carbon, and 
during this time about 200 grains of water will be exhaled by the lungs. 

During this period there would be consumed about 415 grains of oxygen. 

In one hour, then, there would be vitiated 73 cube feet pure air. 

A man, weighing 150 lbs., requires 930 cube feet of air per hour, in order 
that the air he breathes may not contain more than .1 per 1000 of carbonic 
acid (at which proportion its impurity becomes sensible to the nose); he 
ought, therefore, to have 800 cube feet of well ventilated space. 


























432 


ATMOSPHERIC AIR.-ANIMAL POWER. 


An adult human being consumes in food from 145 to 165 grains of carbon 
per hour, and gives off from 12 to 16 cube feet of carbonic acid gas. 

An assemblage of 1000 persons will give off in two hours, in vapor, 8.5 
gallons water, and nearly as much carbon as there is in 56 lbs. of bitumi¬ 
nous coal. 

Proportion of Oxygen, and. Carbonic ^Acicl at following 

Locations. 


Pure Air represented by Oxygen 20.96. 


Street in Glasgow.20.895 

Regent Street, London.20.865 

Centre Hyde Park.21.005 


Metropolitan Railway (underground).. 20.6 


Pit of a Theatre.20.74 

Gallery of a Theatre.20.63 


Carbonic Acid .04 Per cent. 


Open field, Manchester.0383 

Churchyard.0323 

Market, Smithfield.0446 

Factory mills.283 

School-rooms.097 

Pitt of theatre, n P. M.32 

Boxes “ 12 “ .218 

Gallery “ 10 “ .101 


* Roscoe. 


Top of Monument, London.0398 

Hyde Park.0334 

Metropolitan Railway (underground).. .338 

Lake of Geneva.046 

Boys 1 2 3 school.31* 

Girls’ “ .7231- 

Horse stable.7 

Convict prison.045 

t Peltenhoffer. 


Consumption of Atmospheric .A.ir, (Coathupe.) 

One wax candle (three in a lb.) destroys, during its combustion, as much 
oxygen per hour as respiration of one adult. 

A lighted taper, when confined within a given volume of atmospheric air, 
will become extinguished as soon as it has converted 3 per cent, of given 
volume of air into carbonic acid. 


Carbonic Acid Exhaled per Minute by a Man. (Dr. Smith.) 

During sleep 4,99 per cent., lying down 5.91, walking at rate of 2 miles 
per hour 18.1, at 3 miles 25.83, hard labor 44.97. 


ANIMAL POWER. 

Work. 

Work is measured by product of the resistance and distance through 
which its point of application is moved. In performance of work by 
means of mechanism, work done upon weight is equal to work done by 
power. 

Unit of Work is the moment or effect of 1 pound through a distance 
of 1 foot, and it is termed a foot-pound. 

In France a kilogrammetre is the expression, or the pressure of a 
kilogramme through a distance of 1 meter = 7.233 foot-pounds. 

Result of observation upon animal power furnishes the following as maximum 
daily effect: 

1. When effect produced varied from .2 to .33 of that which could be produced 
without velocity during a brief interval. 

2. When the velocity varied from .16 to .25 for a man, and from .08 to .066 for a 
horse, of the velocity which they were capable for a brief interval, and not involv¬ 
ing any effort. 

3. When duration of the daily work varied from .33 to .5 for a brief interval, 
during which the work could be constantly sustained without prejudice to health 
of man or animal; the time not extending beyond 18 hours per day, however lim¬ 
ited may be the daily task, so long as it involved a constant attendance. 

























ANIMAL POWER. 


433 


Men. 

Mean effect of power of men working to best practicable advantage, is 
raising of 70 lbs. 1 foot high in a second, for 10 hours per day = 4200 foot¬ 
pounds per minute. 

Windlass. —Two men, working at a windlass at right angles to each other, 
can raise 70 lbs. more easily than one man can 30 lbs. 

Labor. —A man of ordinary strength can exert a force of 30 lbs. for 10 
hours in a day, with a velocity of 2.5 feet in a second = 4500 lbs. raised one 
foot in a minute = .2 of work of a horse. 

A man can travel, without a load, on level ground, during 8.5 hours a day, 
at rate of 3.7 miles an hour, or 31.25 miles a day. He can carry in lbs. 
11 miles in a day. Daily allowance of water, 1 gallon for all purposes • and 
he requires from 220 to 240 cube feet of fresh air per hour. 

A porter going short distances, and returning unloaded, can carry 135 lbs. 
7 miles a day, or he can transport, in a wheelbarrow, 150 lbs. 10 miles in a 
day. 

Crane. —The maximum power of a man at a crane, as determined by Mr. 
Field, for constant operation, is 15 lbs., exclusive of frictional resistance, 
which, at a velocity of 220 feet per minute = 3300 foot-pounds, and when 
exerted for a period of 2.5 minutes was 17.329 foot-pounds per minute. 

Pile-driving. —G. B. Bruce states that, in average work at a pile-driver, a 
laborer, for 10 hours, exerts a force of 16 lbs., plus resistance of gearing, and 
at a velocity of 270 feet per minute, making one blow every four minutes* 

Rowing. —A man rowing a boat 1 mile in 7 minutes, performs the labor 
of 6 fully-worked laborers at ordinary occupations of 10 hours per day. 

Drawing or Pushing. —A man drawing a boat in a canal can transport 
110 000 lbs. for a distance of 7 miles, and produce 156 times the effect ©f a 
man weighing 154 lbs., and walking 31.25 miles in a day ; and he can push 
on a horizontal plane 20 lbs. with a velocity of 2 feet per second for 10 hours 
per day. 

Tread-mill. —A man either inside or outside of a tread-mill can raise 30 
lbs. at a velocity of 1.3 feet per second for 10 hours, = 1 404000 foot-pounds. 

Pulley. —A man can raise by a single pulley 36 lbs., with a velocity of .8 
of a foot per second, for 10 hours. 

Walking. —A man can pass over 12.5 times the space horizontally that he 
can vertically, and, according to J. Robison, by walking in alternate directions 
upon a platform supported on a fulcrum in its centre, he can, weighing 165 
lbs., produce an effect of 3 984 000 foot-pounds, for 10 hours per day* 

Pump, Crank, Bell , and Roicing. —Mr. Buchanan ascertained that, in work¬ 
ing a pump, turning a crank, ringing a bell, and rowing a boat, the effective 
power of a man is as the numbers 100, 167, 227, and 248. 

Pumping. —A practised laborer can raise, during 10 hours, 1 000 000 lbs* 
water 1 foot in height, with a properly designed and constructed pump. 

Crank. —A man can exert on the handle of a screw-jack of 11 inches ra¬ 
dius for a short period a force of 25 lbs., and continuously 15 lbs., a net 
power of 20 lbs. Mr. J. Field’s tests gave 11.5 lbs. as easily attained, 17.3 as 
difficult, and 27.6 with great difficulty. 

Mowing. —A man can mow an acre of grass in 1 day. 

Reaping. —A man can reap an acre of wheat in 2 days. 

Ploughing. —A man and horse .8 of an acre per day. 

0 o 


434 


ANIMAL POWER. 


Day’s "Worls. ( D. K. Clark.) 


Laborer.— Carrying bricks or tiles, net load 106 lbs.r=6oo lbs. i mile. 

Carrying coal in a mine, net load 95 to 115 lbs. = 342 lbs. 1 mile. 

Loading coke into a wagon, net load 100 lbs. = 270 lbs. 1 mile. 

Loading a boat with coal, net load 190 lbs.= 1230 lbs. 1 mile, or 20 cube yards of 
earth in a wagon. 

Digging stubble land .055 of an acre per day, or 2000 cube feet of superficial earth. 
Breaking 1.5 cube yards hard stone into 2 inch cubes. 

Quarrying .—A man can quarry from 5 to 8 tons of rock per day. 

A foot-soldier travels in 1 minute, in common time, 90 steps = 70 yards. 

He occupies in ranks a front of 20 inches, and a depth of 13, without a knapsack; 
interval between the ranks is 13 inches. 

Average weight of men, 150 lbs. each, and five men can stand in a space of 1 
square yard. 


Effective Dower of HVTen for a, Sliort Deriocl. 


Manner of Application. 


Force. 


Manner of Application. 


Force. 


Bench-vice or Chisel.... 
Drawing-knife or Auger. 

Hand-plane. 

Hand-saw. 


Lbs. 

72 

100 

50 

36 


Screw driver, one-liand 
Small screw-driver 

Thumb and fingers. 

Windlass or Pincers ... 


Lbs. 

84 

14 

M 

60 


The muscles of the human jaw exert a force of 534 lbs. 

Mr. Smeaton estimated power of an ordinary laborer at ordinary work was equiv¬ 
alent to 3762 foot-pounds per minute. But, according to a particular case made by 
him in the pumping of water 4 feet high, by good English laborers, their power was 
equivalent to 3904 foot-pounds per minute; and this he assigned as twice that of 
ordinary persons promiscuously operated with. 

Mr. J. Walker deduced from experiments that the power of an ordinary laborer, in 
turning a crank, was 13 lbs., at a velocity of 320 feet per minute for 8 hours per day. 


A.moviiit of Labor prod viced by a NI an. (Morin.) 


For 10 hours per day. 


MANNER OF APPLICATION. 


Throwing earth with a shovel, a height of 5 feet.. 
Wheeling a loaded barrow up an inclined plane, 

i to ..... 

Raising and pitching earth in a shovel 13 feet 

horizontally. 

Pushing and drawing alternately in a vertical 

direction. 

Transporting weight upon a barrow, and return¬ 
ing unloaded. 

For 8 Hours per Day. 

Ascending a slight elevation, unloaded. 

Walking, and pushing or drawing in a horizontal 

direction. . 

Turning a crank ....... 

Upon a tread mill*... 

Rowing. 

For 7 Hours per Day. 

Walking with a load upon his back. 

For 6 Hours per Day. 

Transporting a weight upon his back, and return¬ 
ing unloaded.. 

Transporting a weight upon his back up a slight 

elevation, and returning unloaded. 

Raising a weight by his hands. 


Power. 

Velocity 

per 

Second. 

Weight 
raised. 
Feet per 
Minute. 

pp 

for 

Period 

given. 

Lbs. 

Feet. 

Lbs. 

No. 

6 

i -33 

480 

8.7 

132 

.625 

4 950 

90 

6 

2.25 

810 

14.7 

13 

2-5 

1950 

35-5 

132 

I 

7 920 

144 

1 43 

•5 

4 290 

62 

26 

2 

3 120 

45-2 

l8 

2-5 

2 79O 

39 

140 

■5 

4 200 

6 l. I 

26 

5 

7 800 

XI 3 

88 

2-5 

13 200 

160.5 

140 

!•75 

14 700 

160.5 

140 

.2 

1680 

19 

44 

•5 

1320 

14.4 


* Morin gives amount, of labor of a man upon tread-mill, in an individual case, at 140 lbs., at a ve¬ 
locity of .5 feet per second for 8 hours per day —70 lbs. at x foot per second; hence 70-r-1.3 feet as 




































ANIMAL POWER. 


435 


To Compute Number of IVLen to Perform Work upon 
a Tread-mill or Tile-driver. 

Rule.—T o product of weight to be raised and radius of crank, add fric¬ 
tion of wheel, and divide sum by product of power and radius of wheel. 

Example.—H ow many men are required upon a tread-mill, 20 feet in diameter, 
to raise a weight of 9233.33 lbs., crank 9 inches in length, weight of wheel and its 
load estimated at 5000 lbs., and friction at .015. 

Weight of a man assumed at 25 lbs. Kadius of crank .75 feet. 

Effect of a man on a treadmill, page 433, 30 lbs. at a velocity of 1.3 feet per second, 
= 1.3 X 60 = 78 feet per minute. 

9 2 33 - 33 X .75 + 500° X .015 = 7000 lbs. resistance of load and wheel , and 7000-7- 
7 ^ 

— — - y X 10 x 30 = 7000 = toad dnd weight -4- product of power increased by its 

20 X 3- I4IO 

velocity over load, radius of tuheel and power — 7000 - 4 - 1.241 X 10 X 30 = 18.8 men. 

Horse. 


Amount of Labor prod.need, by a Horse under different 

C i re umst ances. ( Morin .) 

For 10 hours per day. 


MANNER OF APPLICATION. 

Power. 

Velocity 

per 

Second. 

Weight 
drawn. 
Feet per 
Minute. 

IP 

for 

Period 

given. 


Lbs. 

Feet. 

Lbs. 

No. 

Drawing a 4-wheeled carriage at a walk. 

154 

3 

27 720 

5°4 

With load upon his back at a walk. 

264 

3-75 

59 400 

1080 

Transporting a loaded wagon, and return.ng un- 


loaded at a walk. 

I 54° 

2 

184 800 

3360 

Drawing a loaded wagon at a walk. .... 

1540 

3-75 

346 500 

6300 

For 8 Hours per Day. 





Upon a revolving platform at a walk. 

IOO 

3 

18 COO 

260.8 

For 4.5 Hours per Day. 





Upon a revolving platform at a trot. 

66 

6-75 

26730 

218. 7 

Drawing an unloaded 4-wheeled carriage at a trot. 

97 

7-25 

43 i95 

353-5 

Drawing a loaded 4-wheeled carriage at a trot. 

77° 

7-25 

334 950 

2741 


If traction power of a horse, when continuously at a walk, is equal to 120 lbs., 
and grade of road 1 in 30, resistance on a level being one thirtieth of load, he can 
draw a load of 120 X 30 - 4 - 2 = 1500 lbs. 


Street Rails or Tramways. (Henry Hughes.) 
Cars, 26 lbs. per ton, or 1 to 86 as a mean. 


Performance of Horses in France. (M. Charie-Marsaines.) 




Weight 

Speed 

Work per 

Ratio of 

SEASON. 


Road. 

per 

per 

Hour, drawn 

Pavement to 


Horse. 

Hour. 

One Mile. 

Macadam. 



Tons. 

Miles. 

Ton-miles. 


Winter. 


Pavement 

Macadam 

1.306 

.851 

2.05 

2.677 \ 
1.625 \ 

1.644 I 



1.91 

Summer. 

J 

Pavement 

Macadam 

1-395 

2.17 

3.027) 

2.464) 

1.229 I 


.1 

I. I4I 

2.16 


Average daily work of a Flemish horse in North of France, where country is flat 
and loads heavy, is, on same authority, as follows: 

Winter, 21.82 ton-miles per day. | ^ ean f or year, 25. 

Summer, 27.82 *‘ j 


given in example = 53.8 lbs., from which a deduction is to be made for excess of amount of labor that 
can be performed in 8 hours over 10. Or, as 10 : 8 ;; 53.8 : 43.04 lbs., which does not essentially differ 
from effect of 30 lbs. for that of an average performance. 






































436 


ANIMAL POWER. 


Greatest mechanical effect of an ordinary horse is produced in operating a 
gin or drawing a load on a railroad, when travelling at rate of 2.5 miles per 
hour, where he can exert a tractive force of 150 lbs. for 8 hours per day. 

Horse upon Turnpike Road. 

At a speed of 10 miles per hour, a horse will perform 13 miles per day for 
3 years. In ordinary staging, a horse will perform 15 miles per day. 


To Compute Tractive Power of a Horse Team , see Traction , page 848. 
Assuming maximum load that a horse can draw on a gravel road as a 


standard, he can draw, 

On best-broken stone road. 2 to 3 times. 

On a well-made stone pavement. 3 to 5 “ 

On a stone trackway. 7 to 8 “ 

On plank road. 4 to 12 “ 

On a railway. 18 to 20 “ 


Note.—T rack of an iron railway compared with a plank-road is as 27 to 10. 

To Compute Power of Draught of a Horse at Different 

Elevations. 


R Let. ABC represent an inclined plane, 0 weight 
of a horse which, being resolved into two com¬ 
ponent forces, one of which, w, is perpendicular to 
c plane of inclination, and other, r, is parallel to it. 
Hence, r represents force which horse must over¬ 
come to move his own weight. 

Then, by similar triangles, A C or l : B C or h :: 0 : r. Or, — = r. 

If t represents tractive power of horse, upon a level, of 100 lbs., t tractive 
power upon a plane of inclination, and r that part of force exerted by horse 

which is expended upon his own body, then t = t — r, or t — ~ = t! in lbs. 

Illustration.— If incliuation is 1 in 50. 

Assume t— 100, weight of horse 900 lbs., and l = 50.01. 

m, I X 900 

then, 100-— 100 — 17.99 = 82.01 lbs . 

50.01 ' 

. Assuming load that a horse can draw on a level at 100, he can draw upon 
inclinations as follows : 



x in 

IOO. . . 

.. 91 

1 in 75.. 

.. 88 

1 in 50... 


1 in 

35- • 


I “ 

90... 

..go 

1 “ 70... 

.. 87 

1 “ 45... 


1 “ 

30.. 


I U 

80. . . 

.. 89 

i “ 60.. 

..85 

1 “ 40... 


1 “ 

25... 

.'.64 


On his back a horse can carry from 220 to 390 lbs., or about 27.“; per cent, 
of Ins weight. 1 


Labors -'The work of a horse as assigned by Boulton & Watt, Tredgold, 
Ken me, Beardmore, and others, ranges from 20 000 to 30 320 foot-pounds per 
minute for 8 hours, a mean of 27 750 lbs. 

A hoise can travel, at a w T alk, 400 yards in 4.5 minutes; at a trot, in 2 
minutes; and at a gallop, in 1 minute. He occupies in ranks, a front of 40 
ms., and a depth of 10 feet; in a stall, from 3.5 to 4.5 feet front; and at a 
picket, 3 feet by 9; and his average weight =: 1000 lbs. 

. Cariying a soldier and his equipments (225 lbs.) he can travel 23 miles 
m a day of 8 hours. 

1 ^ draught-horse can draw 1600 lbs. 23 miles a day, weight of carriage in- 





























ANIMAL POWER. 437 

Ordinary work of a horse may be stated at 22 500 lbs., raised 1 foot in a 
minute, for 8 hours per day. 

In a mill, he moves at rate of 3 feet in a second. Diameter of track should not 
be less than 25 feet. 

Rennie ascertained that a horse weighing 1232 lbs. could draw a canal-boat 
at a speed of 2.5 miles per hour, with a power of 108 lbs., 20 miles per day. 
This is equivalent to a work of 23 760 foot-lbs. per minute, He estimated 
that the average work of horses, strong and weak, is at the rate of 22000 
foot-lbs. per minute. 

From results of trials upon strength and endurance of horses at Bedford, Eng., it 
was determined that average work of a horse — 20000 foot-lbs. per minute. A good 
horse can draw 1 ton at rate of 2.5 miles per hour, from 10 to 12 hours per day. 

Expense of conveying goods at 3 miles per hour, per horse teams being 1, expense 
at 4.33 miles will be 1.33, and so on, expense beiDg doubled when speed is 5.125 miles 
per hour. 

Strength of a horse is equivalent to that of 5 men, and his daily allowance of 
water should be 4 gallons. 

-A.mou.iit of Labor a, Horse of average Strength, is capa¬ 
ble of performing, at different Velocities, on Canal, 
Railroad, and Turnpike. 


Traction estimated at 83.3 lbs. 


Veloci- 

Dura- 

Useful Effect, drawn i Mile. 

Veloci- 

Dura- 

Useful Effect, drawn 1 Mile. 







ty per 

tion of 

On a 

On a Rail- 

On a Turn- 

ty per 

tion of 

On a 

On a Rail- 

On a Turn- 

Hour. 

Work. 

Canal. 

road. 

pike. 

Hour. 

Work. 

Canal. 

road. 

pike. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

2-5 

n -5 

520 

115 

14 

6 

2 

3 ° 

48 

6 

3 

8 

243 

92 

12 

7 

i -5 


41 

5 -i 

4 

4-5 

102 

72 

9 

8 

1.125 

12.8 

36 

4-5 

5 

2.9 

52 

57 

7.2 

IO 

•75 

6.6 

28.8 

3-6 


Actual labor performed by horses is greater, but they are injured by it. 

Tractive Power of a horse decreases as his speed is increased, and within limits 
of low speed, or up to 4 miles per hour, it decreases nearly in an inverse ratio. 


For 10 Hours per Day. 


Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction 

Per Hour. 

Lbs. 

Per Hour. 

Lbs. 

Per Hour. 

Lbs. 

Per Hour. 

Lbs. 

75 

330 

1-5 

165 

2.25 

no 

3 

82 

I 

250 

1-75 

140 

2-5 

100 

3-5 

70 

1.25 

200 

2 

125 

2-75 

90 

4 

62 


For Ordinary or Short Periods. (Molesworth.) 

Miles per hour. 2 3 3-5 4 4-5 5 

Power in lbs. 166 125 104 83 62 41 


Miles per hour. 2 3 3-5 4 4-5 5 

Power in lbs. 166 125 104 83 62 41 


jVXnle. (D. K. Clark.) 

Load on back, 1 70 to 220 lbs. day’s work = 6400 lbs. 1 mile; 400 lbs. at 2.9 
miles per hour =: 5300 lbs. 1 mile, and 330 lbs. at 2 miles per hour = 5000 lbs. 
1 mile. 

Upon a revolving platform, at a velocity of 3 feet per second, = n 880 lbs. raised 
one foot per minute, or 172.2 IP for 8 hours per day 

.A.SS, 

Load on back , 176 lbs. carried 19 miles day’s work = 3300 lbs. 1 mile. 

In Syria an ass carries 450 to 550 lbs. grain. 

Upon a revolving platform, at a velocity of 2.75 feet per second, = 5280 lbs. raised 
one foot per minute, or 76.5 IP for 8 hours per day. 

0 O* 











































438 


ANIMAL POWER. 


Ox. 

An Ox, walking at a velocity of 2 feet in a second (1.36 miles per hour), 
exerts a power of 154 lbs., = 18480 lbs. raised one foot per minute, or 
268.8 IP for 8 hours per day. 

A pair of well-conditioned bullocks in India have performed work = 8000 foot-lbs. 
per minute. 

Camel. 

Load on bach , 550 lbs. carried 30 miles per day for 4 days, 4 days’ work 
16500 lbs. 1 mile, for 5 days 13000 lbs. 1 mile = 44 IP for 10 hours per day. 

Load of a Dromedary , 770 lbs. 

Llama. 


Load on bach, no lbs., day’s work 2000 to 3000 lbs. 1 mile = .5 to .75 IP 
for 10 hours per day. 

Birds and Insects. 


Area of their wing surface is in an inverse ratio to their weight. 

Assuming weight of each of the following Birds to be one pound, and each Insect 
one ounce, the relative area of their wing surface proportionate to that of their act¬ 
ual weight would be as follows ( M. De Lucy): 


Sq. ft. 

Swallow_4.85 

Sparrow .... 2.7 
Turtle-dove.. 2.13 


Sq. ft. 


Pigeon.1.27 

Vulture.82 


Crane, Australia, .41 


Sq. ft. 

Gnat.3.05 

Dragon-fly, sm’ll, 1.83 
Lady-bird.1.66 


Sq. ft. 

Cockchafer.. .32 

Bee.33 

Meat-fly.35 


Crocodile and Bog. 

The direct power of their jaws is estimated at 120 lbs. for the former and 
44 for the latter, which, with the leverage, will give respectively 6000 and 
1500 lbs. 

PERFORMANCES OF MEN, HORSES, ETC. 

Following are designed to furnish an authentic summary of the fastest or 
most successful recorded performances in each of the feats, etc., given. 


MAN. W allying. 

1874, Wm. Perkins , London, Eng., .5 mile, in 2 min. 56 sec.; 1, in 6 min. 23 sec.; 
2, in 13 min. 30 sec.; 1876, 8, in 59 min. 5 sec.; 1877, 20, in 2 hours 39 min. 57 sec. 

1880, T. Smith , London, Eng., 12 miles, in 1 hour 31 min. 42.4 sec. 

1881, C. A. Harriman , Chicago, Ill., 530 miles, in 5 days 20 hours 47 min. 

1851, J. Smith , London, Eng., 25 miles, in 3 hours 42 min. 16 sec. 

1878, W. Howes , London, Eng., 50 miles, in 7 hours 57 min. 44 sec.; 1880, 75 miles, 
in 13 hours 7 min. 27 sec., and 100, in 18 hours 8 min. 15 sec. 

1880, John Dobler , Buffalo, N. Y., 150 miles 850 yards, in 24 hours. 

1801, Capt. R. Barclay , Eng., country road, 90 miles, in 20 hours 22 min. 4 sec., in¬ 
cluding rests; 1803, .25 mile, in 56 sec., and Charing Cross to Newmarket, 64, in 10 
hours, including rests; 1806, 100, in 19 hours , including 1 hour 30 min. in rests; 1809, 
1000, in 1000 consecutive hours , walking a mile only at commencement of each hour. 

1877, D. O'Leary, London, Eng., 200 miles, in 45 hours 21 min. 33 sec. 

1818, Jos. Eaton, Stowmarket, Eng., 4032 quarter miles, in 4032 consecutive quar¬ 
ter hours. 

1877, Wm. Gale , London, Eng., 1500 miles, in 1000 consecutive hours , 1.5 miles 
each hour ; and 4000 quarter miles, in 4000 consecutive periods of 10 minutes. 

1882, Chas. Rowell , New York, N. Y., 89 miles 1640 yards, in 12 hours. 

1882, Geo. Hazael, New York, N. Y., 600 miles 220 yards, in 6 days. 

Ft 11 lining. 

1710, Levi Whitehead , Branham Moor, Eng., 4 miles, in 19 min. 

1844, Geo. Seward, of U. S., Manchester, Eng., 100 yards, in 9.25 sec. 

1869, Geo. Forbes, Providence, R. I.. 150 yards, in 15 sec. 

1851, Chas. Westhall, Manchester. Eng., 150 yards, in 15 sec., and 200, in 19.5 sec. 

1864, Jas. Nuttall , Manchester, Eng., 600 yards, in 1 min. 13 sec. 

1881, L. E. Myers, New York, N. Y., 1000 yards, in 2 min. 13 sec. 

1863, Wm. Lang. Newmarket, Eng., 1 mile, in 4 min. 2 sec., descending ground; 
Manchester, 2, in 9 min. 11.5 sec.; 1865, n miles 1660 yards, in 1 hour 2 min. 2.5 sec. 











ANIMAL POWEE. 


439 


1852, Wm. Howitt, “ American Deer,” London, Eng., 10 miles, in 51 min. 34 sec., 
walking last 200 yards, time, if run, 51 min. 20 sec.; and 15, in 1 hour 22 min. 

1863, L. Bennett ,‘‘ Deerfoot,’’Hackney Wick, Eng., 12 m., in 1 hour 2 min. 2.5 sec. 

1879, Patrick Byrnes , Halifax, N. S., 20 miles, in 1 hour 54 sec. 

1880, D. Donovan , Providence, R I., 40 miles, in 4 hours 48 min. 22 sec. 

1879, G. Hazael., London, Eng., 50 miles, in 6 hours 15 min. 57 sec. 

17—, A Courier , East Indies, 102 miles, in 24 hours. 

Jumping, Leaping, etc. 

1848, P. M l Neely, Petersburg, Ky., 10 jumps, standing, no feet 4 ins. 

1854, J. Howard , Chester, Eng., 1 jump, board raised 4 ins. in front, running start, 
with dumb-bells, 5 lbs., 29 feet 7 ins. 

1868, Geo. M. Kelley , Corinth, Miss., running, and from a spring board, leaped over 
17 horses standing side by side. 

1874, J. Lane , Dublin, Ireland, running start, 1 jump, without aid, 23 feet 1.5 ins. 

1878, E. W. Johnson , Baltimore, Md., standing leap, 5 feel 3 ins. 

1879, G. W. Hamilton , Romeo, Mich., dumb-bells, 22 lbs., standing jump, 14 feet 
5.5 ins. ; and 1880, dumb-bells, 12 lbs., 3 standing jumps, 39 feet 1 inch. 

1880, P. Davin, Dublin, Ireland, running leap, 6 feet 2.75 ins. 

Lifting. 

1825, Thomas Gardner , of New Brunswick, N. S., a barrel of pork, 320 lbs., under 
each arm; also transported across a pier an anchor, 1200 lbs. 

1868, Wm. B. Curtis , New York, N. Y., 3239 lbs., in harness. 

1881, D. L. Dowd, Springfield, Mass., by hands, 1317 lbs. 

Throwing "Weights. 

1870, D. Dinnie, New York, N. Y., light stone, 18 lbs., 43 feet; heavy stone, 24 lbs., 
34 feet 6 ins.; heavy hammer, 24 lbs., 8 ^ feet 8 ins.; 1872, Aberdeen, Scotland, light 
hammer , 138 feet; run, 16 lbs., 162 feet. 

1877, M. Davin, Dublin, Ireland, run, 56 lb. weight, 30 feet 2 ins. 

Swimming. 

1835, S. Bruck, 15 miles, in a rough sea, in 7 hours 30 min. 

1846, A Native, off Sandwich Islands, 7 miles at sea, with a live pig under one arm. 

1878, E. T. Jones, London, Eng., 100 yards, in 1 min. 8.5 sec. 

1870, Pauline Bohn , Milwaukee, Wis., 650 feet, still water, in 2 min. 43 sec. 

1881, Wm. Beckwith, London, Eng., 1000 yards, in 15 min. 8.5 sec. 

1872, J. B. Johnson , Hendon, Eng., open water, 1 mile, in 28 min. 24.6 sec. ; Agri¬ 
cultural Hall, London, Eng., remained under water, 3 min. 35 sec. 

1875, Capt. M. Webb, Dover, Eng., to Calais, France, 23 miles, crossing two full 
and two half tides = 35 miles, in 21 hours 45 min. 

1880, J. Strickland, Melbourne, Australia, plunged 73 feet 1 inch. 

Skating. 

1854, Wm. Clark, Madison, Wis., 1 mile, in 1 min. 56 sec. 

1868, John Conyers, Lake Simcoe, Can., 8 miles, in 18 min. 40.5 sec. 

1876, E. St. Clair Milliard, Chicago, III, 50 miles, in 4 hours 57 min. 3 sec. 

1877, John Ennis, Chicago, Ill., 100 yards, calm, in n.75 sec.; 9 laps to a mile, 
100 miles, in n hours 37 min. 45 sec. ; and 145 inside of 19 hours. 

Note.—T he Sporting Magazine, London, vol. ix., page 135, reports a man in 1767 to have skated a 
mile upon the Serpentine, Hyde Park, London, in 57 seconds. 

HORSE. Trotting. 

1878, “Controller,” San Francisco, Cal., 10 miles, harness, in 27 min , 27.25 sec., 
and 20 miles, w r agon, in 58 min. 57 sec. 

1875, “Steel Grey,” Yorkshire, Eng., 10 miles, saddle, in 27 min. 56.5 sec. 

1867, “John Stewart,” Boston, Mass., half-mile track, 20 miles, harness, in 58 
min. 5.75 sec., and 20.5 miles in 59 min. 31 sec. 

1830, “Top Gallant,” Philadelphia, Penn., 12 miles, harness, in 38 min. 

1829, “Tom Thumb,” Sunbury Common, Eng., 16.5 miles, harness, 248 lbs., in 56 
min. 45 sec.; and 100 miles, in 10 hours 7 min., including 37 min. in rests. 

1869, “Morning Star,” Doncaster, Eng., 18 miles, harness (sulky 100 lbs.), in 57 
min. 27 sec. 

1835, “Black Joke,” Providence, R. I., 50 miles, saddle, 175 lbs., in 3 hows 57 min. 


440 


ANIMAL POWER. 


1855, “Spangle,” Long Island, N. Y., 50 miles, wagon and driver 400 lbs., in 3 
hours 59 min. 4 sec. 

1837, “Mischief,” Jersey City, N. J., to Philadelphia, Penn., 84.25 miles, harness, 
very hot day and sandy road, in 8 hours 30 min. 

1853, “Conqueror,” Long Island, N. Y., 100 miles, harness, in 8 hours 55 min. 53 
sec., including 15 short rests. 

1873, M. Delaney's mare, St. Paul’s, Minn., 200 miles, race track, harness, in 44 
hours 20 min., including 15 hours 49 min. in rests. 

1834, “ Master Burke ” and “ Robin,” Long Island, N. Y., 100 miles, wagon, in 10 
hours, 17 min. 22 sec., including 28 min. 34 sec. in rests. 

Stage-coaclaing. 

1750, By the Duke of Queensberry, Newmarket, Eng., 19 miles, in 53 min. 24 sec. 

1830, London to Birmingham , Eng., “Tally-ho,” 109 miles, in 7 hours 50 min., 
including stop for breakfast of passengers. 

Leaping.* 

1821, A horse of Mr. Mane, at Loughborough, Leicestershire, Eng., 173 lbs., over a 
hedge 6 feet iu height, 35 feet. 

1821, A horse of Lieut. Green, Third Dragoon Guards, at Inchinnan, Eng., ridden 
by a heavy dragoon, over a wall 6 feet in height and 1 foot in width at top. 

1839, “ Lottery,” Liverpool, Eng., over a wall, 33 feet. 

1847, “Chandler,” Warwick, Eng., over water, 37 feet. 

Note.—T he maximum stride of a horse is estimated to be 28 feet 9 ins.; “Eclipse” has covered 25 
feet. The maximum stride of an elk is 34 feet, and of an elephant 14 feet. 

Running. 

1701, Mr. Sinclair, on the Swift at Carlisle, a gelding, 1000 miles, in 1000 consecu¬ 
tive hours. 

1731, Geo. Osbaldeston , Newmarket, 156 lbs., 100 miles, by 16 horses, in 4 hours 19 
min. 40 sec., and 200, by 28 horses, in 8 hours 39 min., including 1 hour 2 min. 56 sec. 
in rests; 1 horse, “Tranby,” 16 miles, in 33 min. 15 sec. 

1752, Spedding's mare, 100 miles, in 12 hours 30 min., for 2 consecutive days. 

1754, A Galloway mare of Daniel Corker’s, Newmarket, 300 miles, by one rider, 
67 lbs., in 64 hour's 20 min. 

1761, John Woodcock , Newmarket, too miles per day, by 14 horses, one each day, 
for 29 consecutive days. 

1814, An Officer of 14 th Dragoons , Blackwater, 12 miles. 1 horse, in 25 min. n sec. 

1868, N. H. Mowry, San Francisco, Cal., race track, 160 lbs., 300 miles, by 30 horses 
(Mexican), in 14 hours 9 min. , including 40 minutes for rests; the first 200, in 8 
hours 2 min. 48 sec. , and the fastest mile in 2 min. 8 sec. 

1869, Nell Coher, San Pedro, Texas, 61 miles, in 2 hours 55 min. 15 sec., including 
rests. 

1870, John Faylor, Carson City, Nevada, 50 miles, by 18 horses, in 1 hour 58 min. 
33 sec.; and Omaha, Neb., 56 miles, in 2 hours 26 min., including rests. 

1876, John Murphy , New York, N. Y., 155 miles, by 20 horses, in 6 hours 45 min. 

7 sec. 

1878, Capt. Salvi, Bergamo to Naples, Italy, 580 miles, in 10 days. 

1880, “ Mr. Brown,” Rancocas, N. J., aged. 160 lbs., 10 miles, in 26 min. 18 sec. 

1828, “Chapeau de Paille” (Arabian), India, 1.5 miles, 115 lbs., in 2 min. 53 sec. 

183-, Capt. Horne (Arabians), Madras to Bungalore, India, 200 miles, in less than 
10 hours. 

DOGS. Coursing and Cliasing. 

A Greyhound and Hare ran 12 miles in 30 min. 

1794, A Fox, at Brende, Eng., ran 50 miles in 6.5 hours. 

A Greyhound, at Bushy Park, Eng., leaped over a brook 30 feet 6 ins. 

BIRDS. Flying. 

Vulture, 150 miles; Wild Goose and Swallov>, 90 miles; Crow, 25 miles per hour. 

1870, Carrier Pigeons. Pesth to Cologne, Germany, 600 miles, in 8 hours. 

1875, Carrier Pigeon , Dundee Lake to Paterson, N. J., 3 miles, in 3 min. 24 sec. 


* A Salmon can leap a dam 14 feet in height .—Sporting Magazine, London, vol. xii., page 79. 



HORSE-POWER.-BELTS AND BELTING. 


441 


HORSE-POWER. 

Horse-power .—IP is the principal measure of rate at which work is per¬ 
formed. One horse-power is computed to be equivalent to raising of 33 000 
lbs. one foot high per minute, or 550 lbs. per second. Or, 33 000 foot-lbs. of 
work, and it is designated as being Nominal, Indicated, or Actual. 

A IP in work is estimated at 33000 lbs., raised 1 foot in a minute; but as a horse 
can exert that force for only 6 hours per day, one work IP is equivalent to that of 
4.5 horses. 

Cheval-vapeur of France is computed to be equivalent to 75 kilogram- 
meters of work per second, or 7.233 foot-lbs., or 75 x 7.233 = 542.5 foot-lbs., 
which is 1.37 per cent, less than American or English value. 


BELTS AND BELTING. 

Capacity of belts to transmit power is determined by extent of their 
adhesion to surface of pulley, and it is very limited in comparison with 
tensile strength of belt. 

Resistance of a belt to slipping depends essentially upon character 
of surface of pulley, its degree of tension, and width, and as adhesion 
is in proportion to pressure on surface of pulley, long belts, by having 
greater weight, give greater adhesion. 

Tensile strength of Belting per square inch of section ranges as follows: 

Tanned Leather, .186 inch thick, from 2846 to 5000 lbs., or from 530 to 
930 lbs. per inch of width; when spliced 385 lbs., and when laced 210 lbs. 

Taking .3 as a factor of safety, 70 and 128 lbs. represent resistance per 
sq. inch that belts in operation may be subjected to, and they have been run 
successfully at these tensions. 

Raw hide has a tensile strength of 1.5 times that of tanned. 

By Experiments of H. R. Towne and Mr. KirJcaldy. [England.') 

Tensile strength of Single leather belting per square inch of section. 

Laced, 960 lbs. Riveted, 1740 lbs. Solid, 3080 lbs. 

Norris Sp Co. —Double, 2 ins., 2942 lbs.; 6 ins., 5603 lbs.; 12 ins., 14861 lbs. 
Single, 3.5 ins., 3007 lbs.; 5 ins., 4060 lbs.; 10 ins., 8846 lbs. 

Spill's belting, from flax, saturated with an endurable substance, gave ten¬ 
sile strength per inch of width as follows; 

No. 1, 5 ins. wide, 1254 lbs. No. 2, 5 ins. wide, 1489 lbs. No. 3, 10 ins. 
wide, 1663 lbs. 

At a velocity of 1000 feet per minute, a width of leather belt Of r inch will trans¬ 
mit power of 1 horse, and at a velocity of 1800 feet, .56 of an inch will transmit a 
like power, pulley being fully three feet in diameter, equal to a stress of 33 lbs. per 
inch of width of belt of ordinary thickness. 

To Compute 'Width, of a Leather Belt. 

Assuming a well-defined case (where limit of adhesion was ascertained), 
a belt of ordinary construction (laced), and 9 inches in width, transmitted 
the power of 15 horses over a pulley 4 feet in diameter, at a velocity of 1800 
feet per minute, with an arc of adhesion of 210°, or of .6 or 7.54 feet of cir¬ 
cumference, and with an area of 95 square feet of belt per BP. 

TT 4400 to 5000 BP . . 7 , . . . ... 

Hence,--- — w;w representing width of belt in inches, d di- 

CL V 

ameter of pulley in feet, and v velocity of belt in feet per minute. 

Note.— Thickness of belt should be added to diameter of pulley. Applying these 
elements to the formulas of 13 different authors, the result varies from 7.85 to 13.5 
ins., mean of which is 10.675. For double belting width = .6 w. 




442 


BELTS AND BELTING. 


Illustration. —If E? 25, diameter of pulley 4 feet, and velocity 2250 feet; what 
should be width of belt? 

4 5°° X 2 5 _ j2 i ns , for ordinary thickness of. 1875 in. 

4 X 2250 

To Compute Elements of Belting. 

V w IP 33 000 _ p 33000 IP _ W_ g< A - a _ l 

IOOO ’ v to ’ V 1 t 1 l t 

P representing power or stress transmitted , W weight or stress on belt , 
t ihickness of belt , S stress on belt per inch of width, A and a areas of coil 
and eye , and l length in feet. 

Note.— 70 square feet of good belting are capable of transmitting an indicated IP. 


India Rubber Belting. (Vulcanized.) 

Results of Experiments upon Adhesion of India Rubber and Leather Belting .— 

(J. H. Cheever). 

Leather. 


Lbs. 

Leather belt slipped on iron pulley at 48 
“ “ leather “ 64 

“ “ rubber “ 128 


Rubber. 

Lbs. 

Rubber belt slipped on iron pulley at 90 
“ leather “ 128 

“ “ rubber “ 183 

Hence it appears that a Rubber Belt for equal resistances with a Leather Belt 
may be reduced respectively 46, 50, and 30 per cent. 

Iron Wire .—A wire rope .375 inch in diameter, over a pulley 4 feet in 
diameter, and running at a velocity of 1250 feet per minute, will transmit 

4-5 IP- 

Diameter of pulley should not be less than 140 times diameter of rope, in order 
to avoid undue bending of wires. 

A sheet-iron belt 7 inches in width proved more effective than one of leather of 
like width. 

Greneral Notes. 


Leather Belts —Are best when oak tanned, should be frequently oiled,* and when 
run with hair side over pulley will give greatest adhesion. 

Ordinary thickness .1875 inch, and weight 60 lbs. per cube foot. 

Relative effect of different pulleys and belts: 


Pulleys.— Leather surface. 1. I Turned iron.64 

Rough iron.41 | Turned wood.7 

Tensile strength of calf and sheep skins is about one half that of beeve and horse. 

Morin assigns 50 lbs. as a proper stress per inch of width of good belting. 

Presence of small holes in a belt will prevent its slipping or squealing. 

Rubber Belts .—Best vulcanized rubber is stronger than leather, and its resistance 
is from 50 to 85 per cent, greater. 

To increase adhesion, coat driving surface with boiled oil or cold tallow, and then 
apply powdered chalk. 

When new, cut them .1875 inch short for each foot in length required, to admit 
of the stretch that occurs in their early operation. 

They should be kept free from contact with an animal oil. 

Three ply, .1875 inch thick, has a tensile resistance of 600 lbs. per inch of width. 

Relative slipping of a vulcanized belt, over smooth or turned leather or rubber¬ 
faced iron pulleys is as .5, .7, and 1. 

Rubber, Gutta percha, and Canvas belts will stretch continuously. 


Memoranda. 


Belts should be set as near horizontal as practicable, in order that the sag may 
increase adhesion on pulley, and hence power should be communicated through 
under side. 

The “creeping” or lost speed by belts is about 2 per cent., hence, to maintain a 
uniform or required speed, driver must be increased in diameter pro rata with slip. 


See Cements, etc., page C71, for compositions, etc. 













BELTS AND BELTING.-BLASTING. 


443 


A belt, xi ins. in width, over a driver 4 feet in diameter, running from 1200 to 2250 
feet per minute, will transmit the power from two steam cylinders, 6 ins. in diam¬ 
eter and 11 ins. stroke, averaging 125 revolutions per minute, with a pressure of 
60 lbs. per sq. inch. 

A double belt, 75 ins. in width and 153.5 feet in length, transmitted 650 IIP. 

Pulleys should have a slight convexity of surface. Authorities differ, from .5 inch 
per foot of breadth to .1 of breadth. Belts run at a high speed are less liable to slip 
than at low speed. 

The best speeds for economy are from 1200 to 1500 feet per minute, and the best 
for result not to exceed 1800. 

Belts.— Leather, hair-side.... 1 I Leather, flesh-side... .74 | Rubber.51 

Gutta percha.44 | Canvas.35 

Coefficient of Friction of a Belt in operation is assumed to be .423. 

Smooth surface belts are most endurable and soft most adherent. 

Round belts .25 and .5 inch in diameter are fully equal in operation to flat of 1 
and 3 ins., and grooves in their pulleys should be angular or V shaped. 

The neutral point of a rope belt is at .33 of diameter from inside surface. 

Friction of driving and pulley bearings is about .025. 

A fan-blower No. 6*, driven by a belt 3.875 ins. in width and .18 in thickness, at 
a velocity of 2820 revolutions per minute, requires power of 9.7 horses. 

Area of belts per IP varies essentially, ranging from 25 to 100 square feet; the 
mean is 75. 


BLASTING. 

In Blasting , rock requires from .25 to 1.5 lbs. gunpowder per cube 
) 7 ard, according to its degree of hardness and position. In small blasts 
2 cube yards have been rent and loosened, and in very large blasts 2 to 
4 cube yards have been rent and loosened, by 1 lb. of powder. 

Tunnels and shafts require 1.5 to 2 lbs. per cube yard of rock. 

Gunpowder has an explosive force varying from 40000 to 90000 
lbs. per sq. inch. That used for blasting is much inferior to that used for 
projectiles, the proportion being fully one third less. 

ZSTitro-glyceriiae is an unctuous liquid, which explodes by concussion, 
an extreme pressure (2000 lbs. per sq. inch), or a temperature exceeding 6oo° 
if quickly applied to it; it will inflame, however, and burn gradually. 

At a temperature below 40° it solidifies in crystals. 

Its explosion is so instantaneous that in rock-blasting tamping is not nec¬ 
essary; its explosive power by weight is from 4 to 5 times that of gun¬ 
powder. 

Dynamite is nitro-glvcerine 75 parts, absorbed in 25 parts of a sili¬ 
ceous earth termed kieselguhr; it also explodes so instantaneously as to 
render tamping in blasting quite unnecessary. 

It is insoluble in water,'and may be used in wet holes; it congeals at 40°, 
is rendered ineffective at 212 0 , and has an explosive force by weight of 3 
times that of gunpowder, and by bulk 4.25 times. 

Gun-cotton is insoluble in water, and has an explosive force by 
weight of from 2.75 to 3 times that of gunpowder, and by bulk 2.5 times. 
It may be detonated in a wet state with a small quantity of dry material. 

Tonite is nitrated gun-cotton, and is known also as cotton powder . It 
is produced in a granulated form. 

Litho-fracteur is a nitro-glycerine compound in which a portion of 
the base or absorbent material is made explosive by the admixture therein 
of nitrate of baryta and charcoal. 


* For a table of Belts for Fan-blowers, etc., see J H. Cooper, in “ Jour. Franklin Inst.,” vol. 66, p. 409 









444 


BLASTING, 


Cellulose Dynamite is when gun-cotton is used as the absorbent 
for nitro-glycerine; it will explode frozen dynamite, and is more sensitive to 
percussion than it. 

To Compute Charge of Gunpowder for Rods: Blasting. 

Rule. —Divide cube of line of least resistance by 25, as for limestone, to 
32 for granite, and remainder will give charge of powder in lbs. 

Or, L .3 -5- 32 = lbs. 

Example.— When line of least resistance is 6 feet, what is charge required? 

6 -7-32 = 6.75 lbs. 

Line of least resistance should not exceed . 5 depth of hole. 

Tamping. — Dried clay is the most effective of all materials for tamping; Broken 
Brick the next, and Loose Sand the least. 

Relative Costs of a Tunnel and Shaft in England. (Sir John Burgoyne.) 

Iron and steel.8.98 Powder.29.04 

Smiths and coal.6 Labor.48.8 

Fuses.7.18 IOO 


Weight of Explosive Materials in Holes of Different Diameters. 
Per Inch of Length. 


Diam. 

Powder 
or Gun¬ 
cotton. 

Dynamite. 

Diam. 

Powder 
or Gun¬ 
cotton. 

Dynamite. 

Diam. 

Powder 
or Gun¬ 
cotton. 

Dynamite 

Ins. 

Oz. 

Oz. 

Ins. 

Oz. 

Oz. 

Ins. 

Oz. 

Oz* 

I 

.419 

.67 

1-75 

1.283 

2-053 

2-5 

2.618 

4.189 

1.25 

•654 

1.046 

2 

1-675 

2.68 

2-75 

3.166 

5.066 

i -5 

.942 

1-507 

2.25 

2.12 

3 - 39 2 

3 

3 - 7 6 9 

6.03 


Boring Holes in Granite. 


Diam. 

of 

Jumper. 

Depth 

of 

Hole. 

Men. 

Depth bored 
per Day. 

Ham¬ 

mer. 

Diam. 

of 

Jumper. 

Depth 

of 

Hole. 

Men. 

Depth bored 
per Day. 

Ham¬ 

mer. 

Ins. 

Ins. 

No. 

Feet. 

Lbs. 

Ins. 

Ins. 

No. 

Feet. 

Lbs. 

I 

1 tO 2 

I 

8 

6 

2.25 

5 to 10 

3 

6 

16 

i -75 

2.5 to 6 

3 

12 

14 

2-5 

9 to 12 

3 

5 

l6 

2 

4 *-0 7 

3 

8 

14 

3 

9 t0 15 

3 

4 

18 


Drill .—Width of bit compared to stock .625. 

Charges of Powder. 

Usual practice of charging to one third depth of hole is erroneous, inasmuch as 
volume of charge increases as square of diameter of hole. Hence holes of 1.5 and 
2 inches, although of equal depths, would require charges in proportion of 2.25 and 4. 


Line of 
least re¬ 
sistance. 

Powder. 

Line of 
least re¬ 
sistance. 

Powder. 

Line of 
least re¬ 
sistance. 

Powder. 

Line of 
least re¬ 
sistance. 

Powder. 

Feet. 

Oz. 

Feet. 

Lbs. Oz. 

Feet. 

Lbs. Oz. 

Feet. 

Lbs. Oz. 

I 

•75 

3 

13-5 

5 

3 14-5 

7 

10 11.5 

2 

4 

4 

2 

6 

6 12 

8 

l6 


Effects. 

Gunpowder. — From its gradual combustion, rends and projects rather than 
shatters. 

A hole 5.5 ins. in diameter and 19 feet 7 ins. in depth, filled to 8 feet 10 ins. with 
75 lbs. powder, has removed and rent 1200 cube yards, equal to 2400 tons. The 
iabor expended was that of 3 men for 14 days. 

Temperature of gases of explosion 4000 0 . 

Gun-cotton.— From the rapidity of its combustion, shatters. 

Dynamite.—From the greater rapidity of its combustion over gun-cotton, is more 
shattering in its explosion. 





























































BLASTING.—BLOWING ENGINES. 


445 


Drilling. 

Churn-drilling .—A churn-driller will drill, in ordinary hard rock, from 8 to 12 
feet, 2 inch holes of 2.5 feet depth, per day, and at a cost of from 12 to 18 cents per 
foot, on a basis of ordinary labor at $1 per day. Drillers receiving $2.50. 

One man can bore, with a bit 1 inch in diameter, from 50 to 100 inches per day 
of 10 hours in granite, or 300 to 400 inches per day in limestone. 

Tamping .—Two strikers and a holder can bore, with a bit 2 inches in diameter, 
10 feet in a day in rock of medium hardness. 

Composition for waterproof charger or fuse consists by weight of Pitch, 8 parts; 
Beeswax and Tallow each 1 part. 

IVliniiig. (Lefroy's Handbook.) 

In demolition of walls line of least resistance L = half thickness, and C is a co¬ 
efficient depending on structure. 

Charge in lbs. = C X L 3 . 

In a wall without counterforts, where interval between the charge is 2 L, C = .i5. 

In a wall with counterforts the charge to be placed in centre of each counterfort 
at junction with wall, C =. 2. 

Where the charge is placed under a foundation, having equal support on both 
sides, C =. 4. 

A leather bag, containing 50 to 60 lbs. pow'der, hung or supported against a gate 
or like barrier, will demolish it. 

For ordinary mines in average rock charge in ounces = L 3 -4- 160. 


BLOWING ENGINES. 

For Smelting. 

Volume of oxygen in air is different at different temperatures. Thus, 
dry air at 85° contains 10 per cent, less oxygen than when it is at tem¬ 
perature of 32 0 ; and when it is saturated with vapor, it contains 12 
per cent. less. If an average supply of 1500 cube feet per minute is 
required in winter, 1650 feet will be required in summer. 

Smelting of Iron Ore. 

Cole or Anthracite Coal—18 to 20 tons of air are required for each ton 
of Pig Iron, and with Charcoal 17 to 18 tons are required. 

(1 ton of air at 34 0 = 29 751, and at 6o° = 31 366 cube feet.) 

Pressure .—Pressure ordinarily required for smelting purposes is equal to 
a column of mercury from 3 to 10 inches, or a pressure of 1.5 to 5 lbs. per 
square inch. 

Peservoir .—Capacity of it, if dry, should be 15 to 20 times that of cylin¬ 
der if single acting, and 10 times if double acting. 

Pipes .—Their area, leading to reservoir, should be .2 that of blast cylinder, 
and velocity of the air should not exceed 35 feet per second. 

A smith’s forge requires 150 cube feet of air per minute. Pressure of 
blast .25 to 2 lbs. per square inch. A ton of iron melted per hour in a cu¬ 
pola requires 3500 cube feet of air per minute. A finery forge requires 
100000 cube feet of air for each ton of iron refined. A blast furnace re¬ 
quires 20 cube feet per minute for each cube yard capacity of furnace. 

A Ton of Pig Iron requires for its reduction from the ore 310000 cube 
feet of air, or 5.3 cube feet of air for each pound of carbon consumed. 
Pressure, .7 lb. per square inch. 

P P 



446 


BLOWING ENGINES. 


To Compute Tower E,eq.nired. to Drive a Blowing 

Engine. 

£S # T ^ + ?) & -33<x» = ®. 

. v representing velocity of air in feet per sec- 


d' 


=W: 


93 X .7854 X v 


ond , d and d' diameters of pipe and of nozzle in feet, —\J' - 


35 


•93 X .7854 X 500 

= ‘309- 

Illustration.—W hat should be power of a steam-engine to drive 35 cube feet of 
air at a velocity of 500 feet per second, through a pipe 1 foot in diameter and 300 
feet in length ? 

c = ratio between power employed and effect produced by it — in a well-constructed 
engine .5, and C = .93. ^ = .2974, assumed at .3. 

.0000509 ^ 3 /300 42 \ 6o _^_ 33O0O — 22 6 3i .625 X 60-r-33 000 = 41.15 IP. 

•5 \ l5 - 3 V 

To Compute Required Bower of a Blowing Engine, 
p 1 ^ a v * 

— ——-= IP. P representing pressure of blast in lbs. per sq. inch; 

a area of cylinder in sq. ins.; v velocity of piston in feet per minute; ffric¬ 
tion of piston and from curvatures, etc., estimated at 1.25 per sq. inch of 
piston. 

Note.—I f cylinder is single acting, divide result by 2. 

Illustration.—A ssume area of blast cylinder 5600 sq. ins., pressure of blast 2.25 
lbs. per sq. inch, and velocity of piston 96 feet per second. 

2.254-1.25X5600X96 1881600 , , . 

— “57 horses , the exact power developed in 


this case. 


33000 


3300° 


To Compute Dimensions of a Driving Engine. 

_ Rule i.— Divide power.in lbs. by product of mean effective pressure upon 
piston of steam cylinder in lbs. per sq. inch, and velocity of piston in feet 
per minute, and quotient will give area of cylinder in sq. 'ins. 


2 * Divide velocity of piston by twice number of revolutions, and quotient 
will give stroke of piston in feet. 

Volume of air at atmospheric density delivered into reservoir, in consequence of 
escape through valves, and partial vacuum necessary to produce a current, will be 
about .2 less than capacity of cylinder. 

Example. —Assume elements of preceding case, with a pressure of 50 lbs. steam, 
cut off at .375, and with 12 revolutions of engine per minute, what should be area 
of cylinder of a non-condensing engine? 

Mean eff ective pressure of steam with 5 per cent, clearance = 50 lbs., and 50 — 
/* 4 * T 4-7 = 5 ° — 2 -5 4 - 3-33 + *4-7 = 29.47 lbs., and velocity of piston = 192 feet. 


5600X2.254-1.25X96 1 881600 


29.47 X 192 


5658 


: 332.5 sq. ins., and 


192 


12 X 2 


8 feet stroke. 


Area of cylinder in this case was 324 sq. ins. 


For Volume, Pressure, and Density of Air, see Heat, page 521. 


* See formula and note for power of non-condensing engine, page 733. 

















BLOWING ENGINES. 


447 


To Compute Elements of a Blowing Engine. 
Single Stroke. 

V V -f- io L , D 2 s n 


VP+/ 

230 
D 2 s n 


As»P+ f _ 

33000 “ 

230 IP 


40 v 


IP; 

= V; 




D : 


40 a 


:v; 


= V; and 34 P + 32 = *. 


’ *>+/ . . ' 92 

Y representing volume of air in cube feet per minute, I‘ pressure of air and 
f frictional resistance in lbs. per sq. inch, A area of cylinder and a area 
of its valves in sq. ins., s stroke of piston in feet, n number of single strokes 
of piston per minute, L length of air-pipe from reservoir to discharge in feet, 
d diameter of air or blast pipe and I) diameter of cylinder in ins., v velocity 
of blast in feet per second, and t temperature of blast consequent upon com¬ 
pression in degrees. 

Illustrations. —Assume blowing cylinder 50 ins. in diam., stroke of piston 10 
feet, number of single strokes 10 per minute, pressure by mercurial manometer 
6.12 ins., frictional resistance .4 lb., length of pipe 25.25 feet, and area of valves 
95 sq. ins. 

V = 1363.54 cube feet, P = 3 lbs., A = 1963.5 sq. ins. 


Then 


1363-54 X 3 +-4 


230 

V1363-54+10X25.25 


= 20.16 IP, and 


1963.5 X 10 X xo X 3 +-4 


33000 


- 20.23 IP- 


=13.4 ms. 


5o 2 XioX io 
4 °X 95 


=65.8 feet. 


5 o 2 X 10X io 
40X65.8 


=95 sq. ms. 


To Compute Volume of Air transmitted by an Engine. 


When Pressure, Temperature, etc., are given. 


34-5 V 


/ /i + .004 t 

h 'w+ir 


j C = v. Then av X 6o = V in cube feet per minute. 


PI and h representing height of barometer and pressure of blast in ins. of 
mercury; t temperature of blast; and v velocity in feet per second. 


Illustration.—A furnace having 2 tuyeres of 5 ins. diameter, pressure and tem¬ 
perature of blast 3 ins. and 350 0 , and barometer 30 ins.; what is volume of air trans¬ 
mitted per minute? 

C for a conical opening — .94. 


34.5V 3 ( ,+ 3 7 3 X 0 3500 ) x .94 = 34.5/3 ( f ) = 34.5 x .467 X .,4 = .5. .4 
feet velocity per second. 

Then, area 5 ins.= 19.635, which X 2 = 39.27 ins., and 39.27 X 15.14 X 60 = 144 = 
247.73 cu t> e feet. 


To Compute Pressure of Blast from Water or NTercnrial 

Gauge. 

Rule. —Divide Water and Mercurial Gauge in ins. by 27.67 and 2.04 re¬ 
spectively, and quotient will give pressure in lbs. per sq. inch. 

Fan-blowers. 


Proportions of Parts. Blades .—Their width and length should be at least 
equal to .4 or .5 radius of fan. 

Openings .—Inlet should be equal to radius of fan; and outlet, or dis¬ 
charge, should be in depth not less than .125 diameter, its width being equal 
to width of fan. / 

Eccentricity .—.1 of diameter of fan. Journals, 4 diameters of shaft. 



























448 


BLOWING ENGINES. 


By the experiments of Mr. Buckle, he deduced 

1. That velocity of periphery of blades should be .9 that of their theoretical 
velocity; that is, velocity a body would acquire in falling height of a homo¬ 
geneous column of air equivalent to required density. 

2. That a diminution of inlet from proportions here given involved a 
greater expenditure of power to produce same density. 

3. That greater the depth of blade, greater the density of air produced 
with same number of revolutions. 

To Compute Elements of a Fan-blower. 

/ v \ 2 , /—j a v 60 „ . d a v 

q-- 939-45 = (l ; 244 Vd — v\ = Y • and -= IBP. 

\8.02/ 160 400 

v representing velocity of periphery of fan in feet per second , d inches of 
■mercury , V volume of air in cube feet , and a area of discharge in sq. ins. 

Illustration.—A ssume velocity of periphery of fau 123 feet per second, density 
of blast .25 inch, volume of air 1845 cube feet, and area of discharge 40 sq. ins. 

123-7-8.02-F939.454 — .25 inch. 244 V. 25 = 122 feet. 4° X 123 X 60 ^ cub. ft. 

IOO 

■ 242 X 40 X 123 __ 2 ^ up independent of friction of blast in pipes and tuyeres. 

To Compute Eower of a Centrifugal Fan. 

V 2 -F 97 300 = P. Y representing velocity of tips offan in feet per second. 

Memoranda. 

Operation of a blower requires about 2.5 per cent, of power of attached 
boiler. 

An increase in number of blades renders operation of fan smoother, but 
does not increase its capacity. 

Pressure or density of a blast is usually measured in ins. of mercury, a 
pressure of 1 lb. per sq. inch at 6o° = 2.0376 ins. 

When water is used, a pressure of 1 lb. = 27.671 ins. 

Cupola blast .8 lbs., and Smith's forge .25 to .3 lbs. per sq. inch. 

An ordinary Eccentric Fan, 4 feet in diameter, with 5 blades 10 ins. wide 
and 14 in length, set 1.5 ins. eccentric, with an inlet opening of 17.5 ins. in 
diameter, and an outlet of 12 ins. square, making 870 revolutions per min¬ 
ute, will supply air to 40 tuyeres, each of 1.625 ins. in diameter, and at a 
pressure per sq. inch of .5 inch of mercury. 

An ordinary eccentric fan-blower, 50 ins. in diameter, running at 1000 revolutions 
per minute, will give a pressure of 15 ins. of water, and require for its operation a 
power of 12 horses. Area of tuyere discharge 500 sq. ins. 

A non-condensing engine, diameter of cylinder 8 ins., stroke of piston 1 foot, press¬ 
ure of steam 18 lbs. (mercurial gauge), and making 100 revolutions per minute, will 
drive a fan, 4 feet by 2, opening 2 feet by 2, 500 revolutions per minute. 

Such a blower was applied as an exhausting draught to smoke-pipe of steamer 
Keystone State , cylinder 80 ins. by 8 feet, and evaporation was doubled over that 
of when wind was calm. 

In French blowing engines, volume of air discharged 75 per cent, that of 
volume of piston space in cylinder, stroke equal diameter of cylinder-, and 
velocity of piston from 100 to 200 feet per minute. 

Area of admission valves from .066 to .083 of that of cylinder for speeds 
of 100 to 150 feet per minute, and from .t to .111 for higher speeds. Area 
of exit valves from .066 to .05 of cylinder. (M. Claudel.) 







BLOWING ENGINES.-CENTRAL FORCES. 


449 


By some experiments lately concluded in England with boilers of two 
steamers, to determine relative effects of natural and forced draught furnaces, 
the results were as follows (R. J. Butler): 

Per Sq. Foot of Grate Surface.—Natural Draught, io to 10.87 IEP; Steam 
Blast , 12.5 to 13; Forced or Blast Draught , 15 to 16. 

Heating Surface per IIP. — Natural Draught , 2.44 to 2.61; Steam Blast , 
1.71 to 2.86; Forced or Blast Draught , 1.56 to 2.5. 

Tube Surface per IIP in Sq. Feet—Natural Draught, 2.03 to 2.18; Steam 
Blast , 2.02 to 2.08; Forced or Blast Draught , 1.3 to 2.8. 

IIP per Sq. Foot of Grate in these Trials. — Natural Draught, 10.15 to 
10.87; Steam Blast, 12.76 to 13.1; Forced or Blast Draught, 10.6 to 16.9. 

Root's Rotary Blower —Is constructed from .125 to 14 nominal IP, supplying 
from 150 to 10 800 cube feet of air per minute. Delivery pipe 2.5 to 19 ins. 
in diameter. Efficiency 65 to 80 per cent, of power. 

For Ventilation of Mines —From 40 to 280 revolutions per minute, equal 
to discharge of 12 500 to 200000 cube feet of air per minute. 15.5 to 189 BP. 

Steam cylinder from 14 x 18 ins. to 28 x 48 ins. 

For other details of Blowiug Engines see page 898. 


CENTRAL FORCES. 

All bodies moving around a centre or fixed point have a tendency to 
fly off in a straight line: this is termed Centrifugal Force; it is op¬ 
posed to a Centripetal Force , or that power which maintains a body in 
its curvilineal path. 

Centrifugal Force of a body, moving with different velocities in same 
circle, is proportional to square of velocity. Thus, centrifugal force of 
a body making 10 revolutions in a minute is 4 times as great as centrif¬ 
ugal force of same body making 5 revolutions in a minute. Hence, in 
equal circles, the forces are inversely as squares of times of revolution. 

If times are equal, velocities and force,s are as radii of circle of revolution. 

The squares of times are as cubes of distances of centrifugal force from 
axis of revolution. 

Centrifugal forces of two unequal bodies, having same velocity, and at same dis¬ 
tance from central body, are to one another as the respective quantities of matter 
in the two bodies. 

Centrifugal forces of two bodies, which perform their revolutions in same time, 
the quantities of matter of which are inversely as their distances from centre, are 
equal to one another. 

Centrifugal forces of two equal bodies, moving with equal velocities at different 
distances from centre, are inversely as their distances from centre. 

Centrifugal forces of two unequal bodies, moving with equal velocities at different 
distances from centre, are to one another as their quantities of matter, multiplied by 
their respective distances from centre. 

Centrifugal forces of two unequal bodies, having unequal velocities, and at differ¬ 
ent distances from their axes are in compound ratio of their quantities of matter, 
squares of their velocities, and their distances from centre. 

Centrifugal force is to weight of body, as double height due to velocity is to radius 
of rotation. 

A Radius Vector is a line drawn from centre of force to moving body. 

P P* 



450 


CENTRAL FORCES. 


To Compute Centrifugal Force of any Body. 

Rule i. —Divide its velocity in feet per second by 4.01., also square of 
quotient by diameter of circle; this quotient is centrifugal force, assuming 
the weight of body as 1. Then this quotient, multiplied by weight of body, 
will give centrifugal force required. 

Example.' —What is the centrifugal force of the rim of a fly-wheel having a diam¬ 
eter of 10 feet, and running with a velocity of 30 feet per second? 

30-1-4.01 =7.48, and 7.48 2 - 5 - 10 = 5.59, or times weight of rim. 


n Wn 2 VR 2 + r 2 _ 

Or, - 1 - = C. 

4100 


r representing radius of inner diameter of ring. 


Note.—D iameter of a fly-wheel should be measured from centres of gravity of rim. 

When great accuracy is required, ascertain centre of gyration of body, and 
take twice distance of it from axis for diameter. 

Rule 2.—Multiply square of number of revolutions in a minute by diam¬ 
eter of circle of centre of gyration in feet, and divide product by constant 
number 5217; quotient is centrifugal force when weight of body is 1. Then, 
as in previous Rule, this quotient, multiplied by weight of body, is centrif¬ 
ugal force required. 


Or, — — = W. n representing number of revolutions per minute, d diameter of 
circle of gyration in feet, and AY weight of revolving body in lbs. 

Example. —What is centrifugal force of a grindstone weighing 1200 lbs., 42 inches 
in diameter, and turning with a velocity of 400 revolutions in a minute? 

Centre of gyration = rad. (42- 4 -2) X-7071 = 14.85 ins., which -1-12 and X2 — 


2.475 feet = d iameter of circle of gyration. 


Then 


400 2 X 2-475 

5217 


X 1200 = 91080 lbs. 


R — 


C* 

2930 C 
W n 2 


Formulas 
W v 2 


to Determine 
AY R n 2 


32.166 R 
AY u 2 


2930 


Various 
= WRr'1.225; AY 


Elements. 
C 32.166 R 


C R 32.166 
AY 


6.28 v' R. 


32.166 C ’ 

C representing centrifugal force, W mass or weight of revolving body, both in lbs., 
R radius of circle of revolving body in feet, n number of revolutions per minute, and 
v and v' linear or circumferential and angular velocities of body in feet per second. 

Illustration.— AA T hat is centrifugal force of a sphere weighing 30 lbs., revolving 
around a centre at a distance of 5 feet, at 30 revolutions per second? 


v = 


5 X 2 X 3-1416 X 3° 
60 


= 15- 7 1 f eet - 


Then C 


30 X 15. 71 2 
32.166 X 5 


= 46.04 lbs. 


Centrifugal forces of two bodies are as radii of circles of revolution directly, and 
as squares of times inversely. 

Illustration. —If a fly-wheel, 12 feet in diameter and 3 tons in weight, revolves 
in 8 seconds, and another of like weight revolves in 6, what should be the diameter 
of the second when their centrifugal forces are equal ? 

Then 3:3::^ : ; °r x = — =6.75 feet, x = unknown element. 


Centrifugal forces of two bodies, ivhen weights are unequal, are directly as squares 
of times. 

Illustration. —What should be the ratio of the weights of the wheels in the pre¬ 
ceding case, their forces being equal? 


Then 3 : x :: 6 2 : 


8 2 or x— 


3 _XS 2 
6 2 


3 X 64 

36 


5.333 tons. 


Molesworth gives .000 34 W R n 2 =z C. 


* This is termed the Vis Viva, or living force. 


















CENTRAL FORCES.-FLY-WHEEL. 


451 


FLY-WHEEL. 

A Fly-wheel by its inertia becomes a reservoir as well as a regulator 
of force, and to be effective should have high velocity, and its diameter 
should be from 3 to 4 times that of stroke of driving engine. 

Co-efficient of fluctuation of energy in a machine ranges from .015 
to .035. 

Weight of a fly-wheel in engines that are subjected to irregular mo¬ 
tion, as in a cotton-press, rolling-mill, etc., must be greater than in others 
where so sudden a check is not experienced, and its diameter should 
range from 3.5 to 5 times length of the crank. 

A single acting engine requires a weight of wheel about 2.5 times greater 
than that for a double acting, and 5 times for double engines of double action. 


To Compute 'Weiglit of* Ftim of a Fly—xvlieel. 

Rule.— Multiply mean effective pressure upon piston in lbs. by its stroke 
in feet, and divide product by product of square of number of revolutions, 
diameter of wheel, and .000 23. 

Note.—I f a light wheel is required, multiply by .0003; and if a heavy one, by 
.00016. 

Example i.— A non condensing engine (double acting), having a diameter of cyl¬ 
inder of 14 ins., and a stroke of piston of 4 feet, working full stroke, at a pressure 
of 65 lbs. mercurial gauge, and making 40 revolutions per minute, develops about 
65 IP; what should be the weight of its fly-wheel, when adapted to ordinary work? 

Area of cylinder 154 sq. ins. Mean pressure assumed 50 lbs. per sq. inch. Diam¬ 
eter of wheel 4 feet stroke X 3.5 = 14 feet. 

50 X 154 X 4 — 30 800, which -r- 40 2 X 14 X -ooo 23 =; 5978 lbs. 

2.—If a fly-wheel, 16 feet in diameter and 4 tons in weight, is sufficient to regulate 
an engine (double acting) when it revolves in 4 seconds, what should be the weight 
of a wheel, 12 feet in diameter, revolving in 2 seconds, so that it may have like cen¬ 
trifugal force? 


Note.— The centrifugal forces of two bodies are as the radii of the circles of revo¬ 
lution directly, and as squares of times inversely. 

4X16 x X 12 _ 4 X 16 X 2 2 4X16X4 


Then 


Or, x 


Assume elements of example 1. 

5978 X - x- 


12 X 4 2 12 X 16 

13.25 = 45.12 square ins. 


1.333 tons. 


To Compute Dimensions of Rim. 


Rule.— Multiply weight of wheel in lbs. by .1, and divide product by 
mean diameter of rim in feet; quotient will give sectional area of rim in 
square inches of cast iron. 


Or, —— = W, and — — = A. P representing pressure on piston and W weight of 
.4 D 10 I) 

wheel in lbs., S stroke of piston and D mean diameter of wheel, both in feet, and A 
area of section of r im in sq. ins. 


Or. 


1 16 n I SC q representing coefficient varying from 3 to 4 ordinarily , 

60 D 

and increasing to 6 when great regularity of speed is required , and n number of revo¬ 
lutions per minute. 

Note.—M aximum safe velocity for cast iron is assumed at 80 feet per second. 

For engines at high expansion of steam, or with irregular loads, as with a rolling- 
mill, multiply W by 1.5, or put W 100 lbs. for each IIP. ( Molesworth.) 

In corn or like mills, the velocity of periphery of fly-wheel should exceed that of 
the stones. 








452 CENTRAL FORCES.-GOVERNORS.-PENDJJLUMS. 


GOVERNORS. 


A Governor or Conical Pendulum in its operation depends upon the 
principles of Central Forces. 

When in a Ball Governor the balls diverge, the ring on vertical shaft 
raises, and in proportion to the increase of velocity of the balls squared, 
or the square roots of distances of ring from fixed point of arms, cor¬ 
responding to two velocities, will be as these velocities. 

Thus, if a governor makes 6 revolutions in a second when ring is 16 
ins. from fixed point or top, the distance of ring will be 5.76 ins. w r hen 
speed is increased to 10 revolutions in same time. 

For 10 : 6 :: V 16 : 2.4, which, squared = 5.76 ins., distance of ring 
from top. Or, 6 2 : io 2 : ’. 5.76 : 16 ins. 

A governor performs in one minute half as many revolutions as a 
pendulum vibrates, the length of which is perpendicular distance be¬ 
tween plane in which the balls rpove and the fixed point or centre of 
suspension. 


To Compute Number of Revolutions of* a Ball Governor 
per 3 VLi.nu.te to maintain Balls at any given Height. 
188-4- ^/H = revolutions. H representing vertical height between plane of balls 
and points of their suspension in ins. 

Illustration.—I f the rise of the balls of a centrifugal governor is 22 ins., what 
are the number of revolutions per minute ? 

188-4- -f 22 = 40.09 revolutions. 

To Compute "Vertical Height between Plane of Balls 
and. their Points of* Suspension. 

(188 - 4 - r) 2 = vertical height in ins. r representing number of revolutions per minute. 
Illustration. —If number of revolutions of a centrifugal governor is 100, what 
will be rise of balls ? 

__2 

188- 4 - 100 —i.88 2 = 3.53 ins. 

To Compute Angle of Arms or Plane of Balls with 
Centre Shaft. 

r-^l~sin. f_. r representing distance of balls from plane of centre shaft, and l 
distance between balls and point of suspension measured in plane of shaft. 

Illustration. —Distance of balls from plane of centre shaft is 10 inches, and 
their distance from point of suspension is 25; whatlis the angle? 

10 - 4 - 25 = .4, and sin. .4 = 23 0 35'. 

When Number of Revolutions are given. = cos. 

L 


Illustration.— Revolutions of a governor per minute are 50, and length of its 
arms 2 feet; what is their angle with plane of shaft? 

<54 ' l6 ^ SO) * = il” =; 5 86 S = c„ S . 54 0 6'. 


PENDULUMS. 

Pendulums are Simple or Compound , the former being a material 
point, or single weight suspended from a fixed point, about which it 
oscillates, or vibrates, by a connection void of weight; and the latter, 
a like body or number of bodies suspended by a rod or connection. 
Any such body will have as many centres of oscillation as there are 
given points of suspension to it, and when any one of these centres are 
determined the others are readily ascertained. 





CENTRAL FORCES.-PENDULUMS. 


453 


Thus, sox s g = a constant product , and s r = Vs q X s g, s g o and r 
representing points of suspension , gravity , oscillation , and gyration. 

Or, any body, as a cone, a cylinder, or of any form, regular or irregular, 
so suspended as to be capable of vibrating, is a compound pendulum, and 
distance of its centre of oscillation from any assumed point of suspension is 
considered as the length of an equivalent simple pendulum. 

The Amplitude of a simple pendulum is the distance through which it 
passes from its lowest position to its farthest on either side. 

Complete Period of a pendulum in motion is the time it occupies in making 
two vibrations. 

All vibrations of same pendulum, whether great or small, are performed 
very nearly in same time. 

Number of Oscillations of two different pendulums in same time and at 
same place are in inverse ratio of square roots of their lengths. 

Length of a Pendulum vibrating seconds is in a constant ratio to force of 
gravity. 

Time of Vibration is half of a complete period, and it is proportional to 
square root of length of pendulum. Consequently, lengths of pendulums for 
different vibrations are— 

Latitude of Washington. 

39.0958 ins. for one second. I 4 344 for third of a second. 

9.774 ins. for half a second. | 2.4435 for quarter of a second. 


Lengths of Rendulnms vibrating Seconds at Level of 
tlie Sea in several Places. 


Equator. 

Washington. 


Ins. 

39.0152 New York 
39.0958 Lat, 45 0 ... 


Ins. 

3Q. 1017 
39- 12 7 


Ins. 

Paris. 39.1284 

London. 39 ' 1 393 


To Compute Length of a Simple Pendulum for a given 

Latitude. 

39.127 — .099 82 cos. 2 L — l. L representing latitude. 

Illustration. —Required the length of a simple pendulum vibrating seconds in 
the latitude of 50 0 31'. 

L = 50 0 31' cos. 2 L — 2 X 50 0 31' — cos. 180 0 — 50 0 31' X 2 = cos. 78° 58' =. 191 38 
— 39.127 + .191 38 X -099 82 (two — or negative = an affirmative or -j-) = 39.1461 ins. 

To Compute Length, of a Simple Pendulum for a given 
Number of "Vibrations. 

L V 2 — l. L representing length for latitude , t time in seconds , and l length of pen¬ 
dulum in ins. 

Illustration. —Required vibrations of a pendulum in a minute at New York, are 
60; what should be its length? 

39.1017 X i 2 =39- 1017, Or, ~^ = l. n representing number of vibrations per second. 

To Compute Number of Vibrations of a Simple Pendu¬ 
lum in a given Time. 

—- - — n. — representing time of one vibration in seconds, 
fl n 

To Compute Centre of Grravity of a Compound Pendu¬ 
lum of Two Weights connected in a Right Line. 

When Weights are both on one Side of Point of Suspension. 


I W + V w 


— 0 — distance oj centre of gravity from point of suspension. 


W + w 














454 


CENTRAL FORCES.—PENDULUMS. 


When Weights are on Opposite Sides of Point of Suspension. 

1 W ~ l f HL — 0 — distance of centre of gravity of greater weight from point of sus- 

W-\~ w 

pension. 

Note.— To obtain strictly isochronous vibrations, the circular arc must be sub¬ 
stituted for the cycloid curve, which possesses the property of having an inclina¬ 
tion, the sine of which is simply proportional to distance measured on the curve 
from its lowest point. 

For construction of a Cycloidal pendulum, see Deschaniel’s Physics, Part I., pp. 
71-2. 

To Compute Lengtli of a Simple Pendulum, "Vibrations 
of which will Tie same in Number as Indies in its 
Length. 

f( 6 o \/L) 2 = l in inches. 

Illustration. —What will be length of a pendulum in New York, vibrations of 
which will be same number as the ins. in its length? 

V (1/39.1017 X 60) 2 = 7.211 2 = 52 ins. 

To Compute Time of Vibration of a Simple Pendulum, 
Lengtli Toeing given. 


fl^-h = tin seconds. 

Illustration. —Length of a pendulum is 156.4 ins.; what is the time of its vibra¬ 


tion in New York? 


I 


156.4 


39.1017 


: 2 seconds. 


Or, y/— X 3-1416 — t. I representing length of a pendulum vibrating seconds in 

ins ., g measure of force of gravity, and t time of one oscillation. 

Illustration. —Length of a simple pendulum vibrating seconds, and measure of 
force of gravity at Washington, are 39.0958 ins., and 32.155 feet. 

3.1416^/~ = 3- i 4 i 6 X V1-013 = 3-1416 X -3183 1 second. 

To Compute ISTum'ber of Vibrations of a Simple Pen¬ 
dulum in a given Time. 

y X t — n. n representing number of vibrations. 

Illustration.— The length of a pendulum in New York is 156.4 ins., and time of 
its vibration is 2 seconds; what are number of its vibrations? 

2 53 60 

— - x 2 =. 5X2 = 1 vibration. Hence. 1 X —• = 30 vi- 

.506 '2 


brations per minute. 


To Compute Measure of Grravity, Length of Pendulum 
and Number of its Vibrations "being given. 


.822 46 l n ! 


: g. g representing measure of gravity in feet. 


To Compute Number of Revolutions of a Conical Pen¬ 
dulum per Minute. 

/ 2 933*5 

yf —j-—- = n. h representing distance between point of suspension and plane of 
revolutions in ins. 


Note.— Number of revolutions per minute are constant for any given height, and 
the time of a revolution is directly as square root of height. 












CRANES. 


455 


CRANES. 

Usual form of a Crane is that of a right-angled triangle, the sides 
being post or jib, and stay or strut, which is hypothenuse of triangle. 

When jib and post are equal in length, and stay is diagonal of a square, 
this form is theoretically strongest, as the whole stress or weight is borne by 
stay, tending to compress it in direction of its length; stress upon it, com¬ 
pared to weight supported, being as diagonal to side of square, or as 1.4142 
to 1. Consequently, if weight borne by crane is 1000 lbs., thrust or com¬ 
pression upon stay will be 1414.2 lbs., or as a e to e W, Fig. 1. 

'Wlieia 3 ?ost is Supported at 'both. Head and. Foot, as 

Fig. 1. 

Weight W is sustained by a rope or chain, 
and tension is equal upon both parts of it; that 
is, on two sides of square, i a and e W. Conse¬ 
quently jib, i a, has no stress upon it, and serves 
merely to retain stay, a e. 

If foot of stay is set at n , thrust upon it, as 
compared with weight, will be as an to aw; 
and if chain or rope from i to a is removed, and 
weight is suspended from a, tension on jib will 
be as i a to a W. 

If foot of stay is raised to 0, thrust, as compared with weight, will be as 
line a 0 is to a W, and tension on jib will be as line ar. 

By dividing line representing weight, as a W or a w, into equal parts, to 
represent tons or pounds, and using it as a scale, stress upon any other part 
may be measured upon described parallelogram. 

Thus, as length of a W, compared to ae, is as 1 to 1.4142: if a W is di¬ 
vided into 10 parts representing tons, a e would measure 14.142 parts or tons. 

When I?ost is Supported, at Foot only. 

If post is wholly unsupported at head, and its foot is secured up to line 
0 W, then W, acting with leverage, e W, will tend to rupture post at e, with 
same effect as if twice that weight was laid upon middle of a beam equal to 
twice length of e W, e being at middle of beam, which is assumed to be sup¬ 
ported at both ends, and of like dimensions to those of post. 

Or, force exerted to rupture post will be represented by stress, W, multi¬ 
plied by 4 times length of lever, e W, divided by depth of post in line of 
stress, squared, and multiplied by breadth of it and Value * of its material. 

Post of such a crane is in condition of half a beam supported at one end, 
weight suspended from other; consequently, it must be estimated as a beam 
of twice the length supported at both ends, stress applied in middle. 

To Compute Stress on Jib, and. on Stay or Strut.-Wig. 3 . 

On diagram of crane, Fig. 2, mark off on line of 
chain, a W, a distance, a b, representing weight on 
chain; from point b draw a line, b c, parallel to jib, 
5 a e, and where this intersects stay or strut, draw a 
vertical line, c 0, extending to jib, and distances 
from a to points b c and 0 c, measured upon a scale 
of equal parts, will represent proportional strain. 

Illustration. —In figure, weight being 10 tons, stress 
on stay or strut compressing, a c, will be 31 tons, and 
on jib or tension-rods, a 0, 26 tons. 





* For Value of Materials, see page 779. 















456 


CRANES. 


To Compute Dimensions of iPost of a Crane. 

When Post is Supported at Feet only. Rule. —Multiply weight or stress 
to be borne in lbs. by length of jib in feet measured upon a horizontal 
plane; divide product by Value of material to be used, and product, divided 
by breadth in ins., will give square of depth, also in ins. 

Example. —Stress upon a crane is to be 22 400 lbs., and distance of it from centre 
of post 20 feet; what should be dimension of post if of American white oak? 

Value of American white oak 50. Assumed breadth 12 ins. 


22 400 x 20 
5° 


= 8960, and 


8960 


: 746.67. Then ^746.67 = 27.32 ins. 


When Post is Supported at both Ends. Rule. —Multiply weight or stress 
to be borne in lbs. by twice length of jib in feet measured upon a horizontal 
plane; divide product by Value of material to be used, and product, divided 
by four times breadth in ins., will give square of depth, also in ins. 

Example. —Take same elements as in preceding case. Assumed breadth 10 ins. 


Then 


22 400 x 20 x 2 


= r 7 9 2 °) 


17 920 


1448, and ^448 = 21.166 ins. 



5° " ' 4 X 10 

In Fig. 3, angle ah e and eb c being equal, chain or rope is represented 
by a b c, and weight by W; stress upon stay b d, as 
compared with weight, is as b d to a b or b c. 

In practice, however, it is not prudent to consider 
chain as supporting stay; but it is proper to disregard 
chain or rope as forming part of system, and crane 
should be designed to support load independent of it. 
It is also proper that angles on each side of diagonal 
stay, in this case, should not be equal. If side a b is 
formed of tension-rods of wrought iron, point a should 
be depressed, so as to lengthen that side, and decrease 
angle ab e\ but if it is of timber, point a should be 
raised, and angle ab e increased. 

Fig. 4. g Fig. 4 shows a form of crane very generally used; 

angles are same as in Fig. 3, and weight suspended from 
it, being attached to point d, is represented by line b d. 
The tension, which is equal to weight, is shown by length 
of line b c, and thrust by length of line b a, measured by 
a scale of equal parts, into which line b d, representing 
weight, is supposed to be divided. 

But if b e be direction of jib, then b g will show ten¬ 
sion, and bf the thrust (df being taken parallel to be), 
both of them being now greater than before; line bd 
representing weight, and being same in both cases. 

To Ascertain Stress on Jit>, on Strut 
of a Crane.—Fig. £3. 

Through a draw a s , parallel to jib or tension-rod 
o r, and also s u parallel to strut a r ; then r s is a 
diagonal of parallelogram, sides of which are equal to r a and r u. 

u If then r s represents a stress of 20 lbs., 
the two forces into which it is decom¬ 
posed are shown by r u and r a ; 0 r is 
equal to r u, as each of them is equal to 
a s, and r s is equal to 0 a. Hence, 20 
represented by a 0, stress on jib will be 
represented by 0 r , and that on strut by 
r a. 

Assuming then 0 r 3 feet, a r 3.5, and 
o a 1, stress on jib will be 60 lbs., and on strut 70. 

















CRANES. 


457 


Thus, in all cases, stress on jib or tension-rod and on strut can be deter¬ 
mined by relative proportions of sides of triangle formed. 


To Compute Stress -upon Stmt of* a Crane. 

Rule. —Multiply length of strut in feet by weight to be borne in lbs.; di¬ 
vide product by height of jib from point of bearing of strut in feet, and 
quotient will give stress or thrust in lbs. 


Example. — Length of strut of a crane is 28.284 feet, height of post is 26.457 feet, 
and weight to be borne is 22 400 lbs.; what is stress? 

28.284X22400 633561.6 

— — 23 947 lbs. 


26.457 


26.457 


Chains and. Ropes. 

Chains for Cranes should be made of short oval links, and should not ex¬ 
ceed 1 inch in diameter. 


Short — linked. Crane Chains and Ropes showing X)i— 
• melisions and. Weight of' each, and Proof of Chain 
in Tons. 


Diam. 

of 

Chains. 

Weight 

per 

Fathom. 

Proof 

Strain. 

Circumf. 

of 

Rope. 

Weight 
of Rope 
per Fath. 

Diam. 

of 

Chains. 

Weight 

per 

Fathom. 

Proof 

Strain. 

Circumf. 

of 

Rope. 

Weight 
of Rope 
per Fath. 

Ins. 

Lbs. 

Tons. 

■ Ins. 

Lbs. 

Ins. 

Lbs. 

Tons. 

Ins. 

Lbs. 

• 3 i 2 S 

6 

•75 

2-5 

i -5 

.6875 

28 

6-5 

7 

10.5 

•375 

8.5 

i -5 

3-25 

2-5 

•75 

32 

7-75 

7-5 

12 

■4375 

1 I 

2-5 

4 

3-75 

.8125 

3 6 

9' 25 

8.25 

15 

•5 

14 

3-5 

4-75 

5 

•875 

44 

io -75 

9 

17-5 

•5625 

18 

4-5 

5-5 

7 

•9375 

50 

12.5 

9-5 

i 9-5 

• 625 

24 

5 -25 

6.25 

8.7 

I 

56 

14 

IO 

22 


Ropes of circumferences given are considered to be of equal strength with 
the chains, which, being short-linked, are made without studs. 

A crane chain will stretch, under a proof of 15 tons, half an inch per fathom. 


Machinery of Cranes. 

To attain greater effect of application of power to a crane, the wheel-work 
must be properly designed and executed. 

If manual labor is employed, it should be exerted at a speed of 220 feet 
per minute. 

Proportions. — Capacity of Crane , 5 tons. 

Radius of winch or handle 15 to 18 ins. Height of axle from floor 36 to 39. 

1st pinion, n teeth, 1.25 ins. pitch. I 2d pinion, 12 teeth, 1.5 ins. pitch. 

1st wheel, 89 “ 1.25 “ “ | 2d wheel, 96 “ “ “ “ 

Barrel 8 ins. X n teeth X 12 teeth Xu 200 ZZ>s.= 30800 ,, x .. , 

' ---—-- --= 20.35 lbs. ----- statical re- 

Winch 17 ms. X 89 teeth x 96 teeth X 4 men = 1513 
sistance to each of the 4 men at winches. 

An experiment upon capacity of a crane, geared 1 to 105, developed that 
a strong man for a period of 2.5 minutes exerted a power of 27 562 foot¬ 
pounds per minute, which, when friction of crane is considered, is fully equal 
to the power of a horse for one minute. 

In practice an ordinary man can develop a power of 15 lbs. upon a crane, 
handle moved at a velocity of 220 feet per minute, which is equivalent to 
3300 foot-pounds. 


For Treatise on Cranes, see Weales’ Series, No. 33. 

Q Q 























458 


COMBUSTION. 


COMBUSTION. 

Combustion is one of the many sources of heat, and denotes combi¬ 
nation of a body with any of the substances termed Supporters of Com¬ 
bustion ; with reference to generation of steam, we are restricted to but 
one of these combinations, and that is Oxygen. 

All bodies, when intensely heated, become luminous. When this heat 
is produced by combination with oxygen, they are said to be ignited; 
and when the body heated is in a gaseous state, it forms what is termed 
Flame. 

Carbon exists in nearly a pure state in charcoal and in soot. It com¬ 
bines with no more than 2.66 of its weight of oxygen. In its combus¬ 
tion, 1 lb. of it produces sufficient heat to increase temperature of 14 500 
lbs. of water i°. 

Hydrogen exists in a gaseous state, and combines with 8 times its 
weight of oxygen, and 1 lb. of it, in burning, raises heat of 50 000 lbs. 
of water 1 0 .* 

An increase in the rapidity of combustion is accompanied by a dimi¬ 
nution in the evaporative efficiency of the combustible. 

Mr. D. K. Clark furnishes the following: When coal is exposed to heat in a fur¬ 
nace, the carbon and hydrogen, associated in various chemical unions, as hydrocar¬ 
bons, are volatilized and pass off. At lowest temperature, naphthaline, resins, and 
fluids with high boiling-points are disengaged; at a higher temperature, volatile 
fluids are disengaged; and still higher, olefiant gas, followed by light carburetted 
hydrogen, which continues to be given off after the coal has reached a low' red heat. 
As temperature rises, pure hydrogen is also given off, until finally, in the fifth or 
highest stage of temperature for distillation, hydrogen alone is discharged. What 
remains after distillatory process is over, is coke, w'hich is the fixed or solid carbon 
of coal, with earthy matter or ash of the coal. 

The hydrocarbons, especially those wdiicli are given off at lowest temperatures, 
being richest in carbon, constitute the flame-making and smoke-making part of the 
coal. When subjected to heat much above the temperatures required to vaporize 
them, they become decomposed, and pass successively into more and more perma¬ 
nent forms by precipitating portions of their carbon. At temperature of low red¬ 
ness none of them are to be found, and the olefiant gas is the densest type that 
remains, mixed with carburetted and free hydrogen. It is during these trans¬ 
formations that the great volume of smoke is made, consisting of precipitated car¬ 
bon passing off'uncombined. Even olefiant gas, at a bright red heat, deposits half 
its carbon, changing into carburetted hydrogen; and this gas, in its turn, may 
deposit the last remaining equivalent of carbon at highest furnace heats, and be 
converted into pure hydrogen. 

Throughout all this distillation and transformation, the element of hydrogen 
maintains a prior claim to the oxygen present above the fuel; and until it is satis¬ 
fied, the precipitated carbon remains unburned. 

Summary of 3?rocLo.cts of Decomposition in tire Furnace. 

Reverting to statement of average composition of coal, page 485, it ap¬ 
pears that the fixed carbon or coke remaining in a furnace after volatile 
portions of coal are driven off, averages 61 per cent, of gross weight of the 
coal. Taking it at 60 per cent., proportion of carbon volatilized in com¬ 
bination with hydrogen will be 20 per cent., making total of 80 per cent, of 
constituent carbon in average coal. 

Of the 5 per cent, of constituent hydrogen, 1 part is united to the 8 per 
cent, of oxygen, in the combining proportions to form water, and remaining 
4 parts of hydrogen are found partly united to the volatilized carbon, and 
partly free. 


* Mean effect. 




COMBUSTION. 


459 


These particulars are embodied in following summary of condition of 
elements of ioo lbs. of average coal, after having been decomposed, and prior 
to entering into combustion— 

ioo Lbs. of Average Coal in a Furnace. 

Composition Lbs. Lbs. Decomposition. 


(Fixed.60 

Carbon {y olatilized .... 2Q 

Hydrogen. 5 

Sulphur.. 1.25 

Oxygen. 8 

Nitrogen. 1.2 

Ash, etc. 4.55 


forming 


60 fixed carbon. 

24 hydrocarbons and free hydrogen. 
1.25 sulphur. 

85.25 

9 water or steam. 

1.2 nitrogen. 

4.55 ash, etc. 


100 


IOO 


showing a total useful combustible of 85.25 per cent., of which 25.25 per 
cent, is volatilized. While the decomposition proceeds, combustion proceeds, 
and the 25.25 per cent, of volatilized portions, and the 60 per cent, of fixed 
carbon, successively, are burned. 

It may be added that the sulphur and a portion of the nitrogen are dis¬ 
engaged in combination with hydrogen, as sulphuretted hydrogen and am¬ 
monia. But these compounds are small in quantity, and, for the sake of 
simplicity, they have not been indicated in the synopsis. 

Volume of Air chemically consumed in complete Combustion of Coal. 


Assume 100 lbs. of average coal. Then, by following 


8o + 3 (s-|) 


-f- 4 X 1-25 X 152 —14060 cube feet of air at 62° for 100 lbs. coal. 


For volatilized portion, Hydrogen (H), 4 lbs. x 457 = 1 828 cube feet. 

Carbon (C), 20 “ Xi52= 3040 “ “ 

Sulphur (S), 1.25 “ X 57= 71_ “ “ 

4939 “ “ 

For fixed portion, Carbon, 60 lbs. x 152= 9120 “ “ 

Total useful combustible, 85.25 “ 14059 “ u for com¬ 

plete combustion 0/100 lbs. coal of average composition at 62°. 


To Compute Volume of Air at 62°, under One At¬ 
mosphere, chemically consumed in Complete Com¬ 
bustion of 1 Lb. of a given IVviel. 

Rule.—E xpress constituent carbon, hydrogen, oxygen, and sulphur, as 
percentages of whole weight of fuel; divide oxygen by 8, deduct quotient 
from hydrogen, and multiply remainder by 3; multiply sulphur by .4; add 
products to the carbon, and multiply sum by 1.52. Final product is volume 
of air in cube feet. 

To compute weight of air chemically consumed. —Divide volume thus found 
by 13.14; quotient is weight of air in lbs. 

Or, 1.52 (C —(— 3 (H — ~) +.4 S) = Air. 0 Oxygen. 

Note.—I n ordinary or approximate computations, sulphur may be neglected. 

Example —Assume 1 lb. Newcastle coal. 0 = 82.24, H = 5.42, 0 = 6.44, and 
S = i. 35 - 

^il = .8o5, 5.42 — .805 = 4.615 X 3 = I 3 - 845 , i -35 X -4 = - 54 , 13-845 +-54+82.24 
= 96.625, and 96.625 X 1-52 = 146.87 cube feet. 

Then 146.87-4-13.14 = 11.18 lbs. 














COMBUSTION - . 



To Compute Total W eight of Gaseous IProdncts of Com¬ 
plete Combustion of 1 I^b). of a given Fuel. 

Rule. —Express the elements as per-centages of fuel; multiply carbon by 
.126, hydrogen by .358, sulphur by .053, and nitrogen by .01, and add prod¬ 
ucts together. Sum is total weight of gases in lbs. 

Or, .126 C + .358 H + .053 S + .oi N = Weight. 


Example. —Assume as preceding case. N = 1.61. 

82.24 X 1.26 + 5.42 X -35 8 + i -35 X 053 + 1.61 X .01 = 12.39 lbs. 

To Compute Total Volume, at 62°, of Gaseous Products 
of Complete Combustion of 1 -Lit), of given Enel. 

Rule. —Express elements as per-centages; multiply carbon by 1.52, hy¬ 
drogen by 5.52, sulphur by .567, and nitrogen by .135, and add products 
together. Sum is total volume, at 62° F., of gases, in cube feet. 

Or, 1.52 C + 5.52 H + . 567 S +. 135 N = Volume. 

To Compute Volume of tire several Gases separately 
from their Respective Quantities. 

Rule. —Multiply weight of each gaseous product by volume of 1 lb. in 
cube feet at 62°, as below. 

Volume of 1 Lb. of Gases at 62° under a Pressure of 14.7 Lbs. 

Cube feet. Cube feet. Cube feet. 

Aqueous Vapor or) I Oxygen. 11.887 I Nitrogen.13-501 

Gaseous Steam, j 21-125 | Hydrogen.190 | Carbonic Acid. 8-594 

Air.13-141 cube feet. 


For a lb. of oxygen in combustion, 4.35 lbs. air are consumed; or, by volume, for 
a cube foot of oxygen 4.76 cube feet of air are consumed. 

1 lb. Hydrogen consumes.34.8 lbs., 

1 “ Carbon, completely burned, consumes_n.6 

1 “ “ partially “ “ _ 5.8 

x “ Sulphur consumes.. 4.35 


or 457 cube feet, at 62°. 

152 “ “ “ “ 

76 “ “ “ “ 

27 £t 11 U U 


Composition 


GASES. 


and. Equivalents of Gases, combined in 
Combustion of Enel. 

B y 


Elements. 


ELEMENTS. 


Oxygen ... 
Hydrogen. 
Carbon.... 
Sulphur... 
Nitrogen.. 


COMPOUNOS. 
^Atmospheric Air 
(mech. mixture).. 
Aqueous Vapor or 
Water. 


Equiv¬ 

alents. 

0 . I 

H. 1 
C. 1 
S. i 
N. i 


0. 23 
N- 77 

0. 1 

H. 1 


Weight. 


8 

1 

6 

16 

*4 


8 i 

26.8 1 
8 ! 


GASES. 


COMPOUNDS. 
Light Carburetted 
Hydrogen. 

Carbonic Oxide.... 

Carbonic Acid. 

Olefiant Gas (Bi-car- 
buretted Hyd. 

Sulphurous Acid... 


Elements. 


Equiv¬ 

alents. 

C. 2 
H. 4 
0. 

C. 

0. 
c. 
c. 

H. 

0. 

s. 


By 

Weight. 


11 

'll 

II 


Weights of products in combustion of ] 
C = .c> 366 . H = .c>9. S = ,o 2 . N = 
Cube Feet. 

.0366 X 8.59^: ,315 volume carbonic 


.09 


acid. 

X 190 


i 7 . 


steam. 


lb. of given fuel, are— 

.0893 C + .268 H + .0335 S + .01 N. 

Cube Feet. 

.02 X 5.85 — .117 volume sulph. acid. 
.0893 + .268 + .0335 + .01 X 13-501 = 
5.409 volume nitrogen. 


Volume of Air or Gases at higher temperatures than here given (62°) is ascer¬ 


tained by, V 1 +4<51 = V'. 

• y ’ t+ 461 
and V' at temperature t'. 


V representing volume of air or gas at temperature t, 


* By Volume 1 Oxygen, 3.762 Nitrogen. 






























COMBUSTION, 


46I 


Chemical Composition, of some Compound Com- 

Lmstibles. 


Combustible. 


Carbonic oxide. 

Light carburetted hydrogen... 
Olefiant gas, Bicarburetted hyd. 

Sulphuric ether. 

Alcohol. 

Turpentine. 

Wax... 

Olive oil. 

Tallow. 


Combining equivalents. 

O 

0 

H 

G 

i—t 

parts by weight. 

Car. 

Hyd. 

Oxy. 

Car. 

Hyd. 

Oxy. 




Per Cent. 

Per Cent. 

Per Cent. 

I 

— 

I 

42.9 

— 

57 -i 

2 

4 

— 

75 

25 

— 

4 

4 

— 

8 5 - 7 

14-3 

— 

4 

5 

I 

64.8 

i3-5 

21.7 

4 

6 

2 

52.2 

13 

34- 8 

20 

16 

— 

88.2 

11. 8 

— 

— 

— 

— 

81.6 

I 3-9 

4-5 

— 

— 

— 

77.2 

i3-4 

9.4 

— 

— 

— 

79 

n-7 

9-3 


Heating powers of compound bodies are approximately equal to sum of 
heating powers of their elements. 

Thus, carburetted hydrogen, which consists of two equivalents of carbon and four 
of hydrogen, weighing respectively 2X6 = 12 and 1 x 4 = 4, in proportion of 3 to 1, 
or .75 lb. of carbon and .25 lb. of hydrogen in one lb. of gas. Elements of heat of 


combustion of one lb. are, then— 

Units of heat. 

For carbon. 14 544 X .75 = 10908 

For hydrogen. 62 032 X -25 = 15 508 

Total heat of combustion, as computed. 26416 

Total heat, by direct trial. 23 513 


Heating Powers of ComTonstilDles. 
(MM. Favre and Silbermann , D. K. Clark and others.) * 


1 Lb. of 
Combustible. 


Hydrogen. 

Carbon, making) 
carbonic oxide. j 
Carbon, making) 
carbonic acid.. J 

Carbonic oxide. 

Light carburetted 1 

hydrogen.) 

Olefiant gas. 

Sulphuric ether.... 

Alcohol. 

Turpentine. 

Sulphur. 

Tallow. 

Petroleum. 

Coal (average). 

Coke, desiccated... 
Wood, desiccated .. 
Wood - charcoal, ) 

desiccated.j 

Peat, desiccated.... 
Peat-charcoal, de-) 

siccated.j 

Lignite. 

Asphalt. 


Oxygen 
consumed 
per lb. of 
Com¬ 
bustible. 

Weight and Volume 
of Air consumed per 
lb. of Combustible. 

Total Heat 
of Combus¬ 
tion of 1 lb. 
of Combus¬ 
tible. 

Equivalent evaporative 
Power of 1 lb. of Com¬ 
bustible, under one At¬ 
mosphere. 

Lbs. 

Lbs. 

Cube Feet 
at 62°. 

Units. 

Lbs. of wa¬ 
ter at 62°. 

Lbs. of wa¬ 
ter at 212°. 

8 

34 - 8 

457 

62 032 

55-6 

64.2 

i -33 

5 - 8 

76 

4 452 

4 

4.61 

2.66 

11.6 

152 

14 500 

13 

15 

•57 

CO 

ci 

33 

4 325 

00 

00 

CO 

4 - 4 8 

4 

17.4 

229 

* 235x3 

21.07 

24-34 

3-43 

15 

196 

21 343 

19. 12 

22.09 

2.6 

n -3 

149 

16 249 

14.56 

16.82 

2.78 

12 . I 

159 

12 929 

11.76 

13 - 3 8 

3-29 

14-3 

188 

19 534 

17-5 

20.22 

I 

4-35 

57 

4032 

3.61 

4.17 

2-95 

12.83 

169 

18 028 

16.15 

18.66 

4. 12 

17-93 

235 

27 53i 

— 

28.5 

2.46 

IO.7 

141 

14 133 

12.67 

14.62 

2.5 

10.9 

i 43 

13 550 

12. 14 

14.02 

1.4 

6.1 

80 

7792 

6.98 

8.07 

2.25 

Vp 

CO 

129 

I 3 3°9 

11.92 

13-13 

i -75 

7.6 

IOO 

9951 

8.91 

10.3 

00 

01 

oi 

9.9 

129 

12325 

11.04 

12.76 

2.03 

8.8 S 

Il 6 

11 678 

— 

12.1 

2-73 

11.87 

156 

16655 

— 

17.24 


When carbon is'not completely burned, and becomes carbonic oxide, it produces 
less than a third of heat yielded when it is completely burned. For heating power 
of carbon an average of 14 500 units is adopted. 

Q Q* 















































COMBUSTION". 


462 


To Compute Heating Power of 1 Hi To. of a given Com¬ 
bustible. 

When proportions of Carbon, Hydrogen, Oxygen , and Sulphur are given. 
Rule. —Ascertain difference between hydrogen and .125 of oxygen; multi¬ 
ply remainder by 4.28; multiply sulphur by .28, add products to the carbon, 
multiply sum by 14500, divide by 100, and product is total heating power 
in units of heat. 

Or, 145 (C —)-4.28 H — Ox . 125-J-.28 S) == heat. 

Illustration. —Assume as preceding case. 

5.42 oj 82.28 X -125 X 4.28 -f- 1.35 X .28 -j- 82.28 X 14 500-P 100 — 15 005. 

To Compute Evaporative Tower of 1 Et>. of a Griven 

Com'bnstihle. 


When Proportions of Carbon , Hydrogen , Oxygen , and Sulphur are given. 
Rule.— Ascertain difference between hydrogen and .125 of oxygen, multiply 
remainder by 4.28; multiply sulphur by .28, add products to the carbon, and 
multiply sum by .13, when water is supplied at 62°, and .15 when at 212 0 ; 
product is evaporative power in lbs. of water at 212 0 . 

Or, When total heating power is known, divide it by 1116 when water is 
at 62°, or 996 when at 212 0 . 

Illustration.— By table, heating power of Tallow is 18028 units. 

Hence, 18 028 - 4 - n 16 = 16.15 Lbs. water evaporated at 62°. 


Temperature of ComTomstioia. 

Temperature of combustion is determined by product of volumes and 
specific heats of products of combustion. 

Illustration.— 1 lb. carbon, when completely burned, yields 3.66 lbs. carbonic 
acid and 8.94 of nitrogen. Specific heats .2164 and .244. 

3.66 X .2164 = .792 units of heat for i°. 

8.94 X-244 = 2.181 “ “ “ i°. 

12.6 2.973 “ u “ i°. 

Consequently, products of combustion of 1 lb. carbon absorbs 2.973 units of heat 
in producing i° temperature. 


"Weight and. Specific Heat of Products of Combustion, 
and Temperature of Combustion. (D. K. Clark.) 

Gaseous Products for 1 Lb. of Combustible. 

x Lb. of Combustible. 


Weight. 

Mean 

specific 

Heat. 

Heat to raise 
the Tempera¬ 
ture i°. 

Temperature of 
Combustion. 

Lbs. 

Water = 1. 

Units. 

O 

Ratio. 

35-8 

• 3 ° 2 

10.814 

5744 

IOO 

11.97 

.256 

3.063 

5305 

92 

i 5-9 

•257 

4.089 

5219 

9 1 

13.84 

.256 

3-54 

5093 

88.7 

ix. 94 

.246 

2-935 

4879 

85 

12.6 

.236 

2-973 

4877 

85 

15.21 

•257 

3 - 9 J 4 

4826 

84 

IO. O9 

.27 

2.68 

4825 

84 

18.4 

.268 

4-933 

4766 

83 

5-35 

.211 

1.128 

3575 

62 

12.18 

•257 

3-127 

3470 

60 

22.64 

.242 

5 - 47 8 

2614 

45 


Hydrogen. 

Sulphuric ether. 

Olefiant gas (Bi-carburetted hyd.) 

Tallow. 

Coal(average). 

Carbon, or pure coke. 

Wax. 

Alcohol. 

Light carburetted hydrogen. 

Sulphur... 

Turpentine. 

Coal, with double supply of air.. 

Whence it appears, that mean specific heat of products of combustion, omitting 
hydrogen .302 and sulphur .211, is about .25. 


Hence, To Ascertain Temperature of Combustion. —Divide total heat of 
combustion in units by units of heat for i°, and quotient will give tem¬ 
perature. 























COMBUSTION. 


463 

Illustration.— What is temperature of combustion of coal of average composi¬ 
tion? 

Gaseous products as per preceding table 11.94, which X -246 specific heat=2-935 
units of heat at i°. 

Hence, 14133 units of combustion (from table, page 461) - 4 - 2.935 = 4812° temper¬ 
ature of combustion of average coal. 

If surplus air is mixed with products of combustion equal to volume of air chem¬ 
ically combined, total weight of gases for one lb. of this coal is increased to 22.64. 
See following table, having a mean specific heat ot .242. 

Then 22.64 x .242 = 5.478 units for i°. 

Hence, 14 133 total heat of combustion -4-5.478 = 2614° temperature of combus¬ 
tion, or a little more than half that of undiluted products. 

Taking averages, it is seen that the evaporative efficiency of coal varies 
directly with volume of constituent carbon, and inversely with volume of 
constituent oxygen ; and that it varies, not so much because there is more or 
less carbon, as, chiefly, because there is less or more oxygen. The per-cent- 
ages of constituent hydrogen, nitrogen, sulphur, and ash, taking averages, 
are nearly constant, though there are individual exceptions, and their united 
effect, as a whole, appears to be nearly constant also. 

Heat of Combustion. 

Or, number of times in combustion of a substance, its equivalent weight of water 
would be raised i°, by heat evolved in combustion of substance. 

Alcohol.12930 I Ether.16246 I Olefiant gas.21 340 

Charcoal.14545 | Olive oil.17750 | Hydrogen.62030 

Combustion of Fax el. 

Constituents of coal are Carbon, Hydrogen, Azote, and Oxygen. 
Volatile products of combustion of coal are hydrogen and carbon, the 
unions of which (relating to combustion in a furnace) are Carburetted 
hydrogen and Bi-carburetted hydrogen or Olefiant gas , which, upon com¬ 
bining with atmospheric air, becomes Carbonic acid or Carbonic oxide, 
Steam, and uncombined Nitrogen. 

Carbonic oxide is result of imperfect combustion, and Carbonic acid 
that of perfect combustion. 

Perfect combustion of carbon evolves heat as 15 to 4.55 compared 
with imperfect combustion of it, as when carbonic oxide is produced. 

1 lb. carbon combines tvith 2.66 lbs. of oxygen, and produces 3.66 lbs. 
of carbonic acid. 

Smoke is the combustible and incombustible products evolved in combustion of 
fuel, which pass off by flues of a furnace, and it is composed of such portions of 
hydrogen and carbon of the fuel gas as have not been supplied or combined with 
oxygen, and consequently have not been converted either into steam or carbonic 
acid; the hydrogen so passing away is invisible, but the carbon, upon being sepa¬ 
rated from the hydrogen, loses its gaseous character, and returns to its elementary 
state of a black pulverulent body, and as such it becomes visible. 

Bituminous portion of coal is converted into gaseous state alone, carbonaceous 
portion only into solid state. It is partly combustible and partly incombustible. 

To effect combustion of 1 cube foot of coal gas, 2 cube feet of oxygen are required; 
and, as 10 cube feet of atmospheric air are necessary to supply this volume of oxy¬ 
gen, 1 cube foot of gas requires oxygen of 10 cube feet of air. 

In furnaces with a natural draught, volume of air required exceeds that 
when the draught is produced artificially. 

An insufficient supply of air causes imperfect combustion; an excessive 
supply, a waste of heat. 








COMBUSTION. 


464 

Volume of atmospheric air that is chemically required for combustion of 
1 lb. of bituminous coal is 150.35 cube feet. Of this, 44.64* cube feet com¬ 
bine with the gases evolved from the coal, and remaining 105.71 cube feet 
combine with the carbon of the coal. 

Combination of gases evolved by combustion gives a resulting volume 
proportionate to volume of atmospheric air required to furnish the oxygen, 
as 11 to 10. Hence the 44.64 cube feet must be increased in this proportion, 
and it becomes 44.64 + 4.46 = 49.1. 

Gases resulting from combustion of the carbon of coal and oxygen of the 
atmosphere, are of same bulk as that of atmospheric air required to furnish 
the oxygen, viz., 105.71 cube feet. Total volume, then, of the atmospheric 
air and gases at bridge wall, flues, or tubes, becomes 105.71 + 49.1 = 154.81 
cube feet, assuming temperature to be that of the external air. Conse¬ 
quently, augmentation of volume due to increase of temperature of a fur¬ 
nace is to be considered and added to this volume, in the consideration of the 
capacity of flue or calorimeter of a furnace. 

There is required, then, to be admitted through the grates of a furnace for 
combustion of 1 lb. of bituminous coal as follows : 

Coal containing 80 per cent, of carbon, or .7047 per cent, of colce. 

1 lb. coal X 44-64 cube feet of gas.= 44.64 

.7047 lb. carbon x 150 cube feet of air ... = 105.71 

150.35 cube feet. 

For anthracite, by observations of W. R. Johnston, an increase of 30 per 
cent, over that for bituminous coal is required = 195.45 cube feet. 

Coke does not require as much air as coal, usually not to exceed 108 cube 
feet, depending upon its purity. 

Heat of an ordinary furnace may be safely considered at iooo°; hence air 
entering ash-pit and gases evolved in furnace under general law of expan¬ 
sion of permanently elastic fluids of ^J g ths of its volume (or .002087) f° r 
each degree of heat imparted to it, the 154.81 is increased in volume from 
ioo° (assumed ordinary temperature of air at ash-pit) to iooo 0 — 900° ; then 
900 X .002 0S7 = 1.8783 times, or 154.81 + 154.81X1.8783 = 445.59 cube feet. 

If the combustion of the gases evolved from coal and air was complete, 
there would be required to give passage to volume of but 445.59 cube feet 
over bridge wall or through flues of a furnace; but by experiments it ap¬ 
pears that about one half of the oxygen admitted beneath grates of a furnace 
passes off uncombined; the area of the bridge wall, or flues or tubes, must con¬ 
sequently be increased in this proportion, hence the 445.59 becomes 891.18. 

Velocity of the gases passing from furnace of a proper-proportioned boiler 

may be estimated at from 30 to 36 feet per second. Then 7 = 

60 x6o X36 

.00687 sq. feet, or .99 sq. ins., of area at bridge wall for each lb. of coal con¬ 
sumed per hour. 

A limit, then, is here obtained for area at the bridge wall, or of flues or 
tubes immediately behind it, below which it must not be decreased, or com¬ 
bustion will be imperfect. In ordinary practice it will be found advan¬ 
tageous to make this area .014 sq. feet, or 2 sq. ins. for every lb. of bitu¬ 
minous coal consumed per sq. foot of grate per hour, and so on in proportion 
for any other quantity. 

Volumes of heat evolved are very nearly same for same substance, what¬ 
ever temperature of combustible. 


* By experiment, 4.464 cube feet of gas are evolved from x lb. of bituminous coal, requiring 44.64 
cube feet of air. 







COMBUSTION. 


465 


Relative Volumes of Air required for Combustion of Fuels. 


Lbs. | 

Warlich’s patent_ 13.1 

Charcoal. 11.16 

Coke. 11.28 


Lbs. I 

Anthracite Coal_12.13 I Bitum. Coal, lowest.. 

Bituminous “ _ 10.98 | Beat, dry. 

Bitum. Coal, average ic.7 | Wood, dry. 


Lbs. 

S- 9 2 

7.08 

6 


Perfect combustion of 1 lb. of carbon requires 11.18 lbs. air at 62°, and 
total Aveight= 12.39 lbs. Total heat of combustion of 1 lb. carbon or char¬ 
coal is 14 500 thermal units; mean specific heat of products of combustion 
is .25, which, multiplied by 12.39 as above = 3.0975, and 14 500* -4- 3.0975 = 
4681° temperature of a furnace, assuming every atom of oxygen that was 
ignited in it entered into combination. 

If, however, as in ordinary furnaces, twice volume of air enters, then 
products of combustion of 1 lb. of coal will be 12.39 + 11.18 = 23.57, which, 
multiplied by its specific heat of .25 as before, and if divided into 14 500, 
quotient will be 2641°, which is temperature of an ordinary furnace. 

Ratio of Combustion. —Quantity of fuel burned per hour per sq. foot of 
grate varies very much in different classes of boilers. In Cornish boilers it 
is 3.5 lbs. per sq. foot; in ordinary Land boilers, 10 to 20 lb^; (English) 13 
to 14 lbs.; in Marine boilers (natural draught), 10 to 24 lbs.; (blast) 30 to 
60 lbs.; and in Locomotive boilers, 80 to 120 lbs. 

Volumes of air and smoke for each cube foot of water converted into 
steam, is for coal and coke 2000 cube feet, for wood 4000 cube feet; and for 
each lb. of fuel as follows: 


Coal. 207 | Cannel coal... 315 | Coke.216 | Wood. 173 

Calorific power of 1 lb. good coal = 14 000 x 772 = 10 808 000 lbs. 


Relative Evaporation of* Several Combustibles in LLs. 
of Water, Heated 1° lay 1 Lb. of IVIaterial. 


Combustible. 

Composition. 

Water. 



Lbs. 

Alcohol.812 

(Hyd. .12) 
(Carb. .45 

8 120 

Bituminous coal... 

( Hyd. .04) 
l Carb. .75} 

9830 

Carbon . 


14 220 

Coke. 

Carb. .84 

Hydrogen (mean).. 

50854 

Oak wood, dry .... 

(Hyd. .06I 
(Carb. .53} 

6 018 

“ “ green... 

(Hyd. .08) 
(Carb. . 37 f 

5662 


Combustible. 

Composition. 

Water. 

Olive oil. 

(Hyd. .i 3 | 

Lbs. 

14560 


(Carb. .77 

Peat, moist. 

(Hyd. .04) 
(Carb. .43) 
(Hyd. .06) 

( Carb. . 58) 
(Hyd. .06( 

(Carb. . 7 j 

3 48 i 

3900 

3618 

“ dry. 

Pine wood, dry.... 

Sulphuric ether. .7 

(Hyd. .13) 
(Carb. .6 f 

8680 

Tallow. 

— 

14 560 


1 lb. Hydrogen will evaporate 62.6 lbs. water from 212 0 = 60.509 lbs. heated i°. 

1 lb. Carbon “ 14.6 lbs. “ 212 0 , or raise 12 lbs. w^ater at 

6o° to steam at 120 lbs. pressure. 

1 lb. of Oxygen will generate same quantity of heat whether in combustion with 
hydrogen, carbon, alcohol, or other combustible. 


Relative Volumes of Gases or Products of Combustion per Lb. of Fuel. 



Supply of Air per lb. of Fuel. 


Supply of Air per lb. of Fuel. 

Temp. 

12 lbs. 

18 lbs. 

24 lbs. 

Temp. 

12 lbs. 

18 lbs. 

24 lbs. 

Air. 

Volume 

Volume 

Volume 

Air. 

Volume 

Volume 

Volume 


per lb. 

per lb. 

per lb. 


per lb. 

per lb. 

per lb. 

O 

Cube Feet. 

Cube Feet. 

Cube Feet. 

O 

Cube Feet. 

Cube Feet. 

Cube Feet. 

32 

150 

225 

300 

572 

314 

471 

628 

68 

i6r 

241 

322 

752 

369 

553 

738 

104 

172 

258 

344 

1112 

479 

718 

957 

212 

205 

307 

409 

1472 

588 

882 

1176 

39 2 

259 

3 8 9 

5 i 9 

2500 

906 

1359 

1812 


* Mean of all experiments 13 964. 





































466 COMBUSTION.-EXCAVATION AND EMBANKMENT. 


To Compute Consumption of Fuel to Heat -Air. 

Rule.—D ivide volume of air to be lieated by volume of 1 lb. of it, at its 
temperature of supply; multiply result by number of heat-units necessary 
to raise 1 lb. air through the range of temperature to which it is to be heated, 
and product, divided by number of heat-units of fuel used, will give result 
in lbs. per hour. 

Example.—W hat is required consumption per hour of coal of an average compo¬ 
sition to heat 776400 cube feet of air at 54 0 to 114 0 ? 

Coal of an average composition (Table, page 461) = 14 133 heat-units. Volume of 
1 lb. air at 54 0 (see formula, page 522) = 54 __ I2 94 cube j- eet% x x n 4 — 54 

X -2377 (specific heat of air) —14.262 heat-units. 

776 400 

•- X 14.262-7-14133 = 60.55 lbs. 

12.94 ^ ^ 

Loss of heat by conduction of it to walls of apartment is to be added to this. 

\ - 

EXCAVATION AND EMBANKMENT. 

Labor and. Work upon Excavation and Embankment. 

Elements of Estimate of Work and Cost. 

Per Day of 10 Hours. 

Cart. —One horse. Distance or lead assumed at 100 feet, or 200 feet for 
a trip , at a speed of 200 feet per minute. 

Earths. —Of gravelly, loam, and sandy, a laborer will load per day into a 
cart respectively 10,12, and 14 cube yards as measured in embankment, and 
if measured in excavation, .11 more is to be added, in consequence of the 
greater density of earth when placed in embankment than in excavation. 

Note.—E arth, when first loosened, increases in volume about .2, but when settled 
in embankment it has less volume than when in bank or excavation. 

Carting. —Descending, load .33 cube yard, Level, .28, and Ascending .25, 
measured in embankment; and number of cart-loads in a cube yard of em¬ 
bankment are, Gravelly earth 3, Loam 3.5, and Sandy earth 4. 

Loosening. —Loam, a three-horsed plough will loosen from 250 to 800 cube 
yards per day. 

Trimming. —Cost of trimming and superintendence 1 to 2 cents per cube 
yard. 

Scooping. — A scoop load measures about .1 cube yard in excavation; 
time lost in loading, unloading, and turning, 1.125 minutes per load; in 
double scooping it is 1 minute. Time occupied for every 100 feet of dis¬ 
tance from excavation to embankment, 1.43 minutes. 

Time. —Time occupied in loading, unloading, awaiting, etc., 4 minutes per 
load. 

To Compute dSTnm'ber ofLoads or Trips in Cube Yards 
per Cart per Day. 

(— - 60 —-— \ h-b-y — n. E representing average distance of carting from em- 

\E 100 -f- 4/ 

bankment in stations of 100 feet each , y number of cart-loads to cube yard of excava¬ 
tion , and n number of cube yards in embankment , hauled by a cart per day to dis¬ 
tance E. 






EXCAVATION AND EMBANKMENT. 


467 


Illustration. —What is number of cube yards of loam that can be removed by 
one cart from an embankment on level ground for-an average distance of 250 feet? 
E = 250 -f- 100 == 2.5, and y — 3.5. 

60 60 

-- X 10 - 4 - 3.5 = -— X 10 -r- 3.5 = 26.37 cube yards. 

2 - 5 t 4 0.5 

Substituting for 3, 3.5, and 4 number of cart-loads in a cube yard of embank¬ 
ment, 20,17.14, and 15, = 60 minutes, divided respectively by these numbers. 


h X 20 


= n, in descending carting ; 


E + 4 ’ ^ E + 4 

ascending, h representing number of hours actually at work. 


17.14 X h ,15 X h 

= n, m level, and — = n, m 


E-f-4 


To Compute Cost of Excavating and Embanking per 

Cu.L>e Yard. 

— + — + £ + s = V. L representing pay of laborers, v value or result of loading 

in different earths, as 10, 12, and 14, c of one cart and driver per day, l cost of loosen¬ 
ing material per cube yard, and s cost of trimming and superintendence, both per 
cube yard, and all in cents. 

Illustration.— Volume of excavation in loam 30000 cube yards. Level carting 
650 feet = 6.5 trips or courses. Loosening by plough 1.7 cents per cube yard, 
laborers 106 cents per day, carts 160, and trimming and superintendence 1.5 cents 
per cube yard. 


v = i2, and 
106 , 


17.14 X 10 


16.33, number of loads per day by preceding formula. 


Then 

yard. 


6.5 + 4 
160 

r.7 + r.S = 8.833 + 9.797 + 1.7 + r.5 = 2i .83 cents per cube 
E artbwork. 


By Carts. —A laborer can load a cart with one third of a cube yard of sandy 
earth in 5 minutes, of loam in 6, and of heavy soil in 7. This will give a result, for 
a day of 10 hours, of 24, 20, and 17.2 cube yards of the respective earths, after de¬ 
ducting the necessary and indispensable losses of time, which is estimated at .4. 

It is not customary to alter the volume of a cart-load in consequence of any dif¬ 
ference in density of the earths, or to modify it in consequence of a slight inclina¬ 
tion in the grade of the lead. 

In a lead of ordinary length one driver can operate 4 carts. With labor at $1 
per day, the expense of a horse and cart, including harness, repairs, etc., is $1.25 
per day. 

A laborer will spread from 50 to 100 cube yards of earth per day. 

The removal of stones requires more time than earth. 

The cost of maintaining the lead in good order, the wear of tools, superintend¬ 
ence, trimming, etc., is fully 2.5 cents per cube yard. 

By Wheel-barrows. —A laborer in wheeling travels at the rate of 200 feet per min¬ 
ute, and the time occupied in loading, emptying, etc., is about 1.25 minutes, with¬ 
out including lead. The actual time of a man in wheeling in a day of 10 hours is .9 
or 2.25 minutes per lead of 100 feet. Hence, 


To Compute Number of Barrow-Loads removed b>y a 

Laborer per 33 a y. 


10 x 60 x • q _ n n , re p resen ti n g number of leads of 100 feet. 
1.25 + n' 

A barrow-load is about .04 of a cube yard. 


Rock. 

By Carts.— Quarried rock will weigh upon an average 4250 lbs. per cube yard, 
and a load may be estimated at .2 cube yard, and weighing a very little more than 
a load of average earth. 

Hence, the comparative cost of carting earth and rock is to be computed on the 
basis of a cube yard of earth averaging 3.5 loads and one of rock 5 loads, with the 
addition of an increase in time of loading, and wear of cart. 









468 


EXCAVATION AND EMBANKMENT. 


Labor. 

For labor of a man, see Animal Power, pp. 433-34. 

By Wheel-barrow. — A barrow-load may be assumed at 175 lbs. =2 cube feet of 
space. 

Blasting. —When labor is $1 per day, hard rock in ordinary position may be 
blasted and loaded for 45 cents per cube yard. 

The cost, however, in consequence of condition, position, etc., may vary from 20 
cents to $ 1. 

See Blasting, page 443. 

17 cube yards of hard rock may be carted per day over a lead of 100 feet, at a cost 
of 7.29 cents per yard. 

The preceding elements are essentially deduced from notes furnished by Ellwood 
Morris, C.E., and the valuable treatise of John C. Trautwine, C.E., Phila., 1872. 

Stone. 

Hauling Stone. —A cart drawn by horses over an ordinary road will travel 1.15 
miles per hour of trip = 2.3 miles per hour. 

A four-horse team will haul from 25 to 36 cube feet of stone at each load. 

Time expended in loading, unloading, etc., including delays, averages 35 minutes 
per trip. Cost of loading and unloading a cart, using a horse-crane at the quarry, 
and unloading by hand, when labor is $1 25 per day, and a horse 75 cents, is 25 
cents per perch =124.75 cube feetr^ 1 cent per cube foot. 

Work done by an animal is greatest when velocity with which he moves is .125 
of greatest with which he can move when not impeded, and force then exerted .45 
of utmost force the animal can exert at a dead pull. 


Earthwork. (Molesworth.) 

Proportion of Getters , Fillers, and Wheelers in different soils, Wheelers being cal¬ 
culated at 50 yards run. 



Gett’s. 

Fill’s. 

Wheel’s. 


Gett’s. 

Fill’s. 

Wheel’s. 

In loose earth, sand, etc. 

X 

1 

1 

In Hard clay. 

I 

1.25 

1.25 

“ Compact. 

I 

2 

2 

“ Compact gravel 

I 

I 

I 

“ Marl. 

I 

2 

2 

“ Rock, from.... 

3 

I 

I 


Average Weight of Earths, Rocks, etc. 
Per cube yard. 



Lbs. 

Sand. 


Gravel.... 

... 3360 

Mud. 




Lbs. 

Marl. 


Clay. 

... 3472 

Chalk.... 

... 4032 


Lbs. 

Sandstone... 4368 

Shale. 4480 

Quartz. 4492 


Granite 
Trap... 
Slate... 


Lbs. 

4700 

4700 

47 10 


Bulk of Rock Earthwork, etc., original Excavation as¬ 
sn m eel at 1. 

When in Embankment. 


Rock, large 
Medium... 
Metal. 


i-5 

1.25 to 1.3 
1.2 


Sand and gravel. 1.07 

Clay and earth after subsidence.... 1.08 
“ “ before “ .... 1.2 


































FRICTION. 


469 


FRICTION. 

Friction is the force that resists the bearing or movement of one sur¬ 
face over another, and it is termed Sliding when one surface moves 
over another, as on a slide or over a pin; and Rolling when a body ro¬ 
tates upon the surface of some other, as a wheel upon a plane, so that 
new parts of both surfaces are continually being brought in contact with 
each other. 

The force necessary to abrade the fibres or particles of a body is 
termed Measure of friction ; this is determined by ascertaining what 
portion of the weight of a moving body must be exerted to overcome 
the resistance arising from this cause. 

Coefficient of Friction expresses ratio between pressure and resistance of 
one surface over or upon another, or of surfaces upon each other. 

Angle of Repose is the greatest angle of obliquity of pressure between 
two planes, consistent with stability, the tangent of which is the coefficient 
of friction. 

Experiments and Investigations have adduced the following observations 
and results: 

1. Amount of friction in surfaces of like material is very nearly propor¬ 
tioned to pressure perpendicularly exerted on such surfaces. 

2. With equal pressure and similar surfaces, friction increases as dimen¬ 
sions of surfaces are increased. 

3. A regular velocity has no considerable influence on friction; if velocity 
is increased friction may be greater, but this depends on secondary or inci¬ 
dental causes, as generation of heat and resistance of the air. 

M. Morin’s experiments afford the principal available data for use. Though con¬ 
stancy of friction holds good for velocities not exceeding 15 or 16 feet per second, 
yet, for greater velocities, resistance of friction appears, from experiments of M. 
Poiree, in 1851, to be diminished in same proportion as velocity is increased. 

4. Similar substances excite a greater degree of friction than dissimilar. 
If pressures are light, the hardest bodies excite least friction. 

5. In the choice of unguents, those of a viscous nature are best adapted for 
rough or porous surfaces, as tar and tallow are suitable for surfaces of woods, 
and oils best adapted for surfaces of metals. 

6. A rolling motion produces much less friction than a sliding one. 

7. Hard metals and woods have less friction than soft. 

8. Without unguents or lubrication, and within the limits of 33 lbs. press¬ 
ure per sq. inch, the friction of hard metals upon each other may be esti¬ 
mated generally at about one sixth the pressure. 

9. Within limits of abrasion friction of metals is nearly alike. 

10. With greatly increased pressures friction increases in a very sensible 
ratio, being greatest with steel or cast iron, and least with brass or wrought 
iron. 

it. With woods and metals, without lubrication, velocity has very little 
influence in augmenting friction, except under peculiar circumstances. 

12. When no unguent is interposed, the amount of the friction is, in every 
case, independent of extent of surfaces of contact; so that, the force with 
which two surfaces are pressed together being the same, their friction is the 
same, whatever may be the extent of their surfaces of eontact. 

13. Friction of a body sliding upon another will be the same, whether the 
body moves upon its face or upon its edge. 


470 


FRICTION, 


14. When fibres of materials cross each other, friction is less than when 
they run in the same direction. 

15. Friction is greater between surfaces of the same character than be¬ 
tween those of different characters. 

16. With hard substances, and within limits of abrasion, friction is as 
pressure, without regard to surfaces, time, or velocity. 

17. The influence of duration of contact (friction of rest) varies with the 
nature of substances; thus, with hard bodies resting upon each other, the 
effect reaches a maximum very quickly; with soft bodies, very slowly; with 
wood upon wood, the limit is attained in a few minutes; and with metal on 
wood, the greatest effect is not attained for some days. 

Coefficient of Friction, of Journals. 


Diameters from 2 to 4 ins. Speeds varied as 1 to 4. Pressure up to 2 tons. 
(From data of M. Morin.) 


Surfaces of Contact. 
Journals. Bearings. 

Cast iron on cast iron. 

Cast iron on gun metal. 

Cast iron on lignum-vitse. 

Wrought iron on cast iron. 

Wrought iron on gun metal. 

Wrought iron on lignum-vitse- 

Gun metal on gun metal. 

Lignum-viLe on cast iron. 


Lubrication. 


Coefficient * 
pressure =' i. 

Ordinary 

Lubrication. 


(Olive oil, or tallow. 
(Unctuous and wet. 
( Olive oil, or tallow. 
| Unctuous and wet. 
(Slightly unctuous . 

< Oil, or lard. 

( Lard and plumbago 
Olive oil, or tallow.. 
( Olive oil, or tallow, 
j Unctuous and wet. 
(Slightly unctuous.. 

(Oil...... 

( Unctuous. 

Oil. 

Unctuous. 


.07 to .08 
.14 

.07 to .08 
. 16 
.18 

.14 

.07 to .08 
.07 to .08 
.19 

•25 

.II 

.19 

. I 

• 15 


* Continuous lubrication reduces the coefficients fully one half. 
Surfaces of Contact. 


Oak on oak.. 

Wrought iron on oak . 


Cast iron on oak. 


Leather on oak. 

Leather belt on oak (Hat).. 
“ “ on oak pulley 


Disposition of Fibres and 
Lubrication. 


Parallel 


and soaped 
“ wet 
“ soaped 
“ wet 
“ soaped 
1 £ wet 
dry 


Coefficient 
pressure = r. 


16 

26 

21 

22 

m 

29 

27 

47 


Perpendicular 

Leather belts over wood drums .47 of pressure, and over turned cast-iron pulleys 
.28 of pressure. 


Coefficients of Friction of Motion. 

Condition of Surfaces and Unguents. 


Substances. 


Hemp cords, etc. {SnTrom 

Metal upon wood. Mean ... 

Sole-leather, smooth, upon wood JRaw_ 

or metal. (Dry. 

Wood upon metal. Mean... 

Wood upon wood.. 


b 

Q 

Water. 

Olive-oil. 

Lard. 

* 

O 

0 

in 

>-» 

■m 

Q 

Greasy 
and wet. 

•45 

•33 

— 

— 

— 

— 

_ 

— 

— 

•15 

— 

.19 

— 

— 

.l8 

• 3 i 

.07 

.09 

.09 

.2 

•13 

•54 

•36 

. l6 

— 

.2 

— 


•34 

• 3 i 

.14 

— 

.14 

:- 

— 

.42 

.24 

.06 

.07 

.08 

. 2 

.14 

•36 

•25 

— 

•°7 

.07 

•15 

*12 






































































FRICTION". 


471 


Relative "Value of "Unguents to Recluce Friction. 


Unguents. 

Wood 

upon 

Wood. 

Wood 

upon 

Metals. 

Metals 

upon 

Metals. 

Unguents. 

Wood 

upon 

W ood. 

Wood 

upon 

Metals. 

Metals 

upon 

Metals. 

Dry soap. 

•4 

•32 

•27 

Olive oil. 

— 

I 

I 

Lard. 

.82 

.85 


Tallow. 



8 

Lard and plumbago. 


.67 

.96 

Water. 

.22 

* yo 

.24 

.18 


To Determine Coefficient of Friction of Bodies. 

Place them upon a horizontal plane, attach a cord to them, and lead it in 
a direction parallel to the plane over a pulley, and suspend from it a scale in 
which weights are to be placed until body moves. 

Then weight that moves the body is numerator, and weight of body moved 
is denominator of a fraction, which represents coefficient required. 

Illustration.— If, by a pressure of 320 lbs. friction amounts to 80 lbs., its coeffi¬ 
cient of friction in this case would be 80 -4-320 — .25. 

Hence, if coefficient of friction of a wagon over a gravel road was .25, and the load 
8400 lbs., the power required to draw it would be 8400 X -25 = 2100 lbs. 

Coefficients of Axle Friction. (M. Morin.) 


Condition of Surfaces and Unguents. 


Substances. 

Dry and 
a little 
Greasy. 

Greasy 
and wet 
with 

W ater. 

Oil, Tallov 

In usual 
way. 

v , or Lard. 

Continu¬ 

ously. 

Very soft 
and puri¬ 
fied Car¬ 
riage 
Grease. 

Bell metal upon bell metal. 

.... 


.097 

.... 

.... 

Cast iron upon bell metal. 

.194 

.161 

•075 

•054 

.065 

Cast iron upon cast iron. 

.... 

•°79 

•075 

•054 

.... 

Cast iron upon lignum-vitse. 

.185 


. I 

.092 

.109 

Wrought iron upon bell metal. 

.251 

.189 

•075 

•054 

.09 

Wrought iron upon cast iron. 

.... 

.... 

•075 

•054 


Wrought iron upon lignum-vitse. 

.188 

.... 

.125 

.... 



Friction of a journal of an axle which presses on one side only, as in a 
worn bearing, is less than when it presses at all points, the difference being 
about .005. 0 

Friction of Axles .—With axles, friction of motion has alone been experi¬ 
mented upon. When weight upon axle and radius of its journal is given, 
mechanical effect of friction may be readily determined. 

The mechanical effect absorbed by, or of friction, increases with pressure 
or weight upon journal of axle and number of revolutions. 

Friction of an axle is greater the deeper it lies in its bearing. 

If journal of an axle lies in a prismatic bearing, as in a triangle, etc., 
friction is greater, as there is more pressure on, and consequently greater 
friction in contact: in a triangular bearing it is about double that of a cyl¬ 
indrical bearing. 

To Compute Mechanical Effect of Friction. 011 Journal 

of an Axle. 

p nfW r _ F n representing number of revolutions , and r radius of journal 

30 

in feet. 

Illustration.— Weight of a wheel, with its axle or shaft resting on its journals, 
is 360 lbs.; diameter of journals 2 ins.; and number of revolutions 30; what is me¬ 
chanical effect of the friction, the coefficient of it being .16? 

3.1416 X 30 X.i6X 360 X 1-^12 _ 452-4 _ i08 lbS ' 

30 


3 ° 















































472 


FRICTION. 


By application of friction-wheels (rollers) friction is much reduced, and 
mechanical effect then becomes, when weights of friction-wheels are disre¬ 
garded, 

r' 

- = F. r' representing radii of axles of friction-wheels, 


p n f W r 
F ‘ — X 


3 ° a’ cos. a - 4 - 2 

a' radii of friction-wheels, and a angle of lines of direction between axis of roller 
and axis of friction-wheels. 

2 1) V Tl F 

When a single friction-wheel is used, ——-X /W = F, and — = F'. F' 

representing mechanical effect. 

Illustration.— A wheel and its shaft, making 5 revolutions per minute, weighs 
30000 lbs.; its diameter and that of its journals are 32 feet and 10 ins. The journals 
rest upon a friction-wheel, the radius of which is 5 times greater than its axle. 

1. What is the power at circumference of wheel necessary to overcome friction? 
2. What is mechanical effect of the friction? 3. W^hat is reduction of friction by 
use of the friction-wheel? 


324-2 X 12 


: 38.4, circum. of wheel = 38.4 times that of axle. 


Coefficient of friction assumed at .075. 


30000 X -073 

Hence ---- — =58.59 lbs.—power 


at circum. to overcome friction at axle, 
by friction. 


10 X 3-1416 


38-4 


2.618 feet = distance passed 


Consequently, 


2.618 X 5 
60 


: feet =. distance passed by friction in one second. 


Hence, .2181 X 2250 (30000 X 075) =490.725. 3. 1-4-5 =.2 — radius of friction- 

axle-^ by radius of friction-wheel, and 38.4 X -2 = 7.68 = friction referred to circum. 

of wheel, and — 0,725 = 98.145 — mechanical effect by application of friction-wheel 

= a reduction of four fifths. 

Friction of Rivots. 

Friction on Pivots is independent of their velocity, increases in a greater 
degree than their pressures, and approximates very near to that of sliding 
and axle friction. 

Friction on Conical Bearings is greater than with like elements on plane 
surfaces. 

Figure of point of a pivot, as to its acuteness, affects friction : with great 
pressure the most advantageous angle for the figure ranges from 30° to 45 0 ; 
with less pressure it may be reduced to to 0 and 12 0 . 

Relative Value of Angles of Pivots. 

6 °-. 1 I i5°.66 | 45°. 39 

Relative Values of different ^Materials for use as Rivots. 


Agate. 


.83 [ Granite.1 


Glass.55 | llock crystal.76 


Tempered steel.44 


Friction, and. Rigidity of Cordage. 

Experiments by Amonton and Coulomb, with an apparatus of Amonton’s, 
furnish the following deductions : 

1. That resistance caused by stiffness of cords about the same or like pul¬ 
leys varies directly as the suspended weight. 

2. That resistance caused by stiffness of cords increases not only in direct 
proportion of suspended weights, but also in direct proportion of diameter 
of the cords. 





















FRICTION. 


473 


Consequently, that resistance to motion over the same or like pulleys, 
arising from stiffness of cords, is in direct compound proportion of suspend¬ 
ed weight and diameter of cords. 

3. That resistance to bending varied inversely as diameter of sheave or 
drum. 


s \ CT 

4. That complete resistance is represented by expression — . S rep¬ 
resenting constant for each rope and sheave , expressing stiffness of rope; T 
tension of rope which is being bent , expressed by C T; C constant for each 
rope and sheave; and d diameter of sheave, including diameter of rope. 

5. That stiffness of tarred ropes is sensibly greater than that of white ropes. 


Extending results obtained by Coulomb, Morin furnishes following for¬ 
mulas : 


For White Ropes: 12 n-r-d (.002 15-f-.001 77 n-f-.ooi2 W) = R. For Tarred 
Ropes : 12 n-—d (.010 54 -f- .0025 a-j- .0014 W) = R. R representing rigidity in lbs., 
n number of yarns, d diameter of sheave in ins. and rope combined, and W weight 
in lbs. 

Illustration. —What is value of stiffness or resistance of a dry white rope hav¬ 
ing a diameter of 60 yarns, which runs over a sheave 6 ins. in diameter in the 
groove, with an attached weight of 1000 lbs. ? 

Assume diameter for 60 yarns to be 1.2 ins. Then — 2 X (.002 i5-}--ooi 77 x 

7. ^ 

60 -f- .0012 X 1000) = 100 x 1-308 35 = 130.835 lbs. 


Value of natural stiffness of ropes increases as the square of number of 
threads nearly, and value of stiffness proportional to tension is directly as 
number of threads, being a constant number. Hence, having the rigidity for 
any number of threads, the rigidity for a greater or lesser number is readily 
ascertained. 

Wire Ropes. 

Weisbach deduced from his experiments on wire ropes that their rigidity 
for diameters capable of supporting equal strains with hemp ropes is con¬ 
siderably less. 

Wire ropes, newly tarred or greased, have about 40 per cent, less rigidity 
than untarred ropes. 

Rolling Friction. 

Rolling Friction increases with pressure, and is inversely as diameter of 
rolling body. 

For rolling upon compressed wood, f =.019 to .031. 

When a Body is moved upon Rollers and Power applied at the Base of the Body , 
W 

(f-\-f ')— = F. f and f' representing coefficients of friction of two surfaces upon 
which rollers act. 

When Power is applied at Circumference of Roller, fW - 4 -r = F. 

When Power is applied at Axis of Roller, f W - 4 - r -4- 2 m F. 


Bearings for Propeller Shaft. {Mr. John Penn.) 


Bearings. 

Pressure 

per 

Sq. Inch. 

Time 
of Op¬ 
eration. 

Bearings. 

Pressure 

per 

Sq. Inch. 

Time 
of Op¬ 
eration. 


Lbs. 

Min. 


Lbs. 

Min. 

Babbit’s metal on iron*... 

1600 

8 

Brass on ironf. 

675 

60 

Rnx on brass. 

4480 

5 

Brass on iron t. 

4480 

_ 

Box on iron. 

448 

30 

Lignum-vitae on brass .. 

4000 

5 

Brass on brass. 

448 

30 

Snake-wood on brass ... 

4000 

5 

Brass on iron. 

448 

3 ° 

Lignum-vitae on iron ... 

1250 

2160 


* Rolled out. + Abraded. f Set fast. 

R R* 





















474 


FRICTION. 


Hesult of Experiments upon Friction, of Several Instru- 


mends. (K. <S. 
Instrument. j Friction. 

Ball .) 

Velocity ratio. 

Mechanical 

efficiency. 

Useful 

effect. 

Pulley, single. 

F L 

2.21 +.5453 

2 

1.8 

Per Cent. 
90 

“ 3 sheaves. 

2.36 +.238 

6 

4 

64 

“ differential. 

3.87 +.151 

l6 

6.1 

38 

Screw. 

.0 -(-.014 

193 

70 

3 6 

Inclined plane, angle 17 0 2'.... 

•09 +.55 

3-4 

1.72 

5 i 

Screw Jack. 

.66 -(-.007 

4 J 4 

116 

28 

Wheel and Axle. 

.204+ .043 

3 i 

22 

70 

“ “ Barrel. 

.5 +-169 

5-95 

5-55 

93 

“ “ Pinion. 

2.46 +.21 

8 

4 - 1 

5 i 

Crane. 

.0 +.056 

23 

18 

78 

U 

. 185 -j- .008 

i 37 

87 

63 


F representing friction, and L load. 

Illustration i.—I f it is required to ascertain power necessary to raise 200 lbs. 
2 feet, by a single movable pulley, 200 X .5453-1-2.21 = 111.27 lbs., which must be 
applied as power to raise 200 lbs. 2 feet. 111.27X 2 = 222.54 lbs. Hence, for appli¬ 
cation of 222.54 lbs., 200 or 89.87 per cent, are usefully or effectively employed. 

2. —If it is required to raise 100 lbs. by a three-sheave pulley, then 100 X .238-f- 
2.36 = 26.16 lbs, which must be applied as power to raise 100 lbs. 6 feet (3X2 = 6). 
26.16 X 6 = 156.96 lbs. Hence, for application of 156.96 lbs., 100 or 63.71 per cent, 
are effectively employed. 

3. —The velocity ratio of a crane being 137, and its mechanical efficiency 87, a 
man applying 26 lbs. to it can raise 87 x 26 = 2262 lbs. 


Application of preceding Results. 

Illustration. — If a vessel, including cradle, weighing 1000 tons, is to be drawn 
upon an inclined plane having a rise of 10 feet in 100 of its length, what will be the 
resistance to be overcome, the cradle being supported on wrought-iron axles in cast- 
iron rollers, running on cast-iron rails? 

1000 X io __ tons—power required to draw vessel independent of friction. 

100 

Ratio of friction to pressure of wrought iron on cast, in an axle and its bearing, 
.075. Ratio of ditto of cast iron upon cast, say .005. 

Hence .075-)- .005 = .08 of 1000 tons = 80 tons, which, added to 100 tons before de¬ 
ducted, gives 180 tons, or resistance to be overcome. 

Power or effect lost by friction in axles and their bearing may be ex¬ 
pressed by formula 

W -f n* 

--— = P. f representing coefficient of friction, d diameter of axle in ins., and 

23O 

r number of revolutions per minute. 

Illustration.— Pressure on piston of a steam-engine is 20000 lbs., number of 
revolutions 20, and diameter of driving shaft of wrought iron in a brass journal is 
8 ins.; what is the effect of friction ? 

20000 X .07 X 8 X 20 77 

-—-= 973 - 9 I lbs ■ 

Hence P a- 4 - 33 000 = IP. v representing circumference of shaft infect X by revo¬ 
lutions per minute. 

The power or effect lost by friction in guides or slides may be expressed 
by following formula: 

W fsr 

-- y, — -— = P. s representing stroke of cross-head, and l length ofcon- 

60 X V (5 1 — s ) 
necting rod in feet . 























FRICTION. 


475 


IFT*ictiorLal Resistances. 


Friction of Steam-engines. 

Friction of Condensing Engines in Xfbs. per Sq. Inch. 

of Piston. 


Diameter 

of 

Cylinder. 

Oscillating 

and 

Trunk. 

Beam 

and 

Geared. 

Direct- 
acting and 
Vertical. 

Diameter 

of 

Cylinder. 

Oscillating 

and 

Trunk. 

Beam 

and 

Geared. 

Direct- 
acting and 
Vertical. 

IO 

5 

6 

7 

50 

2-5 

2.7 

3-3 

15 

4 

5 

6 

60 

2.4 

2.6 

3 

20 

3-5 

4 

5 

70 

2 * 3 

2-5 

2.7 

25 

3 

3 - 6 

4 -S 

80 

2 

2-3 

2.6 

30 

3 

3-5 

4 

IOO 

1.6 

2.2 

2-5 

35 

2.6 

3 

3-5 

no 

i -5 

2 

2.1 


Experiments upon different steam-engines have determined that friction, 
when pressure on piston is about 12 lbs. per sq. inch, does not exceed 1.5 lbs., 
or about one tenth of power exerted. 

Friction of double cylinder (50-inch diam.) direct-acting condensing pro¬ 
peller engine is 1.25 lbs. per sq. inch of piston = 10.3 per cent, of total power 
developed; friction of load is .9 lbs. per sq. inch of piston = 7.5 per cent, of 
total pressure; and friction of propeller is 1.3 lbs. per sq. inch of piston = 
10.8 per cent, of total power = 28.6 per cent. 

Friction of double cylinder (70-inch diam.) inclined condensing water¬ 
wheel engine with its load is 15 per cent, of total power developed. 

In general, when engines are in good order, their efficiency ranges from 80 
per cent, for small engines to 93 per cent, for large. 

Power required to work air-pumps is 5 per cent., and to work feed-pumps 
1 per cent. 

Kesialts of Experiments upon Friction of Machinery. 

(Davison.) 

Steam-engine , vertical beam, one tenth its power; 190 feet horizontal, and 
180 feet vertical shafting, with 34 bearings, having an area of 3300 sq. ins., 
with 11 pair of spur and bevel wheels ; 7.65 EP. 

Set of three-throiv Pumps , 6 ins. in diam., delivering 5000 gallons per hour 
at an elevation of 165 feet; 4.7 IP, or about 13 per cent. 

Two pair iron Rollers and an elevator, grinding and raising 320 bushels 
malt per hour; 8.5 BP. 

Ale-mashing Machine , 800 bushels malt at a time; 5.68 EP. 

Archimedes Screw (ninety-five feet), 15 ins. in diameter, and an elevator 
conveying 320 bushels malt per hour to a height of 65 feet; 3.13 IP. 

Friction Clutch . —Driven by a leather belt 14 ins. in width; face of clutch 
5 ins. deep; broke a cast-iron shaft 6.5 ins. in diameter. 

Flax Mill ( 3 /. Cornut , 1872).—Two condensing engines, cylinders, 12.9 
ins. X 44.3 ins. stroke, and 22 ins. X 59.8 ins. stroke. Pressure of steam, 
50 lbs. per sq. inch; revolutions, 25 per minute. Friction of entire machin¬ 
ery, 20 per cent. 

With vegetable oil and hand oiling a steam pressure of 62 lbs. per sq. 
inch was required, and with mineral oil and continuous oiling a pressure of 
50 lbs. only was required. 

By continuous oiling, a saving of 44 per cent, was effected over hand 
oiling. 

















476 


FRICTION. 


!Klax Mill. 


Power required to Drive Engine, Shafting, and. entire 

^Machinery. (M. Cornut.) 


Parts. 


Engines, shafting, and belts. 

4 cards. 

14 drawing frames (29 heads or 156 

slivers). 

4 combing machines. 

6 roving frames (330 spindles). 

20 spinning frames. 

Dry (1480 spindles)... 

Wet (2080 “ ).. 

Total 150.11 IP. 


Indicated Horse-power. 


Total. 

One Machine 

Effect of 

at work. 

empty. 

Machines. 

3 °- 4 x 

— 

— 

— 

8.42 

2.105 

1.423 

32 

7.19 

•°93 4 

•°794 

15 

2.22 

•555 

•151 

78 

7.78 

.026 27* 

2-434 

7-3 

47-5 

.0321* 

2-515 

21.6 

46-59 

.022 4* 

1.613 

19 


* Per 100 spindles. 


Estimate of Horse's Power .—2080 spindles, wet, 34.4 per IP, long fibre. 

640 “ dry, 20.1 “ “ “ “ 

840 “ “ 14.5 “ “ tow. 

3560 “ average, 23.7 il “ 

The IP per 100 spindles varies inversely as sq. root of their number. 


Winding Engine (G. H. Daglisli). 

Shafts 738 to 1740 feet in depth ; cylinder 65 X 84 ins. stroke ; pressure of steam 
19 lbs. per sq. inch; revolutions 12.5 per minute; mean diameter of drum, 26 feet. 
IP 313.4; effect 235 = 75 per cent. 


Single shearing, . i-f- 


Tools. 

n t 2 


26.7 


( Dr. Hartig). 

= IP to drive tool, n representing number of 


cuts per minute, t thickness of plate, and 


a F 


IP to shear, a representing 


1980 000 

area of surf ace cut or punched per hour in sq. ins., and F (1166 -f-1691 t) a factor ex¬ 
pressing work required to cut or shear a surface of 1 inch square. 

Illustration. —A shearing machine cutting 4648 sq. ins. of surface per hour, in 
plates .4 inch thick, required .68 IP to run and 4.3 to operate it, equal to 5 horses. 


Iron Plate-bending. 


85 000 b t 2 l 


= P for cold plates, and 


11 300 b t 2 l 


r ' r 

— P for red-hot plates, b, t, and l representing breadth, thickness, and length of plate, 
r radius of curvature, all in ins., and P net power of bending. 

Power for large rolls when running only .5 to 6 BP. 


Ordinary Cntting Tools, in Metal. 

Materials of a brittle nature, as cast iron, are reduced most economically in power 
consumed, by heavy cuts; while materials which yield tough curling shavings are 
more economically reduced by thinner cuttings. Following formulas apply to light 
cutting work: 

Power required to plane cast iron is— 


Dlaning Cast iron, W (.0155 -(- —- -) == BP. W representing weight of cast 

\ II 000 s/ 

iron removed per hour, in lbs., and s average sectional area of shavings, in sq. ins. 

Steel, Wrought iron, and Gun-metal, with cuts of an average character— 

Steel.112 W = IP | Wrought iron, .052 W=EP | Gun-metal, .0127 W = EP 


Dialling and Molding. — Run without cutting. 
resenting sum of revolutions of all the shafts per minute. 



2900 


N rep- 

























FRICTION. 


477 


.022 68 

Molding. — Pine, .0566 + -—— , and Red Beech, 088 95 -f- - 


007 3 1 

h 


= EP. h rep¬ 


resenting depth of wood cut down to form molding. 

Turning. —Steel, .047 W^IP; Wrought iron, .0327 W = EP; Cast iron, 
.0314 W = EP. 

For turning off metals, power required is less than for planing, and it is ascer¬ 
tained that greater power is required for small diameters than large. 

Light Lathes , .05-)-.0005 n — BP; 1 or 2 shafts, .05+ .0012 n = BP; 3 or 4 shafts, 
• 05 -(- .05 n = IP. Heavy Lathes, .025 -j- .0031 n ; .025 -j- .053 n; .025-]-. 18 n. 

n representing number of revolutions of spindle per minute. 

Drilling. — Power required to remove a given weight of metal is greater than 
in planing. Volume being taken in place of weight. 


Holes from .4 to 2 ins. in diameter. 
Cast iron, dry. V ^.0168 -j- IP. Wrought iron 


- o ' 1 - V (-° ,68 + ;S ?) : 


IP. 


V representing volume removed in cube ins. per hour, and d diameter of hole. 


Without gearing, .0006 n -f- .0005 n'\ with gearing, .0006 n + .ooi n '; radial 
drills without gearing, .0006 n -f- .004 n'\ radial drills with gearing, .04 -j-.ooo6 n -(- 
.004 n'. n representing number of revolutions per minute of gearing shaft, and n' 
of drill. 

Ti S 

Slotting.—Stroke 8 ins. .045 -j-= IP. n representing number of strokes 


per minute, and s stroke in ins. 


4000 


"Wood-sawing, Circular.—A cube foot of soft wood and half a cube 
foot of hard, reduced to sawdust, requires 1 IP. 

Hard wood, = IP'. Soft wood, —- — IP'. A representing area in sq. feet 
and IP' horse-power per sq foot, both cut per hour, and c width of cut in ins. 

From .4 to 4 ins. in diameter. —Pine. V ^ 000125 -(- - °°^ = IP. 


Dry pine timber. .004 28 -}-. 0065 


Sc 

/ 


IP'. S representing stroke of saw in feet, 


and f feed per cut in ins. 

71 > (1 

-= IP for horse power to run only without cutting, d representing diameter 

32 000 

of saw in ins., and n number of revolutions per minute. 

Net power required to cut with a circular saw is proportional to volume of ma¬ 
terial removed. P"or a saw cutting hot iron, at a circumferential speed of 7875 feet 
per minute, and making a cut .14 inch wide, power is expressed by formulas— 

.702 A = EP, for red-hot iron. 1.013 A = IP, for red-hot steel. 

A representing sectional area of surf ace cut through, in sq. feet. 

S c 

Vertical Saw. .004 28 -j- .0065 — r =BP in dry pine timber per sq. foot 

per hour. S representing stroke of saw in feet, c width of cut in ins., and ffeed of 
cut in ins. 

0034 + — - BP' in Pine. .004 83 + - 9 - 57 = IP 'in Oak. 

10000 / 1 10000 / 


Band Saw. 
1.127 c v 


.00576-, 

10000/ 
in feet per minute. 


: IP' in Beech, v representing velocity of saw, and f rate of feed, 


Screw Cutting. 


5 l 

Screws, —— = IP. 
diameter in ins. , and l length cut in feet per hour. 
Machine of medium dimensions, .2 BP. 


Z d 3 

Taps,-— IP. 

29 


d representing 











478 


FRICTION. 


Grindstones. —-= IP. p representing pressure upon stone , v circum- 

3300° 

ferential velocity of stone in feet per minute , and C coefficient of friction. 

Coefficients of Friction between Grindstones and Metals. 

Cast iron, .22 at high speed, .72 at low speed; Wrought iron, .44 at high speed, 
1 at low; Steel, .29 at high speed, .94 at low. 


Power required to run them alone. 

Large...000 040 9 d v = IP I Small.16-f- .000089 5 ^ v — IP 

or.000128 d 2 n = IP | or.16-]--00028 d 2 n =IP 

Gf rain Conveyers. 

Conveyers of Grain horizontally by Screws and Bands .—A 12-inch screw, having 
4 ins. pitch, turning in a trough, with a clearance of .25 inch, revolving with a 
speed of maximum effect, 60 turns per minute, will discharge 6.75 tons of grain 
per hour, expending .04 IP per foot run. Sectional area of body of grain moved 
49 per cent, of that of screw. At speeds above 60 turns per minute, the grain will 
not advance, but will revolve with screw. 

Screw Steamer. (Vice-admiral C. R. Mooi'som, II. N.) 

Moving friction of hull.07 Slip of screw.171 

Moving friction of load.063 Resistance of hull.606 

Moving friction of rotation of 1 - 

blades of screw.j. 09 1 

Sicfe Lever Steam-engine. (J. V. Merrick.) 

In Pressure of Steam. 


Friction to work air-pump.. 

Friction of weight of parts... 

Friction of cylinder packing. 

Friction of air-pump packing.. 

Friction of valves, parallel motion, resistance to air, etc. 


.585 to .7 lb. 

•5 “ -5 “ 

.15 “ .3 “ 

.046 11 .092 “ 
.169 “ .258 “ 


Hence 


i-45 + 1-85 
2 


1.45 1.85 lbs. 

= 1.65 lbs. per sq. inch. If journals are kept constantly lubri¬ 


cated, as with automatic lubrications, friction of weight will be reduced to .33, and 
pressure will be reduced from 1.65 — .33 to 1.32 lbs. per sq. inch of piston to work 
engine without load. Friction of load, according as journals are lubricated, ends 
keyed up, etc., will range from 2 to 5 per cent. 


Locomotives and Railway Trains. See Railways, page 682. 

Friction developed in Launching of 'V'essels. 

Experiments made by a committee of Franklin Institute on friction of launching 
vessels gave, when pressure or weight was from 2280 to 3560 per sq. foot, a co¬ 
efficient of .0335. 

Marine Railway .—To draw 3000 tons upon greased slides a power of 250 tons was 
necessary to move it, but when started 150 tons would draw it. 

Woollen Machinery. (Dr. Hartig.) When running empty 8.15 IIP, and at work 
32-97- 

The efficiency of the various machines averaging 60.5 per cent. 

Friction of a. INTon-condensing Steam-engine. 

Fi'iction of an Engine. Diameter of cylinder 20 ins. by 40 ins. stroke of piston. 


Revolutions, 15 to 70 per minute. 

Engine, unloaded, 2 lbs. per sq. inch.= 1.86 to 8.69 IP. 

Shafting, unloaded, 2.5 to 45 lbs. per sq. inch.= 2.36 to 19.61 “ 

Total 4.5 to 6.5 lbs. per sq. inch... — 4.22 to 28.3 “ 

























FUEL. 


479 


FUEL. 


With equal weights, where each kind is exposed under like advan¬ 
tageous circumstances, that which contains most hydrogen ought, in its 
combustion, to produce greatest volume of flame. Thus, pine wood is 
preferable to hard, and bituminous to anthracite coal. 

When wood is used as a fuel, it should be as dry as practicable. 
To produce greatest quantity of heat, it should be dried by direct ap¬ 
plication of heat; usually it has about 25 per cent, of water combined 
with it, heat necessary for evaporation of which is lost. 

Different fuels require different volumes of oxygen; for different 
kinds of coal it varies from r.87 to 3 lbs. for each lb. of coal. 60 cube 
feet of air is necessary to furnish 1 lb. of oxygen; and, making a due 
allowance for loss, nearly go cube feet of air are required in furnace of 
a boiler for each lb. of oxygen applied to combustion. 


Classifies f Semi-bitumin.. 
tion -1 

of Coal. Bituminous... 


(Cherry. 
1 Splint. 

1 Caking. 
1 Cherry. 
(Splint. 


Hydrogenous or) 
Gas coal.) 

Anthracites. 


1 Shaly. 

(Asphalt. 

(Hard. 

(Semi or gaseous. 


Bituminous Coal. 

Lignite. Brown Coal or Bituminous Wood. —Presents a distinct woody 
structure; is brittle, and burns readily, leaving a white ash, and contains 
and absorbs moisture in some cases fully 40 per cent. 

Caking. —Fractures uneven, and when heated breaks into small pieces, 
which afterwards agglomerate and form a compact body. When the pro¬ 
portion of bitumen is great, it fuses into a pasty mass. This coal is unsuit¬ 
ed where great heat is required, as the draught of a furnace is impeded by 
its caking. It is applicable for production of gas and coke. 

Splint or Hard. —Color black or brown-black, lustre resinous and glisten¬ 
ing. It kindles less readily than caking coal, but when ignited produces a 
clear and hot fire. 


Cherry or Soft. —Alike to splint coal in fracture, but its lustre is more 
splendent. Does not fuse when heated, is very brittle, ignites readily, and 
produces a bright fire with a yellow flame, but consumes rapidly. 

Cannel. —Color jet, or gray or brown-black, compact and even texture, a 
shining, resinous lustre. Fractures smooth or flat, conchoidal in every di¬ 
rection, and polishes readily. 

Experiments upon practical burning of this description of coal in furnace of a 
steam-boiler give an evaporation of from 6 to 10 lbs. of fresh water, under a pressure 
of 30 lbs. per sq. inch per lb. of coal; Cumberland (Md., U. S.) coal being most ef¬ 
fective, and Scotch least. 


, Limit of evaporation from 212 0 for 1 lb. of best coal, assuming all of heat 
evolved from it to be absorbed, would be 14.9 lbs. 

Coals that contain sulphur, and are in progress of decay, are liable to spontaneous 
combustion. 


There are very great variations in the chemical composition and proper¬ 
ties of coals. 


American. 

Carbon, from 75 to 80 per cent. 
Hydrogen, from 5 to 6. 
Oxygen, from 4 to 10. 
Nitrogen, from 1 to 2. 

Sulphur, from .4 to 3. 

Ash, from 3 to 10. 

Coke, from 48.5 to 79.5. 


Bmtish. 

Carbon, from 70 to 91 per cent. 
Hydrogen, from 3.5 to nearly 7. 
Oxygen, from about. 5 to 20. 
Nitrogen, from a mere trace to 2.2. 
Sulphur, from o to 5. 

Ash, trom .2 to 15. 

Coke, from 49 to 93. 


For Volume of Air, etc., see Combustion, page 465. 






480 


FUEL. 


Coal. 

Anthracite. 

Anthracite or Glance Coal , or Culm —Is hard, compact, lustrous, and some¬ 
times iridescent, most perfect being entirely free from bitumen; it ignites 
with difficulty, and breaks into fragments when heated. 

Evaporative power, in furnace of a steam-boiler and under pressure, is 
from 7.5 to 9.5 lbs. of fresh water per lb. of coal. 

Coal from one pit will sometimes vary 6 per cent, in evaporative value. 


Elements of Various American Coals. 



Specific 

Gravity. 

Fixed 

Carbon. 

Volatile 

Matter. 

Water. 

Moist¬ 

ure. 

Ash. 

Earthy 

Matter. 

Illinois, Warren Co. 

I.23 

Per 

Cent. 

S 1, 7 

Per 

Cent. 

43 -i 

Per 

Cent. 

Per 

Cent. 

Per 

Cent. 

Per 

Cent. 

5-2 

Bureau “ . 

1.32 

57 - 6 

28.8 

— 

II .2 

2.4 

— 

Mercer “ . 

1.26 

54-8 

31.2 

— 

8.4 

5-6 

— 

Indiana, Clay “. 

1.28 

56.5 

32-5 

8-5 


2.5 

— 

Coopriders. 

1.28 

50.5 

42-5 

3 

— 

4 

— 

Pennsyl -1 Connellsviile. 

1.28 

65 

24 

4-5 

— 

6-5 


vania ) Youghiogheny ... 

i -3 

58-4 

35 

— 

I 

5-6 

— 

Fayette Co. 

1.29 

58 

34 

3 

— 

5 

— 

Kentucky, Sardric. 

1.32 

5 i 

4 2 -5 

2 

— 

4-5 

— 

Mud River. 

1.28 

57 

37 

3-5 

— 

2-5 

— 

Ohio, Nelsonville. 

1.27 

58-4 

33-°5 

6.65 

•— 

1.9 

— 

Colorado, Carbon City. 

I.21 

56.8 

34-2 

4-5 

— 

4'5 

— 

Washington Territory. 

1.32 

58.25 

31-75 

7 

— 

3 

— 


Coke. 

Coke. —Coking in a close oven will give an increase of yield of 40 per cent, 
over coking in heaps, gain in bulk being 22 per cent. Coals when coked in 
heaps will lose in bulk. 

Cannel and Welsh (Cardiff) coals when coked in retorts will gain from 10 
to 30 per cent, in bulk and lose 36.5 per cent, in weight. 

Relative costs of coal and coke for like results, as developed by an ex¬ 
periment in a locomotive boiler, are as 1 to 2.4. 

Evaporative power in furnace of a steam-boiler and under pressure, is 
from 7.5 to 8.5 lbs. of fresh water per lb. 

Bituminous coal will yield from 60 to 80 per cent, of coke. Averaging 
66 per cent. It is capable of absorbing 15 to 20 per cent, of moisture. 

Heat of combustion lost in coking of bituminous coal 40 per cent. 

Charcoal. 

Charcoal , properly termed, is not made below a temperature of 536°. The 
best quality is made from Oak, Maple, Beech, and Chestnut. 

Wood will furnish, when properly burned, about 23 per cent, of coal. 

Charcoal absorbs, upon an average of the various kinds, from .8 per cent, 
nf water for Beech, to 16.3 for Black Poplar, Oak absorbing about 4.28, and 
Pine 8.9. 

Evaporative power, in furnace of a boiler and under pressure, is 5.5 lbs. 
of fresh water per lb. of coal. 

Volume of air chemically required for combustion of 1 lb. of charcoal is, 
when it consists of 79 carbon, 129 cube feet at 62°. 

138 bushels charcoal and 432 lbs. limestone, with 2612 lbs. of ore, will pro¬ 
duce 1 ton of pig iron. 
























FUEL. 


48I 


Produce of Charcoal from Various Woods dried at 300° and Carbonized 

at 572 0 . (M. Violette.) 


Woor>. 

Weight. 

Wood. 

Weight. 

Wood. 

Weight. 

Cork. 

Per Cent. 

62.8 
46.09 
44-25 
41.48 

40.9 

Larch . 

Per Cent. 
40.31 
36.06 
34-69 
34-59 
34 -i 7 

Maple. 

Per Cent. 
33-75 
33-74 
33 - 6 i 
33-28 
31-88 

Oak. 

Chestnut. 

Willow. 

Beech. 

Apple. 

Black elder. 

Pine. 

Elm. 

Ash. 

Poplar roots. 

Birch. 

Pear. 


Poplar.31-12 per cent. 


In a Green or Ordinary State. (Weightper cent.) 


Apple .... 

-23.8 

Birch .... 


Oak. 


Red Pine .... 

• 23 

Ash. 


Elm. 

-25.1 

“ young.. 

• 33-3 

White Pine . 

• 23.5 

Beech.... 


Maple.... 


Poplar. 


Willow. 

. 18.6 


It appears from this that cork, the lightest of woods, yields largest per centage 
of charcoal, about 63 per cent.; and that poplar yields lowest, about 31 per cent. 
There does not appear to be any definite relation between density of wood and 
volume of yield. 

Produce by a slow process of charring is very nearly 50 per cent, greater than by 
a quick process. 


Lignite. 

Lignite is an imperfect mineral coal. It is distinguished from coal by 
its large proportion of oxygen, being from 13 to 29 per cent. Its specific 
gravity ranges from 1.12 to 1.35. 


Elements of Various American Lignites . (W. M. Barr.) 


Location. 

Spec. 

Grnv. 

Fixed 

Carbon. 

Volatile 

Matter. 

Water. 

Ash. 

Total 

Volatile. 

Coke. 



Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 

Per Cent. 

Kentucky. 

I. 2 

40 

23 

3 ° 

7 

53 

47 

Blandville ... 

1.17 

31 

48 

n -5 

9-5 

59-5 

4°-5 

Washington Terr’y_ 

Vancouver’s Island.... 

— 

58-25 

31-75 

7 

3 

38-75 

61.25 

— 

62 

31 

4 

3 

35 

65 

Colorado, Carbon City.. 

1.27 

41.25 

46 

3-5 

9-25 

49-5 

50.5 

Canon City .. 

1.28 

.56-8 

34-2 

4-5 

4-5 

38-7 

61.3 

Arkansas. 

— 

34-5 

28.5 

32 

5 

60.5 

39-5 

Texas, Robertson Co. .. 

1.23 

45 

39-5 

II 

4-5 

50.5 

49-5 


A splialt. 

Asphalt , alike to Lignite, contains a large proportion of oxygen. 

Wood. 

Wood, as a combustible, is divided into two classes, the hard, as Oak, Ash, 
Elm, Beech, Maple, and Hickory, and soft, as Pine, Cotton, Birch, Sycamore, 
and Chestnut. 

Green wood subjected to a temperature ranging from 340° to 440° will 
. lose 30 to 45 per cent, of its weight. 

At a temperature of 300°, Oak, Ash, Elm, and Walnut, in a comparatively 
seasoned state, lost from 16 to 18 per cent. 

Woods contain an average of 56 per cent, of combustible matter. 

From an analysis of M. Violette it appears that composition of wood is about 
same throughout the tree, and that of the bark also; that wood and bark have about 
same proportion of carbon (49 per cent.), but that bark has more ash than wood. 
Leaves and small roots have less carbon than wood (45 per cent.), and more ash, 
5 and 7 per cent. 

Leaves when dried at 212 0 lost 60 per cent, of water, and branches 45 per cent. 

S s 



























































482 


FUEL. 


Evaporative power of 1 cube foot of pine wood is equal to that of 1 cube 
foot of fresh water; or, in the furnace of a steam-boiler and under pressure, 
it is 4.75 lbs. fresh water for 1 lb. of wood. 

Northern Wood .—One cord of hard wood and one cord of soft wood, such 
as is used upon Lakes Ontario and Erie, is equal in evaporative effects to 
2000 lbs. of anthracite coal. 

Western Wood .—One cord of the description used by the river steamboats 
is equal in evaporative qualities to 12 bushels (960 lbs.) of Pittsburgh coal. 
9 cords cotton, ash, and cypress wood are equal to 7 cords of yellow pine. 

Solid portion ( lignin ) of all woods, wherever and under whatever circum¬ 
stances of growth, are nearly similar, specific gravity being as 1.46 to 1.53. 

Densest woods give greatest heat, as charcoal produces greater heat than 
flame. 

For every 14 parts of an ordinary pile of wood there are 11 parts of space; 
or a cord of wood in pile has 71.68 feet of solid wood and 56.32 feet of voids. 

Trees in the early part of April contain 20 per cent, more water than they 
do in the end of January. 

.A.sli. 

Proportion of Ash in 100 Lbs. of several Woods. 


Woods. 

Wood. 

Leaves. 

Woods. 

Wood, \ Leaves. 


I’er Cent. 

•5 

•35 

•34 

Per Cent. 

5-4 

5 

Elm. 

PerCent. 1 Per Cent. 
1.88 II.8 

.21 4 

•25 3-15 


Oak. 

Birch. 

Pitch Pine. 


Peat. 

Peat is the organic matter, or soil, of bogs, swamps, and marshes—decayed 
moss, sedge* coarse grass, etc.—in beds varying from 1 to 40 feet in depth. 
That near the surface, and less advanced in transformation, is light, spongy, 
and fibrous, of reddish-brown color; lower down, it is more compact, of a 
darker brown color; and, in lowest strata, it is of a blackish brown, or almost 
black, of a pitchy or unctuous surface, the fibrous texture nearly or alto¬ 
gether transformed. 

Peat, in its natural condition, contains fro;n 75 to 80 per cent, of water. 
Occasionally its constituent water amounts to 85 or 90 per cent., in which 
case peat is of the consistency of mire. It shrinks very much in drying; 
and its specific gravity varies from .22 to 1.06, surface peat being lightest, 
and deepest peat densest. 

When peat is milled, so that its fibre is broken up, its contraction in dry¬ 
ing is much increased, and in this condition it is termed condensed. 

When ordinarily air dried, it will contain 20 to 30 per cent, of moisture, 
and when effectively dried at least 15 per cent. 

Products of Distillation of Peat. 

Water 31.4. Tar 2.8. Gas 36.6. Charcoal 29.2. 

The distillation of the tar will yield paraffine, oil, gas, water, and char¬ 
coal, and the water acetic acid, wood spirit, and chloride of ammonia. 

Evaporative power, in furnace of a steam-boiler and under pressure, is 
from 3.5 to 5 lbs. of fresh water per lb. of fuel. 

Tail. 

Tan, oak or hemlock bark, after haying been used in the process of tan¬ 
ning, is combustible as a fuel. It consists of the fibre of the bark, and, 
according to M. Peclet, 5 parts of bark produce 4 parts of dry tan; and 
heating power of it when perfectly dry, or containing but 15 per cent, of 
ash, is 6100 units; while that of tan in an ordinary state of dryness, con¬ 
taining 30 per cent, of water, is 4284. Weight of water evaporated at 212 0 
by 1 lb., equivalent to these units, is 6.31 lbs. for dry, and 4.44 for moist. 


















FUEL 


483 


Relative Values of different Fuels. 


Description. 

Lbs. of Steam from 

Water at 212° 

by x lb. of Fuel. 

Relative Evapora¬ 

tive Power for 
equal Weights. 

Relative Evapora¬ 

tive Power for 
equal Volumes. 

Relative Rapidi¬ 

ties of Ignition. 

Relative Freedom 

from Waste. 

Relative Com¬ 

pleteness of 
Combustion. 

Relative 

Weights. 

Anthracites. 








Peach Mountain, Pa. 

IO.7 

I 

I 

•505 

•633 

• 7 2 5 

•945 

Beaver Meadow. 

9.88 

• 9 2 3 

.982 

.207 

.748 

.6 

I 

Bituminous. 








Newcastle. 

8.66 

.809 

•776 

•595 

.887 

• 34 6 

• 9°4 

Pictou. 

8.48 

• 79 2 

•738 

.588 

.418 

I 

.876 

Liverpool .. 

7.84 

•733 

.663 

.581 

I 

•333 

.852 

Cannelton, Ind. 

7-34 

.686 

.616 

I 

.984 

•578 

.848 

Scotch. 

6 -95 

.649 

.625 

.521 

•499 

.649 

• 9°9 

Pine wood, dry. 

4.69 

•436 

•175 

— 

16.417 

— 

— 


Weights, Evaporative Powers per Weight and. Bulk, 
etc., of different Fuels. (W. R. Johnson and others.) 


Fuel. 

Specific 

Gravity. 

Weight 

per 

Cube Foot. 

Steam from 
Water at 
212° by 1 lb. 
of Fuel. 

Clinker 
from 100 lbs. 

Cube Feet 
in a Ton. 

Bituminous. 


Lbs. 

Lbs. 

Lbs. 

No. 

Cumberland, maximum . 

I- 3 I 3 

52.92 

IO.7 

2.13 

42-3 

“ minimum . 

I - 337 

54-29 

9.44 

4-53 

41.2 

Duffryn. 

1.326 

53-22 

IO. 14 

— 

42.09 

Cannel, Wigan. 

Blossburgh. 

1.23 

48-3 

7-7 

— 

46 -37 

1.324 

53-05 

9.72 

3-4 

42.2 

Midlothian, screened . 

1.283 

45-72 

8-94 

3-33 

49 

“ average . 

1.294 

54-04 

8-39 

8.82 

41.4 

Newcastle, Hartley. 

i -257 

50.82 

8.76 

3 -i 4 

44 

Pictou. 

1.3:18 

49-25 

8.41 

6.13 

45 

Pittsburgh. 

1.252 

46.81 

8.2 

•94 

47.8 

Sydney. 

1-338 

47.44 

7-99 

2.25 

47.2 

Carr’s Hartley. 

1.262 

47.88 

7.84 

1.86 

46.7 

Clover Hill, Va. 

1.285 

45-49 

7.67 

3.86 

49.2 

Cannelton, Ind. 

i -273 

47- 6 5 

7-34 

1.64 

47 

Scotch, Dalkeith. 

i- 5 i 9 

51.09 

7.08 

5-63 

43-8 

Chili . 


5-72 

— 

— 

Japan. 

1-231 

48-3 

— 

— 

— 

Anthracite. 






Peach Mountain. 

1.464 

53-79 

IO. II 

3-°3 

41.6 

Forest Improvement. 

1-477 

53-66 

10.06 

.81 

41.7 

Beaver Meadow. 

1-554 

56.19 

9.88 

.6 

39-8 

Lackawanna. 

I. 421 

48.89 

9-79 

1.24 

45-8 

Beaver Meadow, No. 3. 

I.6l 

54-93 

9.21 

1.01 

40.7 

Lehigh. 

i -59 

55-32 

8-93 

1.08 

4°-5 

Coke. 






Natural Virginia. 

1-323 

46.64 

8.47 

5 - 3 i 

48.3 

Midlothian . 

— 

32-7 

8.63 

10.51 

68.5 

Cumberland . 

— 

31.6 

8-99 

3-55 

7°-9 

Miscellaneous. 
Charcoal, Oak . 

i-5 

24 

5-5 

Ash. 

3.06 

104 

Peat . 

•53 

30 

5 

—: 

75 

Warlich’s fuel . 

I - I 5 

69.1 

IO.4 

2.91 

32.44 

Wylam’s “ . 

— 

65 

8.9 

— 

106.6 

Pine wood, dry . 

— 

21 

4-7 

•31 






































































484 


FUEL. 


Weights and Comparative Values of different Woods. 


Woods. 

Cord. 

Value. 

Shell-bark Hickory ... 

Lbs. 

4469 

I 

Red-heart Hickory ... 

3705 

.8l 

White Oak. 

3821 

.81 

Red Oak. 

3254 

.69 

Virginia Pine. 

2689 

.6l 

Southern Pine. 

3375 

•73 

Hard Maple. 

2878 

.6 


Woods. 

Cord. 

Value. 

New Jersey Pine. 

Lbs. 

2137 

•54 

Yellow Pine. 

1904 

•43 

White Pine. 

1868 

.42 

Beech . 

— 

•7 

Spruce . 

— 

•52 

Hemlock. 

— 

•44 

Cottonwood. 

— 

•33 


Liquid. ITviels. 

Petroleum. 

Petroleum is a hydro-carbon liquid which is found in America and Europe. 
According to analysis of M. Sainte-Claire Deville, composition of 15 petro¬ 
leums from different sources was found to be practically constant. Average 
specific gravity was .87. Extreme and average elementary composition was 
as follows: 


Carbon.82 to 87.1 per cent. Average, 84.7 per cent. 

Hydrogen.n.2toi4.8 “ “ 13.1 “ 

Oxygen. 5 to 5.7 “ “ 2.2 “ 


100 

Its heat of combustion is 20240, and its evaporative power at 212 0 20.33. 

Petroleum Oils —Are obtained by distillation from petroleum, and are com¬ 
pounds of carbon and hydrogen, in average proportion of 72.6 and 27.4. 

Boiling-point ranges from 86° to 495 0 . 

Schist Oil— Consists of carbon 80.3 parts, hydrogen 11.5, and oxygen 8.2. 

Pine Wood Oil —Consists of carbon 87.1 per cent., hydrogen 10.4, and 
oxygen 2.5. 

Coal-gas. 


Coal Gas —As furnished by Chartered Gas Co. of London is composed as 
follows: 



c 

larbon. 

Hydrogen. 

Olefiant Gas, 
Bi-carb. hyd. 
Marsh gas, ) 
Carb. hyd. ) * 
Carbonic oxide. 

>-c 

2 

3.096 

6.445 

3-84 

•434 

8.815 

5 -ii 



Oxygen. 

Hydrogen. 

Nitrogen. 

Hydrogen. 

' — 

Si -8 

— 

Oxygen. 

.08 


— 

Nitrogen. 

— 

— 

•38 

Total. 




Heat of combustion at 212 0 52961 units, and evaporative power 47.51 lbs. 


Coal-gas. (V. Harcourt.) 



Carb. 

Hyd. 

Oxy. 

Nit. 


Curb. 

Hyd. 

Oxy. 

Nit. 

Olefiant gas. 

Marsh gas. 

Carbonic oxide.. 

Per ct. 
10.5 

39-7 

5-9 

Per ct. 
1-7 
13.2 

Per ct. 

7-9 

Per ct. 

Hydrogen. 

Nitrogen. 

Oxygen . 

Per ct. 

Per ct. 
8 .x 

Per ct. 

•3 

Perct. 

5-8 

Carbonic dioxide 

I- 9 

— 

5 

— 

Total. 

58 

23 

13.2 

5-8 


One lb. of this gas had a volume of 30 cube feet at 62° ; heat of combus¬ 
tion 22684 units; and of one cube foot 756 units, which is equivalent to 
evaporation of .68 lb. of water from 62°, or of .78 lb. from 212 0 per cube foot. 





























































FUEL, 


485 


Average Cornposition of Fuels. 


Bituminous Coals. 

Specific 

Grav¬ 

ity. 

Carbon. 

Hydro¬ 

gen. 

Nitro¬ 

gen. 

Oxygen. 

Sul¬ 

phur. 

Ash. 


Per ct. 

Per ct. 

Per ct. 

Per ct. 

Per ct. 

Per ct. 

Australian. 

I- 3 1 

— 

— 

— 

— 

• 5 

8.38 

Borneo. 

1.28 

64.52 

4-74 

.8 

20.75 

i -45 

7-74 

British, lowest. 

— 

68.72 

4.76 

— 

18.63 

i -35 


Boghead,dry,average . 

I. l8 

63-54 

8.86 

.96 

4-7 

•32 

21.22 

Chili, Conception Bay. 

1.29 

70-55 

5-76 

•95 

13.24 

1.98 

7-52 

“ Chiriqui. 

— 

38.98 

4.01 

•58 

13-38 

6.14 

36.91 

Cannel, Wigan. 

1.23 

79- 2 3 

6.08 

1.18 

7.24 

i -43 

4.84 

Cumberland, Md. 

i- 3 i 

93.81 

1.82 

— 

2.77 

— 

1.6 

Coke, Garesfleld. 


97.6 

— 

— 


•85 

i -55 

“ Durham. 

— 

89-5 

— 

— 

— 

1-25 

9- 2 5 

“ Average. 

— 

93-44 

— 

— 

— 

1.22 

5-34 

Duffryn. 

I -33 

88.26 

4.66 

i -45 

.6 

1.77 

3.26 

Formosa Island . 

1.24 

78.26 

5-7 

.64 

10.95 

• 49 

3 - 9 6 

French, hard . 

1.32 

88.56 

4.88 

( 4 - 38 ) 


2. 19 

‘‘ caking . 

1.29 

87-73 

5.08 

( 5 - 65 ) 

— 

i -54 

“ long flame . 

1.3 

82.04 

5-35 

( 8.63) 

— 

3.08 

“ average* . 

i- 3 * 

85 

4-5 

( 7 

) 

— 

3-5 

Indian, average . 

— 

47-3 

— 

— 

— 

— 

22.9 

“ Kotbec . 

— 

9 ° 

— 

— 

— 

— 

4 

Patagonia . 

— 

62.25 

5-05 

•63 

17-54 

i-13 

13-4 

Russian, Miouchif . 

— 

9*-45 

4-5 

( 4-°5 ) 

— 

— 

Sydney, S. W . 

— 

82.39 

5-32 

1.27 

8.32 

.07 

2.04 

Splint, Wvlam . 

— 

74.82 

6.18 

( 5 - 09 ) 

— 

I 3 - 9 r 

“ Glasgow . 

— 

82.92 

5-49 

( 10 

46 ) 

— 

i-i 3 

“ Cannel, Lancashire . 

— 

83-75 

5.66 

( 8.04) 

— 

2-55 

“ “ Edinburgh . 

— 

67.6 

5-4 

( 12.43) 

— 

14-57 

“ Cherry, Newcastle . 

— 

84.85 

5-05 

(8 

43 ) 

— 

1.67 

“ Caking, Garesfleld . 

— 

87-95 

5-24 

( 5 . 42 ) 

— 

i -39 

“ Ebbro Vale, Welsh . 

— 

89.78 

5 -i 5 

2.16 

•39 

1.02 

i -5 

“ Llangenneck “ . 

— 

84.97 

4.26 

i -45 

3-5 

.42 

5-4 

Vancouver’s Island . 

— 

66.93 

5-32 

1.02 

8.7 

2.2 

15-83 

Anthracites. 








Anthracite . 

i -5 

88.54 

— 

— 

— 

• 52 

8.67 

French . 

i -5 

86.17 

2.67 

( 2.85) 

— 

8.56 

Russian . 


96.66 

i -35 

( 1 

99 ) 

— 

— • 

Woods. 








Beech . 

— 

50-17 

6.12 

1.05 

40.38 

— 

1.77 

Birch . 

— 

48.12 

6-37 

1-15 

43-95 

— 

.48 

Oak . 

— 

48.13 

5-25 

.82 

44-5 

— 

I -3 

White Pine . 

— 

49-95 

6.41 

— 

43-65 

— 

■ 3 i 

Woods, average . 

— 

49-7 

6.06 

1.05 

4 i -3 

— 

1.8 

Charcoal. 








Oak . 

— 

87.68 

2.83 

— 

6-43 

— 

3.06 

Pine . 

— 

71-36 

5-95 

— 

22.19$ 

— 

•4 

Maple . . . 

— 

7O.O7 

4.61 

— 

24.89$ 

— 

•43 

Miscellaneous. 








Asphalt . 

x.06 

79.18 

9-3 

( 8.72) 

— 

2.8 

Lignite, perfect . 

1.29 

69.02 

5-05 

(20.12) 

— 

5.82 

“ imperfect . 

1.25 

60.18 

5-29 

(29.03) 

— 

5-57 

“ bituminous . 

I. l8 

74.82 

7-36 

( 13-38 ) 

— 

4-45 

“ Colorado. 

1.28 

56.8 


— 

— 

— 

4-5 

“ Kentucky. 

1.2 

40 

— 

— 

— 

— 

7 

“ Arkansas. 

— 

34-5 

— 

— 

— 

— 

5 

Peat, dense. 

— 

61.02 

5-77 

.81 

32-4 

— 

— 

“ Irish, average. 

.528 

58.18 

5 - 9 6 

1.23 

31.21 

— 

3 - 43 , 

Patent, Warlich. 

115 

90.02 

5-56 

— 

— 

1.62 

2.918 

“ Wylam’s. 

I. I 

79 - 9 1 

5-69 

1.68 

6.63 

1.25 

4.84 


* Heat of Combustion of 1 Lb. 14 723. 4 Heat of Combustion of 1 Lb. 15651. 

X Including Nitrogen. § Including Oxygen. 

S s* 












































































486 


FUEL. 


Average. Composition of C 3 oals and Fuels, Heat of Com¬ 
bustion, and. Evaporative Power. 

Deduced from analysis and experiments of Messrs. De La Beche, Playfair , and Peclet. 



0 

CC 3 S ' 



Composition. 



4 H 

O O 
•O c 

e u 

0 -*-* 

*43 . 

g 

Coals and Fuels. 

b >. 

2 os 

Carbon. 

Hydro- 

Nitro- 

I Sul- 

Oxy- 

Ash. 

«+- 0 

G-£ 

03 # 

d c 
a P 



gen. 

^en. 

pliur. 

gen. 

a 

H 



Per ct. 

Per ct. 

Per ct. 

Per ct. 

Per ct. 

Per ct. 

Units. 

Lbs. 

Derbyshire and) 
Yorkshire .... j 

1. 29 

79.68 

4-94 

1.41 

I.OI 

10.28 

2.65 

13 860 

I 4-34 

Lancashire. 

I.27 

1.26 

77-9 
82.12 

5-32 
5-31 
5.61 

1-3 

i -35 

1 

1.44 

I. 24 

I. II 

9-53 

5-69 

9.69 

4 -i 5 

4.88 

13918 
14 820 
14 164 
14858 

14.56 

15-32 

1 4 - 77 

15- 52 

Newcastle. 

3 - 77 

4 - °3 
4.91 

Scotch. 

1. 26 

7 8 -53 

83.78 

Welsh. 

1.32 

4-79 

•98 

I -43 


Average of British. 
Patent fuels. 

1.28 

1.17 

80.4 

83-4 

5 -i 9 

4 97 

I. 21 
I.08 

1.25 

1.26 

7.87 

2-79 

4 - °5 

5 - 93 

I432C) 
15 OOO 

14.82 
15.66 


Van Diemen’s Land 


65.8 

3-5 

i -3 

1.1 

5 - 5 « 

22.71 

II 320 

11.83 

Chili . 


63.56 

5-43 

.82 

2-5 

14.84 

i 3 - 3 i 

II 030 

11.68 



Lignite, Trinidad.. 

— 

65.2 

4-25 

i -33 

.69 

21.69 

6.84 

10438 

10.87 

“ French Alps 

1.28 

70.02 

5-2 

— 


— 

3 -oi 

II 790 

12. I 

“ Bitum.,Cuba 

I. 2 

75-85 

7- 2 5 

— 

— 

— 

3-94 

14562 

14.96 

“ Wash. Ter. *. 
Asphalt . 

1 06 

67 

79.18 

4-55 

9-3 

— 

I 

— 

3 -i 

2.8 

12538 

16655 

I2.9I 

I7.24 





Petroleum . 

.87 

84.7 

13- 1 

_ 

- - 

2. 2 

_ : 

20 24O 

20.33 





“ oils . 

•75 


— 

— 

— 

— 

— 

27 530 

28.5 

Oak bark Tan, dry. 


— 

— 

— 

— 

— 

15 

6 100 

6.31 

“ “ moist 

— 

— 

— 

— 

- 

— 

15 

4 284 

4.24 

Charcoal at 302° .. 

i -5 

47 - 5 i 

6.12 

( 0 and N 46.29 ) 

.8 

8 130 

8.4 

“ “ 572 °... 

1.4 

73-24 

4-25 

( 0 and N 21.96) 

•57 

11 861 

12.27 

“ “ 8io c ... 

1.71 

81.64 

4.96 

(0 and N 15.24 ) 

I.6l 

14 916 

15-43 

Peat, dry, average. 

•53 

58.18 

5 - 9 6 

1.23 

— 

31.21 

3-43 

995 i 

10.3 

“ moist, t “ 

— 

43 - 1 

4-3 

( 0 and N 21.4 ) 

3-3 

8917 

9.22 

Coal-gas. 

42 

33-38 

66.16 

•S 8 

- 1 

.08 

— 

52961 

47-51 


* Water 7. Oxygen and Nitrogen 17.36. f Moisture 27.8. Sulphur .2. 


Elements of Fuels not included in Preceding TaToles. 


Fuel. 

Heat of 
Combustion 
of 1 lb- 

Evaporative 
Power of 1 
lb. at 212°. 

Colie pro¬ 
duced. 

Weight 
of 1 

Cub. Foot. 

Volume of 
1 Ton. 

Bituminous Goal . 

Units. 

Lbs. 

Per cent. 

Lbs. 

Cube Feet. 

Welsh... 

14 858 

9-°5 

73 

82 

42.7 

Newcastle. 

14 820 

8.01 

6l 

7 8 -3 

45-3 

Lancashire. 

13 9 l8 

7-94 

58 

79-4 

45-2 

Scotch. 

14 164 

7-7 

54 

78.6 

42 

Boghead. 

14478 

7.87 

30-94 

— 


British, average. 

14 i 33 

8.13 

6l 

79-8 

44-52 

Irish, lowest. 


9-85 

90 

99.6 

35-7 

Cumberland, Md. 

— 

— 

83.7 

84-93 

42.4 

American, average. 

— 

— 

82.5 

87-54 

43-49 

French, average. 

14723 

— 

64.2 

— 

40 

Australian. 

— 

68.27 

— 


Anthracite . 






American. 

— 

— 

94.82 

93-78 

42-35 

French . 

14038 

— 

88.83 


Miscellaneous . 






Warlich’s fuel. 

16495 

— 

— 

73-5 

34-5 

Coke.Mickley 

15 600 

— 

— 


80 

Virginia, average. 

13 55 o 

14.02 

— 

45 

69.8 

Charcoal. 

12 325 

— 

— 


12.76 

Lignite, perfect. 

11 678 

12. I 

47 

— 

“ imperfect. 

9 8 34 

10.18 

37-5 

— 

— 

“ Russian. 

15837 

— 


— 

— 

Asphalt. 

16555 

J 7- 2 4 

9 

— 

— 

Woods, dry, average. 

7 792 

8.07 


— 

”4 



































































FUEL.-GRAVITATION. 


487 


Miscellaneous. 

Experiments undertaken by Baltimore and Ohio R. R. Co. determined 
evaporating effect of 1 ton of Cumberland coal equal to 1.25 tons of anthra¬ 
cite, and 1 ton of anthracite to be equal to 1.75 cords of pine wood; also 
that 2000 lbs. of Lackawanna coal were equal to 4500 lbs. best pine wood. 

One lb. of anthracite coal in a cupola furnace will melt from 5 to 10 lbs. of cast 
iron; 8 bushels bituminous coal in an air furnace will melt 1 ton of cast iron. 

Small coal produces about .75 effect of large coal of same description. 

Experiments by Messrs. Stevens, at Bordentown, N. J., gave following results: 

Under a pressure of 30 lbs., 1 lb. pine wood evaporated 3.5 to 4.75 lbs. of water, 
i lb. Lehigh coal, 7.25 to 8.75 lbs. 

Bituminous coal is 13 per cent, more effective than coke for equal weights; and 
in England effects are alike for equal costs. 

Radiation from Fuel. —Proportion which heat radiated from incandescent fuel 
bears to total heat of combustion is, 

From Wood.29 | From Charcoal and Peat.. ..5 

Least consumption of coal yet attained is 1.5 lbs. per IIP. It usually varies in 
different engines from 2 to 8 lbs. 

Volume of pine wood is about 5.5 times as great as its equivalent of bituminous 
coal. 


GRAVITATION. 


Gravity is an attraction common to all material substances, and 
they are affected by it directly, in exact proportion to their mass, and 
inversely, as square of their distance apart. 

This attraction is termed terrestrial gravity , and force with which a 
body is drawn toward centre of Earth is termed the weight of that body. 


Force of gravity differs a little at different latitudes: the law of variation, 
however, is not accurately ascertained; but following theorems represent it 
very nearly: 


g' (1 —.002 837 cos. 2 lat.) I 
g' (1 + . 002 837), at the poles \=g. 
g' (1 —.002 837), at the equator) 


g’ representing force of gravity at lati¬ 
tude 45 0 , and g force at other places. 


Or, 32.171 (lat. 45 0 ) (1 + .005 133 sin. L) ^1 — —^ = g. L representing latitude , 
H height of elevation above level of sea, and R radius of Earth , both in feet. 


Note. _If 2 L exceeds 90 0 , put cos. 180 — 2 L, and R at Equator = 20926062, at 

Poles 20 853 429, and mean 20 889 746. 


Illustration. —What is force of gravity at latitude 45 0 , at an elevation of 209 


feet, and radius = 20900000 feet ? 

32. 171(1 + -005 i33 sin. 45°) (1 - 2090Q —) 


= 32.171 X 1-00363 X .99998 = 32.287. 


Gravity at Various Locations at Level of Sea. 

Equator.32.088 I New York.32.161 I London.32.189 

Washington.32.155 | Lat. 45 0 . 2 >‘ 2 - 1 T L I Poles. 3 2 - 2 53 

In bodies descending freely by their own weight, their velocities are as 
times of their descent, and space/passed through as square of the times. 

Times , then, being 1, 2, 3, 4, etc., Velocities will be 1, 2, 3, 4, etc. 


Spaces passed through will be as square of the velocities acquired at end 
of those times, as x, 4, 9,16, etc.; and spaces for each time as 1, 3, 5, 7, 9, etc. 













488 


GRAVITATION. 


A body falling freely will descend through 16.0833 fed in first second of 
time, and will then have acquired a velocity which will carry it through 
32.166 feet in next second. 

If a body descends in a curved line, it suffers no loss of velocity, and the 
curve of a cycloid is that of quickest descent. 

Motion of a falling body being uniformly accelerated by gravity, motion 
of a body projected vertically upwards is uniformly retarded in same manner. 

A body projected perpendicularly upwards with a velocity equal to that 
which it would have acquired by falling from any height, will ascend to 
the same height before it loses its velocity. Hence, a body projected up¬ 
wards is ascending for one half of time it is in motion, and descending the 
other half. 

Various Formulas here given are for Bodies Projected Upwards or 
Falling Freely , in Vacuo. 

When , however , weight of a body is great compared with its volume , and velocity 
of it is low , deductions given are sufficiently accurate for ordinary purposes. 

In considering action of gravitation on bodies not far distant from surface of the 
Earth, it is assumed, without sensible error, that the directions in which it acts are 
parallel, or perpendicular to the horizontal plane. 

A distance of one mile only produces a deviation from parallelism less than one 
minute, or the 60th part of a degree. 


Relation, of Time, Space, and. Velocities. 


Time from 
Beginning of 
Descent. 

Velocity acquired 
at End of tliat 
Time. 

Squares 

of 

Time. 

Space fallen 
through in that 
Time. 

Spaces 

for 

this Time. 

Space fallen 
through in last 
Second of Fall. 

Seconds. 

Feet. 

Seconds. 

Feet. 

No. 

Feet. 

I 

32.166 

I 

16.083 

I 

16.08 

2 

64-333 

4 

64-333 

3 

48.25 

3 

9 6 -5 

9 

144-75 

5 

80.41 

4 

128.665 

l 6 

257-33 

7 

112.58 

5 

160.832 

25 

402.08 

9 

144-75 

6 

193 

36 

579 

II 

176.91 

7 

225.166 

49 

788.08 

13 

209.08 

8 

257-333 

64 

1029.33 

15 

241.25 

9 

289.5 

8 l 

1302.75 

17 

273.42 

IO 

321.666 

IOO 

1608.33 

*9 

305-58 


and in same manner this Table may be continued to any extent. 


"Velocity- acqnrred due to given Height of Fall and. 

Height due to given Velocity. 

v 2 

8.04 y/h — v, 32.2 t = v] - —and 16.083 t 2 = h. 

64* 4 

h representing height of fall in feet , v velocity acquired in feet per second , and t 
time of fall in seconds. 

To Compute ^Action of Grravity. 

Time. 

When Space is given. Rule. —Divide space by 16.083, and square root 
of quotient will give time. 

Example.— How long will a body be in falling through 402.08 feet? 

V 402.08 - 4 -16.083 — 5 seconds. 

When Velocity is given. Rule. — Divide given velocity by 32.166, and 
quotient will give time. 

Example.—H ow long must a body be in falling to acquire a velocity of 800 feet 
per second ? 800 4 - 32. 166 — 24. 87 seconds. 














GRAVITATION. 


489 


"V elocity. 

When Space is given. Rule. — Multiply space in feet by 64.333, anf l 
square root of product will give velocity. 

Example.—R equired velocity a body acquires in descending through 579 feet. 

V579 X 64.333 = 193 f eet - 

Velocity acquired at any period is equal to twice the mean velocity during 
that period. 

Illustration.—I f a ball fall through 2316 feet in 12 seconds, with what velocity 
will it strike? 

2316 - 4 -12 = 193, mean velocity , which X 2 = 386 feet =. velocity. 

When Time is given. Rule. —Multiply time in seconds by 32.166, and 
product will give velocity. 

Example.—W hat is velocity acquired by a falling body in 6 seconds? 

32.166 X 6 = 192.996 feet. 

Space. 

When Velocity is given. Rule. —Divide velocity by 8.04, and square of 
quotient will give distance fallen through to acquire that velocity. 

Or, Divide square of velocity by 64.33. 

Example. — If the velocity of a cannon-ball is 579 feet per second, from what 
height must a body fall to acquire the same velocity? 

579 -r- 8,04 = 72.0x4, and 72.014 2 = 5x86.02 feet. 

When Time is given. Rule. — Multiply square of time in seconds by 
16.083, an( i ^ will give space in feet. 

Example.—R equired space fallen through in 5 seconds. 

5 2 — 25, and 25 X 16.083 == 402.08 feet. 

Distance fallen through in feet is very nearly equal to square of time in fourths 
of a second. 

Illustration i.—A bullet dropped from the spire of a church was 4 seconds in 
reaching the ground; what w T as height of the spire? 

4 X 4 — 16, and 16 2 = 256 feet. 

By Rule, 4 X 4 X 16.0833 = 257.33 feet. 

2.—A bullet dropped into a well was 2 seconds in reaching bottom; what is the 
depth of the well ? 

Then 2X4 — 8, and 8 2 == 64 feet. 

By Rule, 2 X 2 X 16.0833 = 64.33 feet. 

By Inversion. —In what time will a bullet fall through 256 feet? 

256 = 16, and 16 -r- 4 = 4 seconds. 

Space fallen, tlirongli in last Second of Fall. 

When Time is given. Rule.— Subtract half of a second from time, and 
multiply remainder by 32.166. 

Example.—W hat is space fallen through in last second of time, of a body falling 
for 10 seconds ? 

10 —. 5 X 3 2 - 16 6 = 305.58 feel. 

Promiscuous Examples. 

1. If a ball is 1 minute in falling, how far will it fall in last second ? 

Space fallen through = square of time, and 1 minute = 60 seconds. 

60 2 X 16.083 = 57 898 feet for 60 seconds. 

59 2 X 16.083 = 55984 “ “ 59 “ 

1914 1 

2. Compute time of generating a velocity of 193 feet per second, and whole space 
descended. 

193-7-32.166 = 6 seconds ; 6 2 X 16.083 = 579 feet. 




490 


GRAVITATION. 


3. If a body was to fall 579 feet, what time would it be in falling, and how far 
would it fall in the last second? 


/ 579 X 2 _ / g _ 6 seconds, and 6 — -5 X 32.166 = 5.5 X 32.166 = 176.91 feet. 
v 32.166 


32.166 

Formulas to determine tlae various Elements. 

T: 

V -5 9 ' 0 ' V ' V g ' g ' 

\T 2 VT /iT2 

A=T—.s 


/ s 

V 

2 S 

/ 2 s 

V -5 9 ’ 

~ o' ~ 

V ; 

“V g 

V 2 

VT 

gu 

; = T 2 .5 

2 9 ’ 

2 * 

2 


Vs X 2 g 

; =T g\ 

m 2 

V.5 <7 S ; 


2 S 


T representing time of falling in seconds, V velocity acquired in feet per second, 
S space or vertical height in feet, h space fallen through in last second, g 32.166 and 
.5 g and .25 g representing 16.083 and 8.04. 


Retarded Afotion. 

A body projected vertically upward is affected inversely to its motion 
when falling freely and directly downward, inasmuch as a like cause retards 
it in one case and accelerates it in the other. 

In air a ball will not return with same velocity with which it started. In 
vacuo it would. Effect of the air is to lessen its velocity both ascending and 
descending. Difference of velocities will depend upon relative specific grav¬ 
ity of ball and density of medium through which it passes. Thus, greater 
weight of ball, greater its velocity. 

To Compute Auction of Gfravity- by- a Body projected. 

Upward or Downward witlr a given Velocity. 

Space. 

When projected Upward. Rule. —From the product of the given velocity 
and the time in seconds subtract the product of 32.166, and half the square 
of the time, and the remainder will give the space in feet. 

Or, Square velocity, divide result by 64.33, and quotient will give space 
in feet. 

Example. —If a body is projected upward with a velocity of 96.5 feet per second, 
through what space will it ascend before it stops? 

96.5 - 3 - 32.166 = 3 seconds = time to acquire this velocity. 

Then, 96.5 X 3^ (^2.166 X = 289.5 — 144.75 = 144.75/666 

Time. 

Rule. —Divide velocity in feet by 32.166, and quotient will give time in 
seconds. 

Example.—V elocity as in preceding example. 

96.5 -f- 32.166 =: 3 seconds. 

Velocity. 

Rule. —Multiply time in seconds by 32.166, and product will give velocity 
in feet per second. 

Example. —Time as in preceding example. 

3 X 32.166 = 96.5 feet velocity. 

Space fallen, tlirongh. in last Second.. 

Rule. —Subtract .5 from time, multiply remainder by 32.166, and product 
will give space in feet per second. 

Example. —Time as in preceding example. 

3 — .5 X 32.166 = 2. 5 X 32.166 — 8o-4i6/ee6 






GRAVITATION. 


49 1 


When projected, Downward. 

Space. 

Rule.— Proceed as for projection upwards and take sum of products. 

Example i.—I f a body is projected downward with a velocity of 96.5 feet per sec¬ 
ond, through what space will it fall in 3 seconds? 

9 6 - 5 X 3 + ^3 2 - *66 X ~j = 289. 5 -f 144.75 = 434.25 feet. 

Or, t 2 X 16.083 -f - v X t = s. 

2. — If a body is projected downward with a velocity of 96.5 feet per second, 
through what space must it descend to acquire a velocity of 193 feet per second? 

96- 5 ■= 3 2 -166 = 3 seconds, time to acquire this velocity. 

193 -r- 32.166 = 6 seconds, time to acquire this velocity. 

Hence 6 — 3 = 3 seconds, time of body falling. 

Then 96.5 X 3 = 289.5 = product of velocity of projection and time. 

16.083 X 3 2 — 144-75 =product of 32.166, and half square of time. 

Therefore 289.5 -{-144.75 = 434.25/eeL 

Time. 

Rule. —Subtract space for velocity of projection from space given, and 
remainder, divided by velocity of projection, will give time. 

Example.—I n what time will a body foil through 434.25 feet of space, when pro¬ 
jected with a velocity of 96.5 feet? 

Space for velocity of 96.5 = 144.75/ee^ 

Then, 434.25 — 144.75 - 4 - 96.5 = 289.5 — 96.5 = 3 seconds. 

"V elocity. 

Rule. —Divide twice space fallen through in feet by time in seconds. 

Example.—E lements as in preceding example. 

Space fallen through when projected at velocity of 96.5 feet = 144.75 feet, and 434.25 
feet = space fallen through in 3 seconds. 

Then, 144.75 -|- 434.25 = 579 feet space fallen through, and -v/579 -i-16.083 = 6 
seconds. 

Hence, 579 x 2-^6 = 1158-4-6 = 193 feet. 

Space Fallen, tlirougli in last Second. 

Rule. —Subtract .5 from time, multiply remainder by 32.166, and product 
wdll give space in feet per second. 

Example.—E lements as in preceding example. 

6 — .5 X 32.166 = 5.5 X 32.166 = 176.91 feet. 

Ascending bodies, as before stated, are retarded in same ratio that descending 
bodies are accelerated. Hence, a body projected upward is ascending for one half 
of the time it is in motion, and descending the other half. 

Illustration i. — If a body projected vertically upwards return to earth in 12 
seconds, how high did it ascend? 

The body is half time in ascending. 12-4-2 = 6. 

Hence, by Rule, p. 489, 6 2 X 16.083 = 579 feet—product of square of time and 
16.083. 

2.—If a body is projected upward with a velocity of 96.5 feet per second, it is 
required to ascertain point of body at end of 10 seconds. 

96-5-4-32.166 = 3 seconds , time to acquire this velocity , and 3 2 X 16.083 = 144.75 
feet , height body reached with its initial velocity. 

Then 10 — 3 = 7 seconds left for body to fall in. 

Hence, by Rule, as in preceding example, 7 s X 16.083 = 788.07, and 788.07 — 
144.75 = 643.32 feet = distance below point of projection. 

Or, io 2 X 16.083 = 1608.3 feet, space fallen through under the effect of gravity, and 
96.5 X 10 = 965 feet , space if gravity did not act. Hence 1608.3 — 965 = 643.3 feet. 




49 2 


GRAVITATION. 


3.—A body is projected vertically with a velocity of 135 feet; what velocity will 
it have at 60 feet? 


i35 2 -^-64.33 — 283.3 feet space projected at that velocity , 135 = 32.16 = 4.197 sec¬ 
onds—time of projection, and 283.3 — 60 = 223.3 — space to be passed through after 

attainment of 60 feet. Hence, V223.3 X 64.33 = 119.85 feet Velocity, and 223.3 -f- 60 
= 283.3 feet. 


By Inversion .—Velocity 119.85. 


Hence, 


ii 9 - 8 5 2 

64-33 


= 223.3 feet space , and 283.3 — 


223.3 — 60 feet. 


Formulas to Determine Elements of Retarded IVfotion. 

s gt 


v — V — gt. 

v=i+2i 




s = V t 


g* 


3. V = v + gt. 
6. *~tv + 9 -* 


9 


8. t 


_V__ /\^_2S 

_ 9 V 9 3 9' 


9. h — T — t — t' — .sg. 


9 V g- 9 

v representing velocity at expiration of time, t any less time than T, t' less time thaw 
t, s space through which a body ascends in time t, V, T, S, and h as in previous formulas , 
page 490. 

Illustration.—A body projected upwards with a velocity of 193 feet per second, 
was arrested in 5 seconds. 

T = 6 , t' = 1. 

1. What was its velocity when arrested? (1.) 

2. What was the time of its passing through 562.92 feet of space ? (7.) 

3. What space had it passed through ? (5.) 

4. What was the time of its projection, when it had a velocity of 96.5 feet? (4.) 

5. What was the height it was projected in the last second of time? (8.) 


i- !93 — 3 2 -i 66 X 5 — 32.17 feet. 

193 velocity. 


562.92 32.166X5 


5. 193 X 5 — 


"93 


32.166 X 5 2 


: 562.92 feet. 


. 32.17 -j- 32.166 X 5 = 193 velocity. 
193 — 96.5 


32.166 

193 — 32-17 

32.166 


= 3 seconds. 
— 5 seconds. 


W3 2 2X562.9 2 


32.166 V 32.166 2 32.166 


= 6 — -v/36 — 35 = 5 seconds. 


8. 6 — 5 — 1 — .5 X 32.166 = 48.25 feet. 

Gravity and IMotion at an Inclination. 

If a body freely descend at an inclination, as upon an inclined plane by 
force of gravity alone, the velocity acquired by it when it arrives at ’ter¬ 
mination of inclination is that which it would acquire by falling freely 
through vertical height thereof. Or, velocity is that due to height of in¬ 
clination of the plane. 

Time occupied in making descent is greater than that due to height, in 
ratio of length of its inclination, or distance passed, to its height. 

Consequently, times of descending different inclinations or planes of like 
heights ai e to one another as lengths of the inclinations or planes. 

Space which a body descends upon an inclination, when descending by 
gravity, is-to space it would freely fall in same time as height of inclination 
is to its length; and spaces being same, times will be inversely in this pro¬ 
portion. * 1 

If a body descend in a curve, it suffers no loss of velocity. 

If two bodies begin to descend from rest, from same point, one upon an in¬ 
clined plane, and the other falling freely, their velocities at all equal heights 
below point of starting will be equal. 
















GRAVITATION. 


493 


Illustration. —What distance will a body roll down an inclined plane 300 feet 
long and 25 feet high in one second, by force of gravity alone? 

As 300 : 25 ” 16.083 : 1.34025 feet. 

Hence, if proportion of height to length of above plane is reduced from 25 to 300 
to 25 to 600, the time required for body to fall 1.34025 feet would be determined as 
follows: 

As 25 : 600 " 1.34025 : 32.166, and 32.166=116.083 X 2 = twice time or space in 
which it would fall freely required for one half proportion of height to length. 

„ 300 600 , 

Or, as —— : —— 1.34025 : 32.166, as above. 

Impelling or accelerating force by gravitation acting in a direction paral¬ 
lel to an inclination, is less than weight of body, in ratio of height of in¬ 
clination to its length. It is, therefore, inversely in proportion to length of 
inclination, when height is the same. 

Time of descent, under this condition, is inversely in proportion to accel¬ 
erating force. 

If, for instance, length of inclination is five times height, time of making 
freely descent at inclination by gravitation is five times that in which a 
body would freely fall vertically through height; and impelling force down 
inclination is .2 of weight of body. 

When bodies move down inclined planes, the accelerating force is ex¬ 
pressed b y k-r-1, quotient of height -5- length of plane; or, what is equivalent 
thereto, sine of inclination of plane, i. e., sin. a. 

Illustration. —An inclined plane having a height of one half its length, the space 
fallen through in any time would be one half of that which it would fall freely. 

Velocity which a body rolling down such a plane would acquire in 5 seconds is 
80.416 feet. 

Thus, 32.166 x 5 = 160.833 feet, and an inclined plane, having a height one half 
of its length, has an angle or sine of 30° Hence, sin. 30 0 = .5, and 160.833 X .5 = 
80.416 feet. 

Formulas to Determine various Elements of Gravita¬ 
tion 011 an Inclined. Plane. 

V 2 -— 

1. S = .5 g T 2 sin. a ; =- - 7 - ; =.5 TV. 4. V = v g T sin. a. 

2 g sin. a , 


2. V = g T sin. a ; = fly. g S sin. a) ; = —. 

/ / 2 S \ _ 2 _S. _ If. 

y \gr sin. a) ’ V ’ ~' 2 ^ vH' 


6. H = 
l 

4 Th: 


7 2 


• 5 </T 2 

7. Z = 4 T-y/H. 


5. S = V T . 5 <7 T 2 sin. a. 


Or, 


V 2 


2 g sm. a 

v representing velocity of projection in feet per second, S space or vertical height 
of velocity and projection, a angle of inclination of plane, l length, and II height of 
plane. 

Illustration. — Assume elements of preceding illustration. V =; 80.416, T = 5, 
and H =: 201.04. 

1. .5 X 32.166 X 5 2 X .5 = 201.04/eet. 2. 32.166 X 5 X -5 = ^0.^16 feet. 

3 


A2^ 

6. - ++- - = 201.04 feet. 7. 4 X 5 X -\/ 20I - 0 4 ~ 28 3 - 4 2 / ee ^- 

.5 X 16.083 X 5 2 

If projected downward with an initial velocity of 16.083 feet per second. V + <7. 


4. 16.083 + 32.166 X 5 X .5 = 96-5 feet. 

5. 80.416 + 16.083 X 5 — -5 X 32.166 X 5 2 X -5 = 281.46 feet. 

T T 














494 


GRAVITATION. 


Illustration.— What time will it take for a ball to roll 38 feet down an inclined 
plane, the angle a = 12 0 20', and what velocity will it attain at 38 feet from its start¬ 
ing-point? 


T = a / - 2 . ^ - =' a / - -- = 3.33 seconds. 

V g sin. a V 32.166 X .2136 0 00 

X • 2136 = 22.88 feel per second. 


V = g T sin. a = 32.166 X 3- 33 


When a body is projected upward it is retarded in the same ratio that a 
descending body is accelerated. 

Illustration.— If a body is projected up an inclined plane having a length of 
twice its height, at a velocity of 96.5 feet per second, 

Then. T = 96.5 = 32.166 = 3 seconds. S = .5 32.166 X 3 2 X .5 = 72.375 feet, v — 
32.166 X 3 X • 5 = 4 8 - 2 5 f eet - 


Inclined. iPlane. 

Problems on descent of bodies on inclined planes are soluble by formulas 
1 to 9, page 495, for relations of accelerating forces. As a preliminary step, 
however, accelerating force is to be determined by multiplying weight of 
descending body by height of plane, and dividing product by length of plane. 

Illustration.— If a body of 15 lbs. weight gravitate freely down an inclined 
plane, length of which is five times height, accelerating force is 15 = 5 = 3 h> s - If 
length of plane is 100 feet and height 20, velocity acquired in falling freely from top 
to bottom of plane would be 

v — J * 5 IO - = 8 f 20 = 35.776 feet. 

Time occupied in making descent, 

/15 X 100 

l =.25yi---=.25 VS 00 = 5-59 seconds. 

Whereas, for a free vertical fall through height of 20 feet, time would be, 

35.776 

£ = - — = 1.118 seconds, 

32.166 

which is .2 of time of making descent on inclined plane. 


Velocities acquired by bodies in falling down planes of like height will all be 
equal when arriving at base of plane. 


When Length of an Inclined Plane and Time of Free Descent are given. 


Rule. —Divide square of length by square of time in seconds and by 16; 
the quotient is height of inclined plane. 

Example. —Length of plane is 100 feet, and time of descent is 5.59 seconds; then 
vertical height of descent is 


100 2 

5 - 59 2 X 16.08 


= 20 feet. 


Accelerated, and Retarded Motion. 

If an Accelerating or Retarding force is greater than gravity, that is, 
weight of the body, the constant, g, or 32.166, is to be varied in proportion 
thereto, and to do this it is to be multiplied by the accelerating force, and 
product divided by weight of body. 


Thus, Letf represent accelerating force, and w weight of body. 


Then, 


64- 333 f 
w 


or 


32.166 f 16.083 f , 

———r , or -■— become the constants. 

w IV 


The same rules and formulas that have been given for action of gravity alone 
are applicable to the action of any other uniformly accelerating or retarding force, 
the numerical constants above given being adapted to the force. 












GRAVITATION". 


495 


Average "Velocity" of a Moving Body- uniformly Accel¬ 
erated or Retarded. 

Average velocity of a moving body uniformly accelerated or retarded, 
during a given time or in a given space, is equal to half sum of initial and 
final velocities; and if body begin from a state of rest or arrive at a state of 
rest, its average speed is half the final or initial velocity, as the case may be. 

Thus, in example of a ball rolling, initial speed or velocity is, in either 
case, 60 feet per second, and terminal speed is nothing; average speed is 

therefore 6 °~^° , namely, one half of that, or 30 feet per second. 


"When a cannon-ball is projected at an angle to horizon, there are two forces act¬ 
ing on it at same time—viz., force of charge, which propels it uniformly in a right 
line, and force of gravity, which causes it to fall from a right line with an accel¬ 
erated motion; these two motions (uniform and accelerated) cause the ball to move 
in the curved line of a Parabola. 


Formulas for Flight of a Cannon-ball. 

/P w V 2 

V = 2800 — ; P = —-; 

\ w 7 840 000 

V 2 sin. a, cos. a , V sin. a , V 2 sin. 2 a 
;- - - : t =-; h = 


g ' g 2 9 

w representing weight of hall and P of powder in lbs. ; t time of fight in seconds ; 
b horizontal range , and h vertical height of range of projection of ball in feet. 

Illustration.— A cannon loaded to give a ball a velocity of 900 feet per second, 
the angle a = 45°; what is horizontal range, the time t and height of range 7 t? 

, — 9 °° 2 X siu- 45 ° X cos. 45° _ 900 2 X -5 
32.166 32.166 

9 °° X .7071_o . r _ 900 2 X .7071 2 


b — 


12 590 feet. 


t = 


19.78 seconds; h 


■ 6295 feet. 


32.166 * 2X32.166 

Note. —As distance b will be greatest when angle a — 45 0 , product of sine and 
cosine is greatest for that angle. Sin. 45 0 X cos. 45° = . 5. 

24 lb. ball with a velocity of 2000 feet per second at 45 0 range 7300 feet. 


General Formulas for Accelerating and. Retarding 

Forces. 


V = 


9 ft 


• 5 gft 2 


t = 


w V 


wV 3 


7 - /= 


5- t 
w V 2 


= .25^/ 


V) 

w S 

T 

8. f= 


6 . 

w V 


9 f _ 

= 8./Z?. 


gf 


gft 

v 1 


2 g S t 32.2 

Note i.— When accelerating or retarding force bears a simple ratio to weight of 
body, the ratio may, for facility of calculation, be substituted in the quantities rep¬ 
resenting modified constants, for force and weight. Thus, if accelerating force is a 

, . , 42.166 16.084 

tenth part of weight, then ratio is 1 to 10, and - -— 3.2166; or, - - = = 1.6083, 


6 A O'"* “2 

and— 6.4333; and these quotients may be substituted for 16.083, 32.166, and 
10 

64.333 respectively, in formulas for action of gravity 1 to 9, to lit them for computa¬ 
tion in an accelerating or retarding force one-tenth of gravity. 

2.—Table, page 488, giving relations of velocity and height of falling bodies, may 
be employed in solving questions of accelerating force general. 

Example. —A ball weighing 10 lbs. is projected with an initial velocity of 60 feet 
per second on a level plane, and frictional resistance to its motion is 1 lb. What dis¬ 
tance will it traverse before it comes to a state of rest? By formula 4: 


10 lbs. x 60 2 
64-333 X 1 lb. 


559.59/eef. 



















GRAVITATION. 


496 


Again, same result may be arrived at, according to Note 1, by multiplying con¬ 
stant 64.333, in Rule, page 494, for gravity, by ratio of force and weight, which in 
this case is -jV, and 64.333 X = 6.4333. Substituting 6.4333 f° r 64.333 * n that 
rule, formula becomes 

„ V 2 60 2 
S = -= -= 559.59 feet. 

6.4333 6.4333 


The question may be answered more directly by aid of table for falling bodies, 
page 488. Height due to a velocity of 60 feet per second, is 55.9 feet; which is to 
be multiplied by inverse ratio of accelerating force and weight of body, or -1^, or 10; 

tbat is ’ 55-9 X 10 = 559 feet. 


If the question is put otherwise—What space will a weight move over before it 
comes to a state of rest, with an initial velocity of 60 feet per second, allowing fric¬ 
tion to be one tenth weight? The answer is that friction, which is retarding force, 
being one tenth of weight, or of gravity, space described will be 10 times as great as 
is necessary for gravity, supposing the weight to be projected vertically upwards to 
bring it to a state of rest. The height due to velocity being 55.9 feet; then 

55.9 x 10 = 559/eei!. 

Average velocity of a moving body, uniformly accelerated or retarded during a 
given period or space, is equal to half sum of initial and final velocities. 

To Compute Velocity’- of a, Walling Stream of Water per 
Second, at End. of any- given Time. 

When Perpendicular Distance is given. 

Example. —What is the distance a stream of water will descend on an inclined 
plane 10 feet high, and 100 feet long at base, in 5 seconds? 

5 2 X 16.083 = 402.08 feet = space a body will freely fall in this time. 

Then, as 100 : 10 :: 402.08 : 40.21 feet — proportionate velocity on a plane of these 
dimensions to velocity when falling freely. 


IVIiscellaneons Illustrations. 


1. —What is the space descended vertically by a falling body in 7 seconds. 

S = *5 gxt 2 . Then 16.083 X 7 2 = 788.067 feet. 

2. —What is the time of a falling body descending 400 feet, and velocity acquired 
at end of that time? 

* — g‘ Then ——^ = 4.98 sec. v = V2 g X S. Then V64.333 X 400 = 160.4 feet. 

3. —If a drop of rain fall through 176 feet in last second of its fall, how high was 
the cloud from which it fell? 

h 2 176 2 

S = —. Then - 7 ^ — 482.75 feet. 

2 g 64.166 ^ 

4. —If two weights, one of 5 lbs. and one of 3, hanging freely over a sheave, are 
set free, how far will heavier one descend or lighter one rise in 4 seconds. 

“jp| X 16.083 X 4 2 = ~ X 257.328 = 64.33 feet. 

5-—If length of an inclined plane is 100 feet, and time of descent of a body is 6 
seconds, what is vertical height of plane or space fallen through ? 


6 2 X .5 g 


579 


= 17.27 feet. 


6.—If a bullet is projected vertically with a velocity of 135 feet per second, what 
velocity will it have at 60 feet? 


Formula 7, page 492. 


i35 


32.166 


7 ^ 


2 X 60 


166 2 32.166 


.41 seconds . 












GUNNERY. 


497 


GUNNERY. 

A heavy body impelled by a force of projection describes in its flight 
or track a parabola, parameter of which is four times height due to 
velocity of projection. 

Velocity of a shot projected from a gun varies as square root of 
charge directly, and as square root of weight of shot reciprocally. 

To Compute Velocity of a Shot or Shell. 

Rule. —Multiply square root of triple weight of powder in lbs. by 1600; 
divide product by square root of weight of shot; and quotient will give ve¬ 
locity in feet per second. 

Example. —What is velocity of a shot of 196 lbs., projected with a charge of 9 lbs. 
of powder? 

V 9 X 3 X 1600 y/ig6z—i 8320 -r-14 = 594.3 lbs. 

To Compute Range for a, Charge, or Charge for a Range. 

When Range for a Charge is given. —Ranges have same proportion as 
charges of powder; that is, as one range is to its charge, so is any other 
range to its charge, elevation of gun being same in both cases. Consequently , 

To Compute Range. 

Rule.—M ultiply range determined by charge in lbs. for range required, 
divide product by given charge, and quotient will give range required. 

Example. —If, with a charge of 9 lbs. of powder, a shot ranges 4000 feet, how far 
will a charge of 6.75 lbs. project same shot at same elevation? 

4000 X 6.75 - 4 - 9 = 3000 feet. 

To Compute Charge. 

Rule.—M ultiply given range by charge in lbs. for range determined, 
divide product by range determined, and quotient will give charge required. 

Example.— If required range of a shot is 3000 feet, and charge for a range of 4000 
feet has been determined to be 9 lbs. of powder, what is charge required to project 
same shot at same elevation? 

3000 X 9 -r- 4000 = 6.75 lbs. 

To Compute Range at one Elevation, when. Range for 
another is given. 

Rule.—A s sine of double first elevation in degrees is to its range, so is 
sine of double another elevation to its range. 

Example.— If a shot range 1000 yards when projected at an elevation of 45 0 , how 
far will it range when elevation is 30 0 16', charge of powder being same ? 

Sine of 45 ° X 2 = 100 000; sine of 30 0 16' X 2 — 87 064. 

Then, as 100000 : 1000 ;; 87064 : 870 .64 feet. 

To Compute Elevation at one Range, when Elevation 
for another is given. 

Rule. _As range for first elevation is to sine of double its elevation, so 

is range for elevation required to sine for double its elevation. 

Example —Tf range of a shell at 45 0 elevation is 3750 feet, at what elevation 
must a gun be set for a shell to range 2810 feet with a like charge of powder? 

Sine of 45 0 X 2 = 100000. 

Then, as 3750 : icoooo !! 2810 : 74 933 — sine for double elevation — 24 0 16 . 

Approximate Rule for Time of Flight. 

Under 4000 yards, velocity of projectile 900 feet in one second; under 
6000 yards, velocity 800 feet; and over 6000 yards, velocity 700 feet. 

Guns and Howitzers take their denomination from weights of their solid 
shot in round numbers, up to the 42-pounder; larger pieces, rifled guns, and 
mortars, from diameter of their bore. 

p T * 


498 


GUNNERY. 


Initial "Velocity and. Ranges of Sliot and Sliells. 


The Range of a shot or shell is the distance of its first graze upon a horizontal 
plane, the piece mounted upon its proper carriage. 


Arms and Ordnance. 

Project 

Description. 

le. 

Weight. 

Powder. 

Initial 

Velocity. 

Time of 
Flight. 

Eleva¬ 

tion. 

Range 

Rifle Musket.. 

Elongated. 

Grains. 

5io 

Grains. 

60 

Feet. 

963 

Seconds. 

O t 

Yards. 

Musket, 1841. 

Round. 

412 

HO 

1500 

— 

— 

— 

6-Pounder. 

<{ 

Lbs. 

6.15 

Lbs. 

1.25 


. . - 

5 

1523 

12 “ . 

U 

12.3 

2-5 

1826 

i-75 

I 

575 

24 “ . 

U 

24.25 

6 

1870 

— 

2 

1147 

32 “ . 

u 

32-3 

8 

1640 

— 

1 

7i3 

4 2 “ . 


42-5 

10.5 

— 

— 

5 

1955 

8-inch Columbiad... 

a 

65 

IO 

— 

14.19 

15 

3224 

10 “ “ 

u 

127.5 

15 

— 

14.32 

15 

3281 

10 “ Mortar. 

Shell. 

98 

IO 

— 

3 6 

45 

4250 

13 “ “ ...- 

U 

200 

20 

— 

— 

45 

4325 

15 “ Columbiad... 

u 

302 

40 

— 

— 

7 

1948 

15 “ “ 

u 

315 

50 

— 

23.29 

25 

4680 

RIFLED. 








10-pounder Parrott.. 

u 

9-75 

I 

— 

21 

20 

5000 

20 “ “ .. 


*9 

2 

— 

17-25 

15 

4400 

30 “ “ 

u 

29 

3-25 

— 

27 

25 

6700 

100 “ “ 

Elongated. 

IOO 

IO 

— 

29 

25 

6910 

100 “ “ 

Shell. 

IOI 

IO 

1250 

28 

25 

6820 

200 “ “ 

( ( 

150 

16 

— 

— 

4 

2200 

12-inch Rodman. 

U 


50 

1154 

5-5 

40 

— 

TTall’s Rockets. 

3-inch. 

16 




47 

1720 

I’ei 

aetration 

of SI 

hot a: 

nd ST 

lell. 


Experiments at Fort Monroe, 1839, and at West Point, 1853. 


Ordnance. 

Charge. 

Distance. 

Mean 

© . 
is* 
is 0 

Penetrai 

-g AS 

O* 

Granite, p 

Ordnance. 

Charge. 

Distance. 

Mean 1 

© 

X# 

-q cl 

’enetr 

■g AS 
c-.g 

Em 

Granite, g 


Lbs. 

Yds. 

Ins. 

Ins. 

Ins. 


Lbs. 

Yds. 

Ins. 

Ins. 

Ins. 

32 Lbs. Shot. 

8 

880 


I 5-25 

3-5 

8-inch Howitz. * 

6 

880 

— 

8-5 

I 

32 “ “ 

II 

IOO 

60 



8 “ Columbdf 

12 

200 

— 


— 

42 “ “ 

10.5 

IOO 

54-75 

18 

4 

10 “ “ t 

18 

114 

63-5 

44 

7-75 

42 “ Shell. 

7 

IOO 

4 °- 75 

— 

— 

20 ^ ^ ^ •¥ 

i£T 

IOO 

56-75 

— 

— 


1 24 ins. of Concrete. * Shell. f Shot. 


Solid shot broke against granite, but not against freestone or brick, and general 
effect is less upon brick than upon granite. 

Shells broke into small fragments against each of the three materials. 

Penetration in earth of shell from a 10-inch Columbiad was 33 ins. 


Experiments — England. ( Holley.) 


Ordnance. 

Charge. 

Projectile. 

Weight. 

Velocity. 

Range. 

Target and Effects. 


Lbs. 


Lbs. 

Feet. 

Yards. 


u-inch U. S. Navy. 

30 

Shot. 

169 

1400 

50 

Iron plates, 14 ins. 
—loosened. 

15-inch Rodman... 

RIFLED. 

60 

u 

400 

H 

00 

0 

50 

Iron plates, 6 ins.— 
destroyed. 

7-inch Whitworth.. 

25 

Shot. 

150 

1241 

200 

Inglis’st—destr’d. 

10.5-inch Armstrong 

45 

i t 

307 

1228 

200 

C L U 

13-inch “ 

90 

* U 

344-5 

1760 

200 

Solid plates, n ins. 
thick—destr’d. 


* Steel. f 8-inch vertical and 5-inch horizontal slabs, and 7-inch vertical and 5-in. horizontal 
slabs, 9X5 ins. ribs and 3-inch ribs. 


























































GUNNERY. 


499 


Elements of Report of Board of Engineers for Fortifications , U. S. A. 

Professional Papers No. 25 . ( Brev. Maj.-Gen. Z. B. Tower.) 

Experimental firings for penetration during tlie past twenty years have 
determined that wrought iron and cast iron, unless chilled, are unsuitable for 
projectiles to be used against iron armor; that the best material for that 
purpose is hammered steel or Whitworth’s compressed steel. 

2. That cast-iron and cast-steel armor-plates will break up under the im¬ 
pact of the heaviest projectiles now in service, unless made so thick as to 
exclude their use in ship-protection. 

3. That wrought-iron plates have been so perfected that they do not break 
up, but are penetrated by displacement or crowding aside of the material in 
the path of the shot, the rate of penetration bearing an approximately deter¬ 
mined ratio to the striking energy of the projectile, measured per inch of 
shot’s circumference, as expressed by the following formula: 


2-035/_ 


V 2 P 


—- =penetration in ins. V representing velocity in 
2 g X 2 r 7r X 2240 X-86 ^ * 

feet per second , P weight of shot in lbs., and r radius of shot in ins. 

That such plates can therefore be safely used in ship construction, their 
thickness being determined by the limit of flotation and the protection 
needed. 

4. That, though experiments with wrought-iron plates, faced with steel, 
have not been sufficiently extended to determine the best combination of 
these two materials, we may nevertheless assume that they give a resistance 
of about one fourth greater than those of homogenous iron. 

5. That hammered steel in the late Spezzia trials proved superior to any 
other material hitherto tested for armor-plates. The 19-inch plate resisted 
penetration, and was only partially broken up by 4 shots, three of which had 
a striking energy of between 33 000 and 34 000 foot-tons each. Not one shot 
penetrated the plate. Those of chilled iron were broken up, and the steel 
projectile, though of excellent quality, was set up to about two thirds of its 
length. 


"Velocity and. Ranges of Shot. (Krupp's Ballistic Tables.) 
^Penetration in Wrought Iron. 


V 


V 2 P 


2 g X 2 r n X 2240 X C 


= penetration in ins. C = 2.53. 


Gun. 

Cali¬ 

ber. 

Powder. 

Shot. 

V 

at 

Muzzle 
per Sec. 

elocity 

Rar 

3000 

ge. 

6000 

at 

Muzzle 

Penet 

600 

ration 

Range 

3000 

Tons. 

Ins. 

Lbs. 

Lbs. 

Feet. 

Yds. 

Yds. 

Ins. 

Ins. 

Ins 

Armstrong, 100.. 

17-75 

550 

2022 

1715 

1424 

II 9 I 

34-7 6 

33-2 

27-55 

U U 

17-75 

776 

2000 

1832 

1518 

1259 

37-52 

35-81 

29.66 

Woolwich, 81.. 

l 6 

445 

1760 

1657 

I 393 

1181 

32.6 

3!-23 

26.24 

Krupp, 71.. 

15-75 

485 

1715 

1703 

1434 

1211 

33-52 

32.12 

27.04 

“ 18.. 

9-45 

165 

474 

1688 

r 35 i 

m3 

20.42 

i 9 - 3 i 

15.46 

U. S. * 8-inch_ 

8 

35 

180 

1450 

1036 

840 

10.23 

9.22 

6.72 


* Unchambered. 


6000 

Ins. 
22.04 
23-47 
2i-35 
21.89 
12.14 
5-i7 


Target .—For 100-ton gun, steel plate 22 ins. thick, backed with 28.8 ins. of wood, 
2 wrought-iron plates 1.5 ins. thick, and the frame of a vessel. 

Effect .—Total destruction of steel plate, and backing entered to a depth of 22 ins., 
but not perforated. 
























500 GUNNERY. 


Summary of Record, of Practice in Europe "with. Heavy 
Armstrong, Woolwich, and Krupp Guns. 

Board of Engineers for Fortifications , U. S. A., Professional Papers No. 25. 


Gun. 

Powder. 

Projectile. 

Charge of 

Powder. 

Weight of 

Projectile. 

Initial Velocity 

per Second. 

V. 


Er 

4 

N 

N 

per inch of 5 

circumference 3 

of shot. 

P V 2 

£ 

01 

IN 

b 

5 - 

Cl 

<N 




Lbs. 

Lbs. 

Feet. 

Ft.-tons. 

Foot-tons. 

Armstrong, 1 

1.5-inch cubes.. 

Shot... . 

330 

2000 

1446 

28 990 

544-05 

100 Tons, caliber | 

Waltham Abbey 

u 

375 

2000 

1543 

33 000 

623 


17 ins., bore 30.5 j 

Fossano. 

u 

400 

2000 

1502 

31 282 

585-74 

feet. J 

U 

it 

776 

2000 

1832 

46580 

835-32 

Woolwich, 81 1 

.75-inch cubes. 

a 

170 

1258 

1393 

16 922 

37 I -5 


Tons, caliber 14.5 ^ 

1.5 “ “ 

it 

220 

1450 

1440 

20 842 

457-57 

ins., bore 24 feet. J 

2 

t c 

250 

1260 

1523 

20259 

444.78 

caliber 16 ins. 

1.5 “ 

u 

310 

1466 

1553 

24 508 

520.4 


38 Tons, 1 

1.5 “ “ 

Pall, shell 

130 

800 

i 45 i 

11 668 

297.64 

caliber 12.5 ins., j- 

1.5 “ <£ 

it 

200 

800 

1421 

II 210 

285.4 


bore 16.5 feet. J 

1.5 “ “ 

11 

0 

00 

M 

800 

1504 

12545 

3 I 9-4 


Krcpp, 71 Tons, 1 

Prism A. 

Plain ... 

298 

1707 

00 

H 

H 

16 602 

335-42 

caliber 15.75 ins., > 

“ H. 

Shrapnel 

485 

w 2 5 

1703 

34 503 

697.91 


bore 28.58 feet. J 

“ 2 inch... 

Shell.... 

44 1 

1419 

1761 

30 484 

616.14 

18 Tons, 1 

“ 1 hole... 

Plain ... 

132 

3 °° 

1873 

7 298 

246.03 

caliber 9.45 ins., V 

“ 2 inch... 

Shrapnel 

145 

474 

1688 

9367 

3 i 5-66 

bore 17.5 feet. J 

U 

Shell.... 

165 

300 

1991 

8 244 

277.69 


Penetration in Ball Cartridge Paper , No. 1 . 


Musket, with 134 grains, at 13-3-yards. 653 sheets. 

Common rifle, 92 grains, at 13.3 yards. 500 sheets. 


Penetration of Lead Ralls in Small Arms. 


Experiments at Washington Arsenal in 1839, and at West Point in 1837. 


Arm. 

Diameter 
of Ball. 

Charge 

Powder. 

Distance. 

Weight 
of Ball. 

Peneti 
White Oak. 

•ation. 

White Pine. 


Inch. 

Grains. 

Yards. 

Grains. 

Ins. 

Ins. 

Musket. 


[•64 

134 

9 

397-5 

1.6 

— 



I.64 

144 

5 

397-5 

3 

— 

Common Rifle. 


— 

IOO 

5 

219 

2.05 

— 



— 

92 

9 

— 

1.8 

— 

Hall’s rifle. 


— 

IOO 

5 

219 

2 

— 



— 

70 

9 

219 

.6 

— 




70 

5 

— 

i -7 

— 

Hall’s carbine, musket 



80 

5 

219 

.8 

— 

caliber. 

i 

•5775 

90* 

5 

— 

I. I 

— 




IOO* 

5 

— 

1.2 

— 

Pistol . 

— 

51 

5 

219 

•725 

— 

Rifle musket. 

•5775 


200 

500 


II 

Altered musket. 

.685 

60 

200 

73 ° 

— 

10.5 

Rifle, Harper’s Ferry.. 

•5775 

70 

200 

500 

— 

9-33 

Pistol carbine. 

•5775 

40 

200 

450 

— 

5-75 

Sharpe’s carbine. 

•55 

60 

30 

463 

— 

7.17 

Burnside’s “ . 

•55 

55 

30 

350 

— 

6.15 


* Charges too great for service. 


Musket discharged at 9 yards distance, with a charge of 134 grains, 1 hall and 3 
buckshot, gave for hall a penetration of 1.15 ins., buckshot, .41 inch. 
























































GUNNERY. 


501 


Loss cf Force by Windage. 

A comparispn of results shows that 4 lbs. of powder give to a ball without wind¬ 
age nearly as great a velocity as is given by 6 lbs. having .14 inch windage, which 
is true windage of a 24-lb. ball; or, in other words, this windage causes a loss of 
nearly one third of force of charge. 

Vents. —Experiments show that loss of force by escape of gas from vent 
of a gun is altogether inconsiderable when compared with whole force of 
charge. 

Diameter of Vent in U. S. Ordnance is in all cases .2 inch. 


Effect of different Waddings with a Charge of 77 Grains of Powder. 


Wad. 

Velocity of Ball 
per Second. 

Ball wrapped in cartridge paper and crumpled. 

Feet. 

1377 

1346 

1482 

1 felt wad. upon powder and 1 upon ball. 

2 felt wads upon powder and 1 upon ball.. 

1 elastic wad upon powder and 1 upon ball. 

2 pasteboard wads upon powder. 


2 elastic wads upon powder. 

I IOO 


Felt wads cut from body of a hat, weight 3 grains. 

Pasteboard wads .1 of an inch thick, weight 8 grains. 

Cartridge paper 3X4-5 ins., weight 12.82 grains. 

Elastic wads, “Baldwin’s indented,” a little more than .1 of an inch thick, 
weight 5.127 grains. 

Most advantageous wads are those made of thick pasteboard, or of or¬ 
dinary cartridge paper. 

In service of cannon , heavy wads over ball are in all respects injurious. 

For purpose of retaining the ball in its place, light grommets should be used. 

On the other hand, it is of great importance, and especially so in use of small 
arms, that there should be a good wad over powder for developing full force of 
charge, unless, as in the rifle, the ball has but very little windage. (Capt. Mordecai.) 

Weight and. Dimensions of Lead Balls. 

Number of Balls in a Lb., from 1.3125 to .237 of an Inch Diameter. 


Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Diam. 

No. . 

Diam. 

No. 

Diam. 

No. 

Ins. 


Inch. 


Inch. 


Inch. 


Inch. 


Inch. 


1.67 

I 

•75 

II 

•57 

25 

.388 

SO 

• 3 01 

170 

■259 

270 

1.326 

2 

•73 

12 

•537 

30 

•375 

88 

•295 

180 

.256 

280 

i-i57 

3 

• 7 1 

13 

• 5 i 

35 

•372 

9 ° 

.29 

190 

.252 

290 

1-051 

4 

•693 

14 

•505 

36 

•359 

IOO 

.285 

200 

•249 

300 

•977 

5 

.677 

15 

. 488 

40 

•348 

no 

.281 

210 

.247 

310 

.919 

6 

.662 

l6 

.469 

45 

•338 

120 

.276 

220 

•244 

320 

•873 

7 

.65 

17 

•453 

50 

•329 

130 

.272 

230 

.242 

330 

•835 

8 

•637 

18 

.426 

60 

.321 

140 

.268 

24O 

•239 

340 

.802 

9 

.625 

19 

•405 

70 

• 3 i 4 

150 

.265 

250 

•237 

350 

•775 

IO 

.615 

20 

■395 

75 

•307 

160 

.262 

260 




Heated shot do not return to their original dimensions upon cooling, but retain 
a permanent enlargement of about .02 per cent, in volume. 


Number of Pellets in an Ounce of Lead Shot of the different Sizes. 


A A. 

A. 
B B. 


40 

So 

58 


No. 


75 

82 


2.112 


No. 3. 
4 - 
, 5 - 


i 35 

177 

218 


No. 6.280 


34i 

600 


No. 9. 
10. 
12. 


984 

1726 

2140 


No. 14. 


3150 



























































502 GUNNERY. 

Proportion of Powder to Slaot for following Numbers 

of Sliot. 


No. 

Shot. 

Powder. 

No. 

Shot. 

Powder. 

No. 

Shot. 

Powder. 


Oz. 

Drams. 


Oz. 

Drams. 


Oz. 

Drams. 

2 

2 

i -5 

4 

i -5 

I -875 

6 

1.25 

2-375 

3 

i -75 

1.625 

5 

i -375 

2.125 

7 

1.125 

2.625 


Note.—2 oz. of No. 2 shot, with 1.5 drams of powder, produced greatest effect. 

Increase of powder for greater number of pellets is in consequence of increased 
friction of their projection. 


Numbers of Percussion Caps corresponding with Birmingham Numbers. 


Eley’s. 

5 

6 

7 

8 

9 

24 

10 

II 

18 

12 

13 

14 

Birmingham.. 

43 

44 

46 

48 

49 

50 

51 and 52 

53 and 54 

55 and 56 

57 

58 

58 


Where there are two numbers of Birmingham sizes corresponding with only one 
of Eley’s, it is in consequence of two numbers being of same size , varying only in 
length of caps. 


Comparison of Force of a Cliarge in. various Arms. 


Aem. 

Lock. 

Powder, 

A 5 - 

Windage. 

Weight 
of Ball. 

Velocity. 

Ordinary rifle. 

Percussion. 

Grains. 

100 

Inch. 

.015 

Grains. 

219 

Feet. 

2018 



70 

.015 

219 

1755 

Hall’s rifle. 

Flint. 

70 

.O 

219 

149° 

Hall’s carbine. 

Percussion. 

70 

.O 

219 

1240 

Jenks’s carbine. 

11 

70 

.O 

219 

1687 

Cadet’s musket. 

Flint. 

70 

•°45 

219 

1690 

Pistol. 

Percussion. 

35 

.015 

218.5 

947 


Ranges for Small Arms. 

Musket .—With a ball of 17 to pound, and a charge of no grains of powder, etc., 
an elevation of 36' is required for a range of 200 yards; and for a range of 500 
yards, an elevation of 3 0 30' is necessary, and at this distance a ball will pass through 
a pine board 1 inch in thickness. 

Rifle .—With a charge of 70 grains, an effective range of from 300 to 350 yards is 
obtained; but as 75 grains can be used without stripping the ball, it is deemed better 
to use it, to allow for accidental loss, deterioration of powder, etc. 

Pistol .—With a charge of 30 grains, the ball is projected through a pine board 
1 inch in thickness at a distance of 80 yards. 


Gunpowder. 

Gunpowder is distinguished as Musket , Mortar , Cannon , Mammoth , and 
Sporting powder; it is all made in same manner, of same proportions of 
materials, and differs only in size of its grain. 

Bursting or Explosive Energy.—By the experiments of Captain Rodman, U. S. 
Ordnance Corps, a pressure of 45 000 lbs. per square inch was obtained with 10 lbs. 
of powder, and a ball of 43 lbs. 

Also, a pressure of 185000 lbs. per sq. inch was obtained when the powder was 
burned in its own volume , in a cast-iron shell having diameters of 3.85 and 12 ins. 

Proof of Powder. (U. S. Ordnance Manual.) 

Powder in magazines that does not range over 180 yards is held to be unservice¬ 
able. 

Good powder averages from 280 to 300 yards; small grain , from 300 to 320 yards. 

Restoring Unserviceable Powder. — When powder has been damaged by being 
stored in damp places, it loses its strength, and requires to be worked over. If 
quantity of moisture absorbed does not exceed 7 per cent., it is sufficient to dry it 
to restore it for service. This is done by exposing it to the sun. 

When powder has absorbed more than 7 per cent, of water it should be sent to a 
powder mill to be worked over. 





























































GUNNERY. 


503 


Properties and. Results of Gunpowder, determined by 
Experiments. (Captain A. Mordecai , U. S. A.) 

Musket Pendulum. 


24-PouNDER Gun. 

Weight of ball and wad_ 24.25 lbs. 

“ u powder. 6 “ 

Windage of ball.135 inch. 


Weight of ball. 307.5 grains. 

“ “ powder. 120 “ 


Windage of ball. 


.09 


inch. 


Grain. 

Cc 

Sal t- 
petre. 

>mpositi< 

Char¬ 

coal. 

>n. 

Sul¬ 

phur. 

Manufacture. 

Where from. 

Number of 

Grains in 10 

Troy Grains. 

Relative 

Quickness of 

Burning. 

Water ab¬ 

sorbed by ex¬ 
posure to Air. 

Relative 

Force. 

. 

Cannon, large... 

“ small... 

Musket. 

1 




* Dupont’s Mills, 
Del. 

77 

569 

1 134 

6 174 

5 344 

1 642 

13 152 

166 
103 
72 808 
295 
2378 

275 

314 

214 

142 

282 

Per c’t. 
2.77 

3-35 

.677 

•72 

.808 

Rifle. 


-76 

H 

12 

_ 

• 9°7 
. 728 

•834 

•943 

.788 

•756 

I 

Rifle. 


3-55 

Musket. 







Rifle. 






_ 

_ 

Cannon, uneven. 

“ large... 

Sporting. 


>75 

77 

70 

12.5 

13 

15 

12.5 

IO ) 

t Dupont’s Mills, 
Del. 

* Dupont’s Mills, 
Del. 

Loomis, Hazard, 
& Co., Conn. * 

183 

l82 

IOO 

2.09 

1.91 
4.42 

Blasting, uneven 
Rifle. 


15 J 

1 

212 

204 

.82 

Sporting. 

| 76 

i 5 

9 1 

_ 

.888 

Rifle. 

15 

10 } 

Waltham Abbey, 
England.* 

11 600 



.865 







* Glazed. 


4 Rough. 


Manufacture of Powder. —Powder of greatest force, whether for cannon or small 
arms, is produced by incorporation in the “cylinder mills.” 

Effect of Size of Grain. —Within limits of difference in size of grain, which occurs 
in ordinary cannon powder, the granulation appears to exercise but little influence 
upon force of it, unless grain be exceedingly dense and hard. 

Effect of Glazing. —Glazing is favorable to production of greatest force, and to 
quick combustion of grains, by affording a rapid transmission of flame through 
mass of the powder. 

Effect of using Percussion Primers. —Increase of force by use of primers, which 
nearly closes vent , is constant and appreciable in amount, yet not of sufficient value 
to authorize a reduction of charge. 

Ratio of Relative Strength of different Powders for use under water differ 

hut little from the reciprocal of the ratio between the sizes of the grains, 

showing that the strength is nearly inversely proportional thereto.* 

Mammoth, .08; Oliver, .09; Cannon, .18; Mortar, 1; Musket, 1.57; 
Sporting 2.61, and Safety Compound 30.62. 

Duialin. is nitro-glycerine absorbed by Schultze’s powder. 

For other powders and explosive materials see Gunnery, page 443. 

Heat and. Explosive IPower. (Capt. Noble and F. A. Abel.) 

One gram of fired powder evolves a mean temperature of 730°. Temper¬ 
ature of explosion 3970°. Volume of permanent gas (which is in an in¬ 
verse ratio to units of heat evolved) at 32 0 = 250°. 

The explosive power of powder, as tested in Ordnance, ranges, for volumes 
of expansion of 1.5 to 50 times, from 36 to 170 foot-tons per lb. burned. 

A charge of 70 lbs. gave to an 180 lbs. shot a velocity of 1694 feet per 
second, equal to a total energy of 3637 foot-tons, and a charge of 100 lbs. 
gave a velocity of 2182 feet, and an energy of 5940 foot-tons. 


* Report of Experiments and Investigations to develop a system of submarine mines. 
Papers, U. S. E., No. 23. 


Professional 




































504 


HEAT. 


HEAT. 

Heat, alike to gravity, is a universal force, and is referred to both as 
cause and effect. 

Caloric is usually treated of as a material substance, though its claims 
to this distinction are not decided; the strongest argument in favor of 
this position is that of its power of radiation. Upon touching a body 
having a higher temperature than our own, caloric passes from it, and 
excites the feeling of warmth; and when we touch a body having a 
lower temperature than our own, caloric passes from our body to it, and 
thus arises the sensation of cold. 

To avoid any ambiguity that may arise from use of the same expres¬ 
sion, it is usual and proper to employ the word Caloric to signify the 
principle or cause of sensation of heat. 

Heat Unit. —For purpose of expressing and comparing quantities of 
heat, it is convenient and customary to adopt a Unit of heat or Thermal 
unit , being that quantity of heat which is raised or lost in a defined 
period of temperature in a defined weight of a particular substance. 

Thus, a Thermal unit, Is quantity of heat which corresponds to an interval of x° in 
temperature of i lb. of pure liquid water , at and near its temperature of greatest 
density. 

Thermal unit in France, termed Caloric , Is quantity of heat which corresponds 
to an interval of i° C. in temperature of i Icilogramme of pure liquid water , at and 
near its temperature of greatest density. 

Thermal unit to Caloric, 3.96832; Caloric to Thermal unit, .251996. 

One Thermal unit or i° in 1 lb. of water, 772 foot-lbs. 

One Caloric or i° C. in 1 kilogramme of water, 423.55 kilogrammetres. 

i° C. in 1 lb. water, 1389.6 foot-lbs. 

Ratio of Fahrenheit to Centigrade, 1.8; of Centigrade to Fahrenheit, .555. 

Absolute Temperature , Is a temperature assigned by deduction, as an 
opportunity of observing it cannot occur, it being the temperature corre¬ 
sponding to entire absence of gaseous elasticity, or when pressure and vol¬ 
ume =0. By Fahrenheit it is—461.2 0 , by Reaumur—229.2 0 , and by Cen¬ 
tigrade— 2 74 0 . 

Heat is termed Sensible when it diffuses itself to all surrounding 
bodies; hence it is free and uncombined, passing from one substance 
to another, affecting the senses in its passage, determining the height 
of the thermometer, etc. 

Temperature of a body, is the quantity of sensible heat in it, present 
at any moment. 

Heat is developed by water when it is violently agitated. 

Heat is developed by percussion of a metal, and it is greatest at the first 
blow. 

Quantities of heat evolved are nearly the same for same substance, with¬ 
out reference to temperature of its combustion. 

Mechanical power may be expended in production of heat either by fric¬ 
tion or compression, and quantity of heat produced bears the same propor¬ 
tion to quantity of mechanical power expended, being 1 unit for power 
necessary to raise 1 lb. 772 feet in height. This number of 772 is termed 
the mechanical equivalent of heat (Joules). 


HEAT. 


505 


Specific Heat. 

Specific Heat of a body signifies its capacity for heat, or quantity re¬ 
quired to raise temperature of a body i°, or it is that which is ab¬ 
sorbed by different bodies of equal weights or volumes when their 
temperature is equal, based upon the law, That similar quantities of 
different bodies require unequal quantities of heat at any given tempera¬ 
ture. It is also the quantity of heat requisite to change the tempera¬ 
ture of a body any stated number of degrees compared with that which 
would produce same effect upon water at 32° 

Quantity of heat , therefore, is the quantity necessary to change the tem¬ 
perature of a body by any given amount (as i°), divided by quantity of 
heat necessary to change an equal weight or volume of water 32 0 by same 
amount. 

Note.—W ater has greater specific heat than any known body. 

Every substance has a specific heat peculiar to itself, whence a change of 
composition will be attended by a change of its capacity for heat. 

Specific heat of a body varies with its form. A solid has a less capacity 
for heat than same substance when in state of a liquid; specific heat of 
water, for instance, being .5 in solid state (ice), .622 in gaseous (steam), 
and 1 in liquid. 

Specific heat of equal weights of same gas increases as density decreases; 
exact rate of increase is not known, but ratio is less rapid than diminution 
in density. 

Change of capacity for heat always occasions a change of temperature. 
Increase in former is attended by diminution of latter, and contrariwise. 

Specific heat multiplied by atomic weight of a substance will give 
the constant 37.5 as an average, which shows that the atoms of all 
substances have equal capacity for heat. This is a result for which as 
yet no reason has been assigned. 

Thus: atomic weights of lead and copper are respectively 1294.5 and 395.7, and 
their specific heats are .031 and .095. Hence 1294.5 X .031=340.129, and 395.7 X 
•°95 — 37 - 59 1 - 

It is important to know the relative Specific Heat of bodies. The most conve¬ 
nient method of discovering it is by mixing different substances together at dif¬ 
ferent temperatures, and noting temperature of mixture; and by experiments it 
appears that the same quantity of heat imparts twice as high a temperature to 
mercury as to an equal quantity of water; thus, when water at ioo° and mercury 
at 40 0 are mixed together, the mixture will be at 8o°, the 20 0 lost by the water 
causing a rise of 40 0 in the mercury; and when weights are substituted for meas¬ 
ures, the fact is strikingly illustrated; for instance, on mixing a pound of mercury 
at 40 0 with a pound of water at 160 0 , a thermometer placed in it will fall to 155° 
Thus it appears that same quantity of heat imparts twice as high a temperature to 
mercury as to an equal volume of water, and that the heat which gives 5 0 to water 
will raise an equal weight of mercury 115 0 , being the ratio of 1 to 23. Hence, if 
equal quantities of heat be added to equal weights of water and mercury, their 
temperatures will be expressed in relation to each other by numbers 1 and 23; or, 
in order to increase the temperature of equal weights of those substances to the 
same extent, the water will require 23 times as much heat as the mercury. 

Capacity for Heat is relative power of a body in receiving and re¬ 
taining heat in being raised to any given temperature; while Specific 
applies to actual quantity of heat so received and retained. 

Specific Heat of Air and. other Gases. 

Specific heat, or capacity for heat, of permanent gases is sensibly constant 
for all temperatures, and for all densities. Capacity for heat of each gas is 

U u 


HEAT. 


506 


same for each degree of temperature. M. Regnault proved that capacity 
for heat for air was uniform for temperatures varying from —22 0 to 
-[-437°; consequently, specific heat for equal weights of air, at constant 
pressure, averaged .2377. 


Specific Heat. Water at 32 0 = 1. 


Metals from 32 0 to 
212 °. 

Antimony... .0508 

Bismuth.0308 

Brass.°939 

Copper.092 

Cast iron.1298 

Gold.°3 2 4 

Lead.0314 

Mercury.0333 

Nickel.1086 

Platinum.0324 


Silver.056 

Steel.1165 

Tin.0562 

Wrought iron .1138 
Zinc. 0955 

Stones. 

Chalk. 2149 

Limestone... .2174 

Masonry.2 

Marble, gray. .2694 
“ white. .2158 


Woods. 

Oak.57 

Pear.5 

Pine.65 

Mind Substances. 

Charcoal.2415 

Coal.2411 

Coke.203 

Glass.. 1977 

Gypsum.1966 

Phosphorus.. 2503 


Sulphur.2026 

Liquids. 

Alcohol...6588 

Ether.4554 

Linseed oil .. .31 

Olive oil.3096 

Steam.365 

Turpentine .. .416 
Vinegar.92 


Solid. 


Ice. 


•504 


Air. 


Gases. 

.2377 | Hydrogen....2356 


Hydrogen. 2.4096 

Carbonic Acid.1714 


Oxygen.2412 | Carbonic Acid.3308 

» For Equal Weights. 

Air.1688 

Oxygen.1559 

Metals have least, ranging from Bismuth .0308 to Cast Iron .1298. Stones and 
Mineral Substances have .2 that of water, and Woods about .5. Liquids, with ex¬ 
ception of Bromine, are less than water, Olive oil being lowest and Vinegar highest. 

Illustration.— If 1 lb. of coal will heat 1 lb. of water to ioo°, —— = —of a lb. 

• 033 30.3 

will heat 1 lb. of mercury to ioo°. 

To Compote Temperature of a AT ix tore of lilie Sub¬ 
stances. 


WT-frot 


= f'; 


tv (l' — () 


W; 


w (t‘ — t) 

1 -' + «' = T. 


W representing weight 


W -f- w T — t' " ’ W 

or volume of a substance of temperature T, w tveight or volume of a like substance of 
temperature t, and t' temperature of mixture W +- w. 

Illustration i. — When 5 cube feet of water (W) at a temperature of 150 0 (T) is 
mixed with 7.5 cube feet (w) at 50 0 (/), what is the resultant temperature of the 
mixture? 

' 5 X 150°+7.5 X 50 0 1125 

--j- =-— 90 0 . 

5 + 7-5 12.5 

2.—How much water at (T) ioo° should be mixed with 30 gallons (tv) at 6 o c , for 
a required temperature of 8 o°? 

30 (8o° — 6o°) 600 


ioo° — 8o° 


= 30 gallons. 


To Compote Temperature of a Mixture of TUolilte 

Substances. 


W S T + w s t 


t'- 


w s (t — t') 


= W: 


t' (W S-fw«)u)MS t 


= T. W and w 


W S + w s " ’ S (T — t) ■ " 1 W S 

representing weights, and S and s specific heat of substances. 

Illustration.— To what temperature should 20 lbs. cast iron (W) be heated to 
raise 150 lbs. ( w } of water to a temperature ( t ) of 50° to 6o°? 

6 o° (20 X . 1298 + 150X1)^150X1X50° 1655.76 

20 X .1298 2.596 


s—iy and S =. 1298. 


= 638°. 



















































HEAT. 


507 


To Compute Specific Heat at Constant "Volume. 
When Specific Heat at Constant Pressure is known. S P 


H 


= s. S represent- 


Or, 


mg specific heat at constant pressure , p proportion of heat absorbed at constant vol¬ 
ume, H total heat absorbed at constant pressure, and s specific heat at constant volume. 
S (t' — t) — 2.742 (V — v) 

-p---= s. t and t representing initial and final tempera¬ 
ture of the gas and that to which it is raised, and V and v initial and final volumes 
of the gas under 14.7 lbs. per sq. inch, and of it heated under constant pressure in 
cube feet. 

Illustration.— Assume 1 lb. air at atmospheric pressure and at 32 0 , doubled in 
volume by heat. S = .2377*, t — t' — 32 0 <x 525 0 = 493 0 and V — v = 12.387* cube 
feet. 

■»3 77 >< 493 -( 2 . 74 »X.2.387) = l68g ^ ^ 

493 

For comparative volumes of other gases, see Table, page 506. 


To Compute Specific Heat for Ecpial Volume of Gras 

and. Air-. 

Rule.—M ultiply specific heat of the gas for equal weights of gas and air 
by specific gravity of gas, and product is specific heat for equal volume. 
Example.— What is specific heat of air at equal volume with hydrogen? 

Specific heat of hydrogen for equal weights at constant volume, 2.4096, and speci¬ 
fic gravity of the gas, .0692. (See Table, page 506.) 

Then, 2.4096 X .0692 =. 1667 specific heat for equal volumes at constant volume. 
Specific heat of steam, air at unity = 1.281. 


Capacity for Heat. 

When a body has its density increased, its capacity for heat is di¬ 
minished. The rapid reduction of air to .2 of its volume evolves heat 
sufficient to inflame tinder, which requires 550°. 


Relative Capacity for Heat of Various Bodies. {Water at 32° = 1.) 


Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Water.. 

I 

I 

Gold.... 

•°5 

.966 

Mercury 

.036 

— 

Brass... 

.Il6 

.971 

Ice. 

•9 

— 

Silver .. 

.082 

•833 

Copper.. 

.114 

1.027 

Iron.... 

. 126 

•993 

Tin. 

.06 


(Hass... 

.187 

.448 

Lead... 

•°43 

.487 

Zinc.... 

. 102 

— 


To Ascertain Relative Capacities of Different Bodies, 
combined witli experiment. 

Rule.—M ultiply weight of each body by number of degrees of heat lost 
or gained by mixture, and capacities of bodies will be inversely as products. 

Or, if bodies be mingled in unequal quantities, capacities of the bodies 
will be reciprocally as quantities of matter, multiplied into their respective 
changes of temperature. 

Illustration.— If 1 lb. of water at 156° is mixed with 1 lb. of mercury at 40 0 , 
resultant temperature is 152 0 . 

Thus, 1 x 156° — 152° = 4 0 , and 1 X 40 0 152°^ 112 0 . Hence capacity of water 

for heat is to capacity of mercury as 112 0 to 4 0 , or as 28 to 1. 

Sensible Heat. 

Sensible heat or temperature to raise water from 32 0 to 212° = 180.9°, or 
heat units. 


* See Tables, pages 506 and 520-21. 


























5oS 


HEAT. 


Latent Heat. 

Latent Heat is that which is insensible to the touch of our bodies, 
and is incapable of being detected by a thermometer. 

When a solid body is exposed to heat, and ultimately passes into the 
liquid state under its influence, its temperature rises until it attains the 
point of fusion, or melting point. The temperature of the body at this 
point remains stationary until the whole of it is melted; and the heat mean¬ 
time absorbed, without affecting the temperature or being sensible to the 
touch or to the indications of a thermometer, is said to become latent. It is, 
in fact, the latent heat of fusion, or the latent heat of liquidity, and its func¬ 
tion is to separate the particles of the body, hitherto solid, and change their 
condition into that of a liquid. When, on the contrary, a liquid is solidified, 
the latent heat is disengaged. 

If to a pound of newly-fallen snow were added a pound of water at 172 0 , 
the snow would be melted, and 32 0 would be resulting temperature. 

When a body is fusing, no rise in its temperature occurs, however great 
the additional quantity of heat may be imparted to it, as the increased heat 
is absorbed in the operation of fusion. The quantity of heat thus made 
latent varies in different bodies. 

A pound of water, in passing from a liquid at 212 0 to steam at 212 0 , re¬ 
ceives as much heat as would be sufficient to raise it through 966.6 ther¬ 
mometric degrees, if that heat, instead of becoming latent, had been sensible. 


If 5.5 lbs. of water, at temperature of 32 0 , be placed in a vessel, communicating 
with another one (in which water is kept constantly boiling at temperature of 212 0 ), 
until former reaches temperature of latter quantity, then let it be weighed, and 
it will be found to weigh 6.5 lbs., showing that one lb. of water has been received 
in form of steam through communication, and reconverted into water by lower 
temperature in vessel. Now this pound of water, received in the form of steam, 
had, w T hen in that form, a temperature of 212 0 . It is now converted into liquid 
form, and still retains same temperature of 212 0 ; but it has caused 5.5 lbs. of water 
to rise from the temperature of 32 0 to 212 0 , and this without losing any tempera¬ 
ture of itself. Now this heat w T as combined with the steam, but as it is not sensible 
to a thermometer, it is termed Latent. 

Quantity of heat necessary to enable ice to resume the fluid state is equal 
to that which would raise temperature of same weight of water 140° ; and an 
equal quantity of heat is set free from water when it assumes the solid form. 


Swm of SensiTole and. Latent Heats. 
From Water at 32 0 . 


Press¬ 

ure. 

Latent. 

Sum. 

Press¬ 

ure. 

Latent. 

Sum. 

Press¬ 

ure. 

Latent. 

Sum. 

Press¬ 

ure. 

Latent. 

Sum. 

Lbs. 

O 

O 

Lbs. 

O 

O 

Lbs. 

O 

O 

Lbs. 

O 

O 

14.7 

964-3 

1146.1 

26 

943-7 

H 55-3 

55 

912 

1169 

120 

873-7 

1185.4 

16 

962.1 

ii' 47 - 4 

27 

942.2 

1155-8 

60 

908 

1170.7 

130 

869.4 

1187.3 

17 

959 - 8 

1148.3 

28 

940.8 

1156.4 

65 

904.2 

1172-3 

I40 

865.4 

1189 

iB 

957-7 

II49. 2 

29 

939-4 

U 57 -I 

70 

900.8 

1173.8 

15° 

861.5 

1190.7 

19 

955-7 

1150.1 

3 ° 

937-9 

1157.8 

75 

897-5 

1175-2 

160 

857-9 

II92.2 

20 

952.8 

1150.9 

32 

935-3 

1158.9 

80 

894-3 

1176.5 

I70 

854-5 

" 93-7 

21 

95 i -3 

1151-7 

35 

931.6 

1160.5 

85 

891.4 

II 77-9 

180 

85 I -3 

ii 95 -i 

22 

949-9 

1152-5 

37 

9 2 9-3 

1161.5 

90 

888.5 

1179.1 

I90 

848 

1196.5 

23 

948-5 

1153-2 

40 

920 

1162.9 

95 

885.8 

1180.3 

200 

845 

1197.8 

24 

946.9 

H 53-9 

45 

920.9 

1164.6 

IOO 

883.1 

1181.4 

220 

829.2 

1200. 3 

25 

945-3 

1154.6 

5 ° 

9 i6 -3 

1167.1 

no 

878.3 

ii83-5 

250 

831.2 

1203.7 


Latent Heat of Vaporization, or Number of Degrees of Heat required to con¬ 
vert following Substances from their respective Solidities to Vapor at 
Pressure of Atmosphere. 


Alcohol.364° 

Ammonia.86o° 

Ether (Sulph.).163° 


Ice.142.6° 

Mercury.157° 

Carbonic Acid.298° 


Water.966.6° 

Zinc.493O 

Oil of Turpentine.. 124° 






























HEAT. 


509 


Latent Heat of Fusion of Solids. (Person.) 


Substances. 

Melt¬ 

ing 

Point. 

Specific Heat. 
Liquid. I Solid. 

In Heat- 
units of 

1 lb. 

Substances. 

Melt¬ 

ing 

Point. 

Specific 

Liquid. 

Heat. 

Solid. 

In Heat- 
units of 
1 lb. 

Tin. 

O 

44 2 

O 

.0637 

O 

.0562 

25.6 

Ice. 

c 

32 

O 

I 

O 

• 5°4 

142.8=; 

Bismuth.. 

5 °7 

•0363 

.0308 

22. 7 

Phosphorus _ 

112 

.2045 

. 1788 

9 

Lead. 

Zinc. 

617 

773 

.0402 

.0314 

.0956 

9.86 
50.6 

Spermaceti. 

Wax. 

120 

142 


— 

148 

175 

Silver .... 

1873 

— 

•°57 

37-9 

Sulphur. 

239 

•234 

. 2026 

17 

Mercury.. 

39 

•°333 

.0319 

5 

Nitrate of soda.. 

59 1 

•413 

.2782 

113 

Cast iron.. 

3400 

— 

. 129 

233 

Nit. of potassia . 

642 

• 33*9 

.2388 

85 


To Compute Latent Heat of Fusion of a INon-metallic 

Substance. 

C y c (t -}- 256°) = L. C and c representing specific heats of substance in solid and 
liquid stale , t temperature of fusion, and L latent heat. 

Illustration. —What is latent heat of fusion of ice? 

C —.504; c = 1; and t = 32°. 

.504 x 1 X 32 -f 256 = 142.85° units. 

Note.— For Latent Heat of Fusion of some substances, see Deschanel’s. New York, 
1872. Heat, part 2. 


Radiation of Heat. 

Radiation of Heat is diffusion of heat by projection of it in diverging right 
lines into space, from a body having a higher temperature than space sur¬ 
rounding it, or body or bodies enveloping it. 

Radiation is affected by nature of surface of body; thus, black and rough 
surfaces radiate and absorb more heat than light and polished surfaces. 
Bodies which radiate heat best absorb it best. 

Radiant heat passes through moderate thicknesses of air and gas without 
suffering any appreciable loss or heating them. When a polished surface 
receives a ray of heat, it absorbs a portion of it and reflects the rest. The 
quantity of heat absorbed by the body from its surface is the measure of 
its absorbing power, and the heat reflected, that of its reflecting power. 

When temperature of a body remains constant it is in consequence of 
quantity of heat emitted being equal to quantity of heat absorbed by body. 
Reflecting power of a body is complement of its absorbing power; or, sum 
of absorbing and reflecting powers of all bodies is the same. 


Thus, if quantity of heat which strikes a body = 100, and radiating and reflecting 
powers each 90, the absorbent would be 10. 

Radiating or ^Alosorbent and Reflecting Rowers of 

Substances. 


Substances. 

Radiating 
or Ab¬ 
sorbing. 

Reflect 

ing. 

Lamp Black. 

IOO 

— 

Water. 

IOO 

— 

Carbonate of Lead. 

IOO 

— 

Lead, white. 

IOO 

— 

Writing Paper. 

98 

2 

Ivory, Jet, Marble. 

93 t0 98 

7 to 2 

Resin. 

96 

4 

Glass. 

90 

IO 

India Ink. 

85 

15 

Ice. 

85 

I 5 

Shellac. 

72 

28 

Lead. 

45 

55 

Cast Iron, bright pol'shed 

25 

75 

Platinum, a little polish'd 

24 

76 

Mercury. 

23 

77 


Substances. 

Radiating 
or Ab¬ 
sorbing. 

Reflect¬ 

ing. 

Wrought Iron, polished.. 

23 

77 

Lead, polished. 

19 

81 

Zinc, polished. 

19 

81 

Steel, polished. 

17 

83 

Platinum, in sheet. 

17 

83 

Tin. 

15 

8.5 

Copper, varnished. 

14 

86 

Brass, dead polished.... 

II 

89 

“ bright polished... 

7 

93 

Copper, ham’ered or cast 
“ deposited on iron 

7 

93 

7 

93 

Gold, plated. 

5 

95 

“ polished . 

3 

97 

Silver, polished. 

s 

97 

“ cast, polished ... 

3 

97 


U u* 





























































5 io 


HEAT. 


Radiating and .Absorbing Rower of various Bodies, in 
Units of Beat per Sq. Boot per Hour for a Difference 
of lo. (Peclet.) 


Unit. 

Silver, polished.0266 

Copper.0327 

Till.°439 

Brass, polished.0491 

Iron, sheet.092 


Unit. 


Iron, ordinary.5662 

Glass.5948 

Iron, cast.648 

Wood sawdust.7225 

Stone, Brick, etc.7358 


Unit. 

Woollen stuff.7522 

Oil paint..7583 

Paper.7706 

Lamp-black.8196 

Water.1-0853 


To Compute Doss of Beat Toy- Radiation per Sq. Boot. 

^ -— = R. T representing temperature of pipe, which is assumed to he .05 

d v 

less than that of steam ; t temperature of air ; l length of pipe, and v velocity of heat 
in feet per second ; d diameter in ins., and R radiation in degrees per second. 


Illustration. —Assume temperatures of a steam-pipe, steam, 212 0 , 200°, and air 
6o°, length of pipe 20 feet, velocity of heat (steam) 15 feet per second, and diameter 
of pipe 16 ins.; what will be loss of heat by radiation? 


1.7X20(200 — 60) 

- —--- r- — i5.oo u . 

16 X 15 


Reflection. 


Reflection of Heat is passage of heat from surface of one substance 
to another or into space, and it is the converse of radiation. 

Heat is reflected from surface upon which its rays fall in same manner as 
light, angle of reflection being opposite and equal to that of incidence. Met¬ 
als are the strongest reflectors. 


Reflecting Power of various Substances. 


Silver. 

. 97 

Specular metal... 

.. .86 

Zinc. 

.81 

Gold. 

. 95 

Tin. 

.. .85 

Iron. 


Brass. 

. 93 

Steel. 

.. .83 

Lead. 

.6 


Communication and Transmission of Beat. 

Communication of Heat is passage of heat through different bodies 
with different degrees of velocity. This has led to division of bodies 
into Conductors and Non-conductors ; former includes such as metals, 
which allow caloric to pass freely through their substance, and latter 
comprise those that do not give an easy passage to it, such as stones, 
glass, wood, charcoal, etc. 

Velocity of cooling, other things being equal, increases with extent of sur¬ 
face compared with volume of substance; and of two bodies of same mate¬ 
rial, temperature, and form, but differing in volume. 

Transmission of Heat is passage of heat through different bodies with dif¬ 
ferent degrees of intensity. Gaseous bodies and a vacuum are highest in 
order of transmittents. 


Relative Power of various Substances to Transmit Heat. 

All bodies capable of transmitting heat are more or less translucent, 
though their powers of transmitting heat and light are not in same rela¬ 
tive proportions. 


Air. 

1 l Flint-glass .. 

. .67 

Nitric acid .... 

•15 

Alcohol. 

.15 Gypsum. 


Rock-crystal .. 

.62 

Crown-glass. 

.49 1 Ice. 


Rape-seed oil. 

•3 


Sulphuric acid. .17 

Turpentine.31 

Water.n 


Heat which passes through one plate of glass is less subject to absorption 
in passing through a second and a third plate. Of 1000 rays, 451 were in¬ 
tercepted by 4 plates as follows : 

1st. 381. 2d. 43. 


3 d. 18. 


4th. 9. 








































HEAT. 5 i r 

Average Results of Heating and. Evaporating Water by 
Steam in Copper Bipes and Boilers. (D. K. Clark.) 

Steam condensed Heat transmitted 

Per sq. foot for i° difference per hour. 



Heating. 

Evaporating. 

Heating. 

Evaporating 


Lbs. 

Lbs. 

Units. 

Units. 

Cast-iron-plate surface. 

.077 

.105 

82 

IOO 

Copper-plate surface. 

.248 

• 4 8 3 

276 

534 

Copper-pipe surface. 

.291 

1.07 

312 

1034 


Whence.—Efficiency of copper-plate surface for evaporation of water is 
double its efficiency for heating; for copper-pipe surface efficiency is more 
than three times as much; and for cast-iron-plate surface, a fourth more. 

Efficiency of pipe surface is a fifth more than that of plate surface for 
heating, and more than twice as much for evaporation. 

Generally, copper-plate surface condenses .5 lb. of steam, copper-pipe 
1 lb., and cast-iron-plate surface .1 lb. per sq. foot per i° of temperature per 
hour, for evaporation. 

Quantity of heat transmitted is at rate of about 1000 units per lb. of steam 
condensed. 


Transmission of Heat through Glass of different Colors. 
Direct = 100. 


Plate. 

- 65.5 

Blue, deep. 

.19 

Yellow. 

. 40 

Window. 

.... 52 

“ light. 

. 42 

Orange. 


Violet, deep. 

••••53 

Green. 


Red. 

. 53 


M. Peclet defines law of transmission of heat as : The flow of heat which 
traverses an element of a body in a unit of time is proportional to its sur¬ 
face, and to difference of temperature of the two faces perpendicular to direc¬ 
tion of flow, and is in inverse of thickness of element. 

Q 

Or, (t — t') - = H. t, and t' representing temperatures of surfaces, C constant for 

material 1 inch thick, or quantity of heat transmitted per hour for i° difference of 
temperature through 1 unit of thickness, T thickness, and H qua?itity of heat in units 
passed through plate per sq. foot per hour. 

Quantities of Beat transmitted from Water to Water 
through Plates or Beds of HVfetals and otlier Solid 
Bodies, 1 Inch thick, per Sq. Eoot. 

For i° Difference of Temperature between the two Faces per Hour. 

Selected from M. Peclet’s tables. (D. K. Clark.) 


Substance. 

C or 
Quantity 
of Heat. 

Substance. 

C or 
Quantity 
of Heat. 

Substance. 

C or 

Quantity 
of Heat. 

Gold 

Units. 

620 

604 

59 6 

555 

Iron. 

Units. 

225 

225 

177 

112 

Marble. 

Units. 

24 

2.6 

6.56 

2. l6 


Zinc. 

Plaster. 


Tin. 

Glass . 

Copper. 

Lead. 

Sand. 


The conditions are, that the surfaces of conducting material must be per¬ 
fectly clean, that they be in contact with water at both faces of different 
temperatures, and that the water in contact'with surfaces be thoroughly and 
constantly changed. M. Peclet found that when metallic surfaces became 
dull, rate of transmission of heat through all metals became very nearly 
the same. 

To Compute TJnits of Beat Transmitted. 

Illustration i. —If 2000 lbs. beet root juice at 40 0 are contained in a copper 
boiler with a double bottom, and heated to 212 0 , with a heating surface of 25 sq. feet, 
and subjected to steam at a temperature of 275 0 , for a period of 15 minutes, what 
will be the total heat, and heat per degree of difference transmitted per sq. foot per 
hour? 


















































512 


HEAT. 


2i2° — 40 0 x 604-15 = 688 0 per hour , aud 2000 X 688 = 25 = 55040 units per sq. 
foot per hour. 

(212 0 -f- 40 0 ) 4 - 2 = 126° mean temperature of juice, and 275 0 —126° = 149 0 mean 
difference of temperature. 

Hence, 550404- 149 = 369.4 units per sq. foot per degree of difference per hour. 

2.—If 48.2 sq. feet of iron pipe 1.36 ins. in diameter, is supplied with steam at 275 0 , 
and it raises temperature of 882 lbs. water from 46° to 212 0 in 4 minutes, what will 
be total heat per sq. foot per hour, total heat per sq. foot per degree, and quantity 
condensed per sq. foot per degree per hour ? 

212 0 — 46° X 60 4 - 4 = 2490 0 in an hour; 46°-)-212 0 4 -2 = 129 0 mean temper¬ 
ature, and 275 0 — 129 0 = 146° difference of temperature. 

2 49 ° — X 882 _ units per sq. foot per hour, 455634-146 = 312.1 units per sq. 

48.2 

foot per degree, and total heat of steam above 129 0 = 1068 0 . 

012. I 

Hence — — .292 lbs. steam condensed per sq. foot per degree per hour. 

1068 

Evaporation. 

Evaporation or Vaporization is conversion of a fluid into vapor, and 
it produces cold in consequence of heat being absorbed to form vapor. 

It proceeds only from surface of fluids, and therefore, other things equal , 
must depend upon extent of surface exposed. 

When a liquid is covered by a stratum of dry air, evaporation is rapid, 
even when temperature is low. 

As a large quantity of heat passes from a sensible to a latent state during 
formation of vapor, it follows that cold is generated by evaporation. 

Fluids evaporate in vacuo at from 120° to 125 0 below their boiling-point. 


Heat reciviirecl to Evaporate 1 lt>. Water at Temperatures 
below S12° from a, Vessel in open air* at 32°. 
(Thomas Box.) 


Tempera¬ 

ture. 

Water evapor’d 
per sq. foot of 
surface p’rhour. 

lost by radia¬ 
tion from 
surface. 

lost in air. 

a 

> 

to evaporate h 
1 lb. of wa¬ 
ter. 

Total lost 
per hour. 

< 

Ed 

£ 

s s 
w * 

H 

Water evapor’d 
per sq. foot of 
surface p’rhour. 

lost by radia¬ 
tion from 
surface. 

HE 

£ 

’3 

C0 

O 

> 

to evaporate "4 
i lb. of wa¬ 
ter. 

Total lost 
per hour. 

0 

Lbs. 

Units. 

Units. 

Units. 

Units. 

0 

Lbs. 

Units. 

Units. 

Units. 

Units. 

32 

.027 

— 

— 

IO9I 

29 

132 

. 706 

182 

202 

i5°6 

1068 

42 

.04 

270 

424 

1788 

7 i 

142 

.916 

158 

162 

1445 

1326 

52 

.058 

375 

58 x 

2052 

JI 9 

152 

1.178 

137 

127 

1392 

1637 

62 

.083 

405 

605 

2110 

i 74 

162 

I- 5°5 

118 

97 

1346 

2039 

72 

.117 

386 

566 

2055 

239 

172 

X -895 

106 

72 

1312 

2475 

82 

.162 

358 

504 

1968 

319 

182 

2-373 

92 

50 

1279 

3045 

92 

•223 

3 i 9 

434 

1862 

4*5 

192 

2.947 

8l 

32 

1253 

3 6 85 

102 

•303 

280 

366 

1758 

533 

202 

3-633 

7 i 

14 

1228 

4465 

112 

.406 

245 

304 

1664 

671 

212 

4.471 

63 

— 

1209 

5397 

122 

.528 

211 

250 

1580 

849 

— 


— 

— 

— 

— 


To Compute Surface of a Refrigerator. 

Illustration of Table. —If it is required to cool 20 barrels, of 42 gallons each, of 
beer, from 202 0 to 82° in an hour. 

Result to be attained is to dissipate 42 X 8.33 (lbs. U. S. gallons) X 20 X 202 — 82 
= 840000 units of heat per hour. 

At 202 0 , 4465 units are lost, and at 82°, 319, hence, average loss for each temper¬ 
ature between extremes = 1850 units per sq. foot per hour. 

840 000 

Then —-—— = 454 sq.feet m a still air. 

1850 

The volume of air required per hour in this case wrnuld be about 100 000 cube feet. 
























HEAT. 


513 


To Compute Area, of* G-rate and. Consumption of Eu.el 
for Evaporation. 

Illustration of Table .—If it is required to evaporate 6 Beer gallons (282 cube ins.) 
of liquid per hour, at a temperature not exceeding 152 0 . 

6 gallons = 50 lbs. At 152 0 , water evaporated as per table = 1.178 lbs. per hour. 


50 

1.178 


— 42 sq. feet. Heat required to effect this = 1392 X 50 = 69 600 units. 


Assuming 6000 units as average economic value of coals, then ^ = 11.6 lbs. 

coal, on a grate of 1 sq.foot. 

When it is practicable to evaporate at a high temperature, as at or above 212 0 , it 
is most economical. 


Thus, water requires only 1209 units per lb. if surface is exposed, but if enclosed, 
heat is reduced (1209 — 63) to 1146 units. 

Evaporative Powers of Different Tubes per Degree of Heat, per Sq. Foot of 

Surface.—In Units. 

Vertical tube, 230; Double-bottomed vessel, 330; Horizontal tube or Worm, 430. 

To Compute Volume of Water Evaporated in a given 

Time. 

Illustration. —What is volume evaporated at 212 0 , in 15 minutes per sq. foot of 
surface, in a double-bottomed vessel having an area of heating surface of 17 feet, 
and subjected to steam at a pressure of 25 lbs. ? 

Temperature of steam at 25 -f- 14-7 lbs. = 269°. 269° — 212 0 = 57 0 , and latent 

heat — 927. 4 

Then 330 x S7 X .7 X . 5 = 86 . g lbs y Mer . 

927 X DO 


When Water is at a Lower Temperature than 212 0 . 

If 120 gallons or 1000 lbs. of water were to be evaporated from 42 0 in an 
hour, from same vessel and under like pressure as preceding : 

There would be required 1000 X (212 0 — 42 0 ) 170000 units of heat. Mean tempera- 

A2p —I— 212^ 

ture of water while being heated = --- = 127 0 . 

2 

Difference between temperature of steam and watfrr = 267° — i27°=:i4o 0 . 


Then, 


170000 , 

- —..210 hour— time to raise water to 212 0 ; hence 1—.216 = 

X 14° X 17 

.784 hour left for evaporation, and quantity evaporated _ i y^X 57 * 17 — 

927 

270.4 lbs., or 32.44 gallons. 


Dessiccation. 


Dessiccation , or the drying of a substance, is best effected in a drying 
chamber, and it is imperative that to attain greatest effect the hot air 
should be admitted at highest point of exposed substance and dis¬ 
charged at its lowest. 


Wood, submitted to an average temperature of 300° in an enclosed space 
for a period of 2.5 days, will lose its moisture at a consumption of 1 lb. of 
wood for 10.5 lbs. of wood dried, and evaporating 4 lbs. of water, equal to 
2.66 lbs. of water per lb. of undried wood. 

Limit of temperature for drying of wood is 340°. 








514 


HEAT. 


Evaporation of Water per Sq. Eoot of Surface per Hour, 

{Dr. Dalton.) 


Temperature 
of Water. 

Calm. 

Evaporation 

Light 

Air. 

. 

Brisk 

Wind. 

Temperature 
of Water. 

1 

Calm. 

Evaporation 

Light 

Air. 

Brisk 

Wind. 

O 

Lbs. 

Lbs. 

Lbs. 

0 

Lbs. 

Lbs. 

Lbs. 

32 

•0349 

.0448 

•055 

IOO 

.3248 

.4169 

.5116 

40 

•0459 

.0589 

.0723 

125 

.6619 

.8494 

1.043 

50 

•0655 

.0841 

.1032 

150 

1.296 

1.663 

2.043 

60 

.0917 

•1175 

.1441 

*75 

2.378 

3-053 

3-746 

70 

•1257 

. 1616 

.1983 

200 

4.128 

5. 298 

6.502 

SO 

.1746 

.2241 

•2751 

212 

5-239 

6.724 

8.252 


The rates of evaporation for these conditions of the air when perfectly dry are as 
i, 1.28, and 1.57. 


To Compute Quantity of Water exposed to Air that would he evaporated as 
above. —Subtract tabulated weight of water corresponding to dew-point from 
weight of water corresponding to temperature of dry air, and remainder is 
weight of water that would be evaporated per sq. foot of surface per hour. 

Distillation. 

Distillation is depriving vapor of its latent heat, and, though it may 
be effected in a vacuum with very little heat, no advantage in regard to 
a saving of fuel is gained, as latent heat of vapor is increased propor¬ 
tionately to diminution of sensible heat. 

A temperature of 70° is sufficient for distillation of water in a vessel ex¬ 
hausted of air. 

Conduction or* Convection of Heat. 

Air and gases are very imperfect conductors. Heat appears to be 
transmitted through them almost entirely by conveyance, the heated 
portions of air becoming lighter, and diffusing the heat through the 
mass in their ascent. Hence, in heating a room with air, the hot air 
should be introduced at lowest part. The advantage of double win¬ 
dows for retention of heat depends, in a great measure, upon sheet of air 
confined between them, through which heat is very slowly transmitted. 

Convection of heat refers to transfer and diffusion of heat in a fluid mass, 
by means of the motion of the particles of the mass. 


Relative Internal Conducting Powers of Wei 1 *!. 0113 
' Substances. 

Metals. 


Brass .... 

... .76 

Gold.x 

Porcelain... 

.012 

Tin.. 

• 3 

Cast Iron. 

... .517 

Lead.18 

Silver. 

•97 

Wrought Iron 

•44 

Copper... 

... .89 

Platinum.98 

Terra Cotta. 

.Oil 

Zinc. 

•36 



Minerals. 




Cement... 


Coal, anth’cite 1.92 

Fire brick.. 

. .61 

Gypsum. 

.2 

Chalk .... 

.6 

“ bitumin. 1.68 

Fire clay.... 


Lime. 

.24 

Charcoal.. 

. °7 

Coke.1.98 

Glass. 

. .96 

Marble. 

1.22 


Slate... 


Wood ash.08 




Woods tenth Birch 

=. 41 with Silver. 



Apple .... 

.68 

Birch.1 

Ebony. 

• -5 

Oak. 

•73 

Ash. 

. 73 

Chestnut.7 

Elm. 

• -73 

Pine. 

•73 



Hair and Fui 

with Air 1. 




Cotton.... 

. 55 

Flannel.2.44 

Hair. 


Silk. 

• 4 - 

Eiderdown... .44 

Hemp Canvas. .28 

Hare’s fur.. 

• -43 

Wool. 

•5 



Liquids with Water. 




Alcohol.. 


. .93 I Proof spirit.. 

. 85 

Turpentine. 

3.x 

Mercury . 


.2.8 | Sulphuric acid .1.7 

Wafer. 

1 































































HEAT. 


515 


Practical Deductions from preceding Remits. 

Asphalt best.composition for resisting moisture, and, being a slow con¬ 
ductor of heat, it is best adapted for economy of heat and dryness. 

nn f la J e is a very dry material, but, from its quick conducting power, it is 
not adapted for retention of heat. 1 ’ 

Cements. — Plaster of Paris and Woods are well adapted for lining of 
rooms having low conductive powers, while Hair and Lime , being a quick 
conductor, is one of the coldest compositions. 

Fire-brick ^abs^bs much heat, and is well adapted for lining of fire-places 
etc.; while Iron , being a high conductor of heat, is one of the worst of sub¬ 
stances for this purpose. Common brick is not a very slow conductor of heat. 


Communication. 

Communication of Heat is passage of heat through different bodies 
with different degrees of velocity. This has led to the division of 
bodies into Conductors and Non-conductors of caloric; the former in¬ 
cludes such as metals, which allow caloric to pass freely through their 
substance, and the latter comprise those that do not give an easy pas¬ 
sage to it, such as stones, glass, wood, charcoal, etc. 

The velocity of cooling, other things being equal, increases with the extent 
of surface compared with volume of substance; and of two bodies of same 
material, temperature, and form, but differing in volume. 


Condensation. 


TredgoM ascertained by experiment that steam at pressure (absolute) 
of 17.5 lbs. per sq. inch, 221 0 , produced 1 cube foot of water per hour 
by condensation in 182 sq. feet of cast-iron pipe, at a uniform and qui¬ 
escent temperature of 6o°. Hence, condensation .352 lb. water per 
hour, or .0022 lbs. per degree of difference of temperature (221—60). 

From experiments of Mr. B. G. Nichol in England, 1875, it was deduced: 

That rates of transmission of heat, between temperature of steam and 
that of water of condensation at its exit, at the rate of 150 feet per minute, 
may be taken as 380 units for vertical tubes and 520 for horizontal. 


Condensation of* Steam in Cast-iron Pipes. (M. Burnat.) 


Average 
Press, per 
Sq. Inch. 

Steam. 

remperaturi 

Air. 

3 . 

Difference. 

Condens 

Bare. 

ation per sq 

Straw. 

. foot of ext 
per hour. 
Pipe. 

.erual surfai 

Waste. 

;e of pipe 

Plaster. 

Lbs. 

22 

O 

233 

O 

. 36-5 

0 

i 9 6 -5 

Lb. 

.581 

Lb. 

.2 

Lb. 

.229 

Lb. 

.286 

Lb. 

•324 


From these data, following constants are deduced for an absolute pressure of 
22 lbs. per sq. inch of steam condensed, and heat passed o(T per sq. foot of external 
surface of pipe per hour of i° difference of temperature. 


t Surface of Pipe. 

Steam 
condensed 
per Sq. Foot. 

Heat 

passed 

off. 

Surface of Pipe. 

Steam 
condensed 
per Sq. Foot. 

Heat 

passed 

off. 

Bare pipe. 

Lb. 

. on "a 

Units. 

2.812 

.968 

1.108 

Cotton waste 1 inch.. 
E/irth anrl hair 

Lb. 

.001 46 
.001 65 
.001 56 

Units. 

1.384 

-r r-CQ 

Straw coat... 

• vV "S 

.OOI 02 

Cased with clay pipe... 

.OOI 15 

White paint. 

I. 500 

1.486 



























516 


HEAT. 


Pipes were 4.72 ins. diameter, .25 inch thick, and had area of 58.5 sq. feet. 
Bare —rough surface as cast. Straw coat —laid lengthwise .6 inch thick and bound. 
ripe —laid in clay pipe with an air space between them, the whole covered with 
loam and straw. Waste cotton —1 inch thick and bound with twine. Plaster — 
laid in clay and hair 2.36 ins. thick. 

A wrought-iron pipe 3.75 ins. in external diameter, .25 inch thick, and lagged 
with felt and spun yarn .5 inch thick, condensed steam at 245 0 at rate of .262 lb. 
per sq. foot per hour, in an external temperature of 6o°. 

Steam CoiicLensecl per Sep Foot and. per Degree per HTonr. 

Mean Results of several Experiments with bare Cast-iron Pipes, with Steam 
at Absolute Pressure of 20 lbs. per Sq. Inch. 

.4 lb. per sq. foot, and .002 39 lb. per degree. 

Hence, to ascertain quautity of heat lost by condensation of .002 39 lb. = —*- of a lb. 

Difference of total and sensible heats of 1 lb. steam at 20 lbs. absolute pressure = 
ii5i°-|-32 0 — 228° = 955 units, and 955-1-420 = 2.274 units — heat condensed. 

The loss of heat from a naked boiler in air at 62°, under an absolute pressure of 50 
lbs. per sq. inch, was 5.8 units. 

Congelation and. ICiqnefaction. 

Freezing water gives out 140° of heat. All solids absorb heat when 
becoming fluid. 

Particular quantity of heat which renders a substance fluid is termed 
its caloric of fluidity, or latent heat. 


Temperature of Solidification of Several Gases. (Faraday.) 


Cyanogen.31 0 

Carbonic Acid.72 0 


Ammonia.103° 

Sulphurous Acid... 105 0 


Sulphuretted Hydrogen, 123 0 
Protoxide of Nitrogen.. 148° 


Mixtures. 


Sea salt. 

Nitrate of ammonia ... 
Snow, or pounded ice.. 
Muriate of ammonia ) 
Nitrate of potash j 

Snow, or pounded ice 
Phosphate of soda... 
Nitrate of ammonia. 
Dilute mixed acids .. 

Snow. 

Crystallized muriate | 

of lime.J 

Snow. 

Dilute sulphuric acid . 

Phosphate of soda.... 
Nitrate of ammonia .. 

Dilute nitric acid. 

Snow. 

Dilute nitric acid. 


FrigorifLc IVTixtcires. 


| Fall of 
Parts. | Temperature. 


.1 

3 ) 

II 


8! 

IO] 

5 

3 

4 

3 1 
2 


-18 to —25 


'5 to —18 


—34 t0 —50 


—40 to —73 


—68 to 


- 9 1 


0 t0 —34 
o to —46 


Mixtures. 


d 


Nitrate of ammonia 

Water. 

Snow. 

Dilute sulphuric ac 

Sulphate of soda.. 
Diluted nitric acid 
Nitrate of ammonia 
Carbonate of soda. 

Water. 

Sulphate of soda.. 
Muriate of ammonia 
Nitrate of potash.. 
Dilute nitric acid. 
Phosphate of soda. 
Dilute nitric acid. 

Snow. 

Muriate of lime... 

Potash . 

Snow. 


Parts 

Fall of 
Temperature. 


O 

1 1 
if 

-{-50 to 4-4 


— IO to —60 

1 1 


3 ! 

2 ^ 

+50 to —3 


II 

6 

4 

2 

4 j 

I! 

1! 

1} 


-(-50 to 


-(-50 to —IO 


-j-50 to —12 

—1-20 to -48 

4-32 to —51 


A Mixture of Solid Carbonic Acid and Sulphuric Ether, under receiver of an air- 
pump, under pressures of .6 lbs. to 14 lbs., exhibited a temperature ranging from 
—107 0 to —166 0 , which is the most intense cold as yet known. (Faraday.) 





































HEAT, 


517 


IViel t ing-p o in t s. 


Metals. 


Aluminium at red heat. 

Antimony. 

Arsenic. 

Bismuth. 

Bronze. 

Calcium at red heat.... 
Copper. 

Gold, pure. 

“ standard . 


Iron, cast 


“ 2d melting. 


Wrought. 


“ malleable forge. 

Lead. 

Lithium. 

Mercury. 

Platinum. 

Nickel, highest forge heat, 
Potassium. 

Silver. 

Sodium. 

Steel.. 

Tin. 

Zinc. 


Alloys. 

Lead 2, Tin 3, Bismuth 5. 

44 T 44 0 44 - 

*5 O) D * 


8 lO 

365 

476 

1692 

1996 
(2282 
(2590 
2156 
2000 


2250 

3479 * 

( 2200 
2450 

( 37 °°* 
2700 
2912 
3509* 


608 

356 

—39 

3 ° 8 o 

136 
[ 1250 
ti8 73 
194 
2500 
446 
680 


212 

210 


Alloys. 


Lead 1, Tin 4, Bismuth 5. 

“ 2 , “ 3. 

“ 3, “ 2, Bismuth 5. 

“ 3, “ 1 . 

“ 2, “ 1 (solder). 

“ 1, “ 2 (soft solder). 

1, 41 x . 

“ 1, “ 1, Bism. 4, Cadm. 1 

Tin 1, Bismuth 1. 

“ 1. 

> “ 1. 

Zinc 1, Tin 1. 


2, 

“ 8 , 


ITvLsiljle IPlmgs. 

Lead 2, Tin 2. 

“ 6, “ 2. 

“ 7, “ 2. 


Various SYilostanees. 

Ambergris. 

Beeswax. 

Carbonic acid. 

Glass. 

Ice. 

Lard. 

Nitro-Glycerine. 

Phosphorus. 

Pitch... 

Saltpetre. 

Spermaceti. 

Stearine. 

Sulphur. 

Tallow. 

Wax, white. 

* Rankine. 


240 

334 

199 

552 

475 

360 

368 

155 

286 

336 

39 2 

399 


372 

383 

388 

410 


i 45 

151 

—108 
2377 
32 
95 
45 
112 

9 1 

606 

xi2 

114 

239 

92 
142 


Volume of Water, Antimony, and Cast iron, in the solid state, exceeds 
that of the liquid, as evidenced by the floating of ice on water, and of cold 
iron on iron in a liquid state. 


Boiling-points. 

Liquids. 0 


Alcohol, S. g. 813. 

Ammonia. 

Benzine.. 

Chloroform. 

Ether. 

Linseed oil. 

Mercury. 

Milk. 

Nitric acid, s. g. 1.42. 

4 4 4 4 4 4 j ^. 

Oil of Turpentine. 

Petroleum, rectified. 

Phosphorus. 

Sea water, average. 

Sulphur. 

Sulphuric acid, s. g. 1.848. 

U U i ( J 2 

“ ether. 


W 3 

140 

J 73 

146 

.100 

597 

648 

2x3 

248 

2X0 

315 

316 

554 

213.2 

570 

59 ° 

240 

100 


(Under One Atmosphere.) 
Liquids. 


Turpentine. 

Water...... 

“ in vacuo... 

Whale oil.. 

Saturated Solutions. 

Acetate of Soda. 

“ “ Potash. 

Brine. 

Carbonate of Soda. 

“ “ Potash. 

Nitrate of Soda. 

“ “ Potash. 

Salt, common. 

Various Substances. 

Coal Tar. 

Naphtha... 

X 


3 i 5 

2X2 

98 

630 


255-8 

33 6 

226 

220.3 

275 

250 

240.6 

227.2 


325 

186 












































































































SIS 


HEAT. 


Boiling-points of Saturated Vapors under Various 
Pressures. (Regnault.) 


Temper¬ 

ature. 

Water. 

' 

Alcohol. 

Ether. 

Chloro¬ 

form. 

Temper¬ 

ature. 

Water. 

Alcohol. 

Ether. 

Chloro¬ 

form. 

0 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

O 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

32 

.089 

.246 

3-53 

— 

212 

14.7 

32.6 

95-17 

45-54 

50 

. 178 

.466 

5-54 

2.52 

230 

20.8 

45-5 

120.9 

58.42 

68 

•337 

.851 

8.6 

3.68 

240.8 

25-37 

— 

137 

Turp’tine. 

86 

.609 

1.52 

12.32 

5-34 

248 

29.88 

62.05 

— 

4-97 

104 

1.06 

2-59 

17.67 

7.04 

266 

39-27 

83-8 

— 

6.71 

122 

1.78 

4.26 

24-53 

10.14 

276.8 

46.87 

— 

— 

— 

140 

2.88 

6.77 

33-47 

14.27 

284 

52-56 

IO9. I 

— 

8.94 

158 

4 - 5 i 

10.43 

44.67 

18.88 

302 

69.27 

140.4 

— 

n -7 

176 

6.86 

15-72 

57 -oi 

26.46 

305.6 

73-07 

147-3 

— 

— 

194 

10.16 

23.02 

75-41 

35-03 

320 

89.97 

— 

— ’ 

i 3 -1 


Boiling-points of Water corresponding to Altitudes of Barometer between 

62 and 31 Ins. 


Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

26 

O 

2O4.9I 

27-5 

O 

207.55 

29 

O 

210.19 

30-5 

O 

212.88 

26.5 

205.79 

28 

208.43 

29-5 

211.07 

32 

213.76 

27 

206.67 

28.5 

209.31 

30 

212 




Boiling-point of Salt water, 213.2 0 . Water may be heated in a Digester 
to 400° without boiling. 

Fluids boil in a vacuum with less heat than under pressure of atmosphere. 
On Mont Blanc water boils at 187° ; and in a vacuum water boils at 98° to 
ioo°, according as it is more or less perfect. 

Water may be reduced to 5 0 if confined in tubes of from .003 to .005 inch in diam¬ 
eter: this is in consequence of adhesion of water to surface of tube, interfering with 
a change in its state. It may also be reduced in its temperature below 32 0 if it is 
kept perfectly quiescent. 

Effect upon "Varions Bodies by Beat. 

Wedgewood’s zero is 1077° (Fahrenheit), and each degree = 130°. 


In designation of degrees of temperature, symbol -f- is omitted when temperature 
is above o; but when below it, symbol — must be prefixed. 


Degrees. 

Acetification ends.... 88 

Acetous fermen -1 o 

tation begins.. J ’ ' * 7 

Air Furnace.3300 

Ammonia (liq.) freezes — 46 
Blood (hum.), heat of. 98 
“ freezes. 25 

Brandy freezes. —7 

Charcoal burns. 800 

Cold, greatest artific. —166 
“ “ natural —56 

Common fire. 790 

Fire brick... .4000 to 5000 
Gutta-percha softens.. 145 

Heat, cherry red.1500 

“ “ (Daniell) 1141 

“ bright red.i860 

“ red, visible by) 

day.j 1077 

“ white.2900 


11 7 


Degrees. 

Highest natural tem-) 
perature, Egypt.. j 
India-rubber and j 
Gutta-percha vul- / 293 

canize.) 

Iron, bright red in( 

the dark. J 752 

Iron, red hot in twi-1 no 

light.j 64 

Iron, wrought, welds. .2700 
Ignition of bodies .... 750 
Combustion of do... 800 
Mercury volatilizes... 680 

Milk freezes. 30 

Nitric Acid (sp. grav.) _ 

1.424) freezes_j 45 

Nitrous Oxide freezes—150 

Olive-oil freezes. 36 

Petroleum boils. 306 

Proof Spirit freezes... —7 


Degrees. 

Sea-water freezes_ 28 

Snow and Salt, equal 1 

parts.| 0 

Spirits Turpen. freezes 14 
Steel, faint yellow.... 440 
“ full “ .... 470 

“ purple. 530 

“ blue...... 550 

“ full blue. 560 

“ dark “ . 600 

“ polished, blue .. 580 

“ “ straw color 460 

Strong Wines freeze.. 20 

Sulph. Acid (sp. grav.) _ 

1.641) freezes_j 45 

Sulph. Ether freezes..—46 

Vinegar freezes. 28 

Vinous ferment. ..60 to 77 

Zinc boils.1872 

Wood, dried. 340 


Volume of Several Biq/uicls at th.eir Boiling-point. 

Steam, j ^ Steam. I Steam. I Steam. 

1 Water.1700 | 1 Alcohol.528 | 1 Ether.298 | 1 Turpentine.. 193 

































































HEAT. 


519 


Heiglat corresponding to Boiling-point of Pure Water. 
Boiling-point at Level of Sea = 212 0 . 


Degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

Degree. 

Feet. 

211 

52 i 

207 

2625 

203 

4761 

199 

6929 

195 

9129 

210 

1044 

206 

3156 

202 

5300 

198 

7476 

194 

9 684 

209 

1569 

205 

3689 

201 

5841 

197 

8025 

*93 

IO 241 

208 

2096 

204 

4224 

200 

6384 

196 

8576 

192 

10 800 


Correction for temperature of air same as given at page 428 for Elevation 
by a Barometer by multiplying by C. 

Illustration.—I f water boils at a temperature of 200 0 and C = 136°, 

Then 6384 X 1-08 = 6894.72 feet. 

Underground Temperature. 

Mean increase of underground temperature per foot, from observations in 
36 mines in various and extended localities, is .01565° = x° in 64 feet. 

Linear Expansion or Dilatation of a Bar or Prism P>y 

Heat. 


For i° in a Length of 100 Feet. 


Metals. Minerals, etc. 


Inch. 

Antimony.007 22 

Bismuth. .009 28 

Brass.012 5 

“ yellow.0126 

Brick.00144 

Cast Iron.0074 

Cement.009 56 

Copper from o° to 212 0 .on 5 

u from 32 0 to 572 0 .004 18 

Fire brick.003 3 

Glass.005 74 

“ flint.00541 

“ tube.06x2 

Gold—Paris standard annealed.. .0101 

“ “ “ unannealed .0103 

Granite.....005 25 

Gun Metal—16 copper-|-i tin... .0127 
“ “ 8 copper-j-1 tin... .0121 

Ice.0333 

Iron, forged.008 14 

“ from o° to 212 0 .007 88 


Inch. 

Iron, from 32 0 to 572°.003 26 

Iron wire.008 23 

Lead.019 

Marble.005 66 

Palladium.006 67 

Platinum.005 71 

“ from 32 0 to 572 0 .00204 

Sandstone.013 

“ .008 14 

Silver.012 7 

Speculum metal. 013 

Steel, rod.. , .007 63 

“ cast.0072 

“ tempered.00826 

“ not tempered.007 19 

Tin.014 5 

Water.000 222 9 

White Solder—tin 1 + 2 lead.. .016 7 

Zinc, forged.0207 

“ sheet.0196 

“ 8 —j— 1 tin. 017 9 


Superficial expansion is twice linear, and cubical, three times linear. 

To Compute Linear Expansion of a Substance. 

Divide 1 by decimal given in above Table, and quotient will give pro¬ 
portion. 

Illustration i. —A rod of copper 100 feet in length will expand between tem¬ 
peratures of 32° and 212 0 . 212 — 32 = 180 X .0115 = 2.07 ins. 

2 ._A cube of cast iron of 1 foot will expand in volume between temperatures of 

62° and 212 0 . 212 — 62 = 150, and 150 X .0074 —1.11, which 100 for 1 foot = 

.01x1 inch, and i2-f-.oiix X 3 = 12.0333 ins. 

Some solids, as ice, cast iron, etc., have more volume when near to their melting- 
point than when melted. This is illustrated in floating of solid metal in the liquid. 


Expansion of Water. 

Water expands from temperature of maximum density (see page 520), 
39.1 °, to 46°, at which degree it regains its initial volume of 32°, and from 
thence it expands under one atmosphere to 212 0 ; and its cubical expansion 
is .0466, that is, its volume is dilated from 1 at 32° to 1.0466 at 212 0 . 

Its expansion increases in a (greater ratio than that of temperature. 





























































520 


HEAT. 


To Compute Density of Water at a given Temperature. 

62.5 X 2 


t-\- 461 


500 


500 


t -j- 461 


= approximate density , t representing temperature of water. 


6 2.5 X 2 


Illustration. -What is density -- = 57 . 42 lbs. or iveight of 

ot pure water at 298° ? 298 + 461 ^ 500 x cube foot. 


500 


298 -J- 461 


Expansion of 'Water. (Dalton.) 


Temp. 

Expansion. 

Temp. 

Expansion. 

Temp. 

Expansion. 

Temp. 

Expansion 

O 


O 


O 


O 


22 

I.OOO9 

52 

I OOO 21 

112 

1.008 8 

172 

1-02575 

32 

I 

72 

1.001 8 

132 

1.01367 

192 

1.032 65 

*46 

I 

92 

1.004 77 

152 

1-01934 

212 

1.046 6 


* Greatest density 39.i°. 


Hence, at 72 0 , water expands 


- — = 555.55th part of its original bulk. 

.0018 


Expansion of Eicpnicls from 3S2° to 21S°. Volume at 32 ° = 1 . 


Liquids. 

Volume at 212 0 . 

Liquids. 

Volume at 212 0 

Alcohol. 

1.11 

1.08 

1-015 4 
1.018433 1 

1.018 867 9 

1.11 

Olive oil. 

1.08 

1.06 

1.07 

1.07 

1.046 6 
1.05 

Linseed oil. 

Mercury. 

“ 212° tO 392° . 

u 392 0 to 572 0 . 

Nitric acid. 

Sulphuric acid. 

“ ether. 

Turpentine. 

Water. 

Water sat. with salt. 


Expansion of Gases from 32° to SIS 0 . Volume at 32 0 = 1 . 


Gases. 

Volume 
at 212 0 . 

Gases. 

Volume 
at 212 0 . 

Air..1 Atmosphere.. 

3-45 “ 

Hydrogen_1 “ 

3-35 “ 

Carbonic acid, 1 “ 

3-32 “ 

1.367 06 
1.369 64 
1.366 13 
1. 366 16 
1.37099 
1-384 55 

Nitrous oxide ... 1 Atmosphere.. 
Sulphurous acid, 1 “ 

1.16 11 

Carbonic oxide ..1 “ 

C3umogen.1 “ 

I - 3 I 79 
i- 3903 
!■ 398 
I.3669 
!- 3 8 77 


Expansion of Gases is uniform for all temperatures. 


Volume of One Pound of Various Gases at 32 0 under one Atmosphere. 


Cube feet. 


Air.12.387 

Carbonic acid. 8.101 

Ether, vapor. 4-777 


Cube feet. 

Hydrogen.178.83 

Nitrogen. 12.753 

Olefiant.. 12.58 


Cube feet. 


Oxygen.12.205 

Mercury. 1-776 

Steam.19.913 


Expansion of Air. (Dalton.) 


Temp. 

Expan¬ 

sion. 

Temp. 

Expan¬ 

sion. 

Temp. 

Expan¬ 

sion. 

O 


O 


O 


32 

I 

40 

1.021 

60 

1.066 

33 

1.002 

45 

1.032 

65 

1.017 

34 

1.004 

50 

1.043 

70 

1.089 

35 

1.007 

55 

1-055 

75 

1.099 


Temp. 

Expan¬ 

sion. 

Temp. 

Expan¬ 

sion. 

Temp. 

Expan¬ 

sion. 

O 


O 


O 


80 

1. no 

IOO 

1-152 

392 

J -739 

85 

1.121 

200 

i -354 

482 

1.912 

9 ° 

1-132 

212 

1.376 

680 

2.028 

95 

1 142 

302 

i -558 

772 

2.312 


To Compute Volume of a Constant Weiglit of Air or 
Permanent Gas for any Pressure. 

When volume at a given pressure is known , temperature remaining con¬ 
stant. Rule.— Multiply given volume by given pressure and divide by 
new pressure. 

Example. —Pressure at 212° = 18.92 lbs. per sq. inch, and volume 16.91 cube feet- 
what is volume at pressure of 13.86 lbs. 

16.91 X 13.86-1-18.92 = 12.39 cube feet. 




























































































HEAT. 


521 


Relative Densities of some Vapors. 

Water 1. Alcohol 2.59. Ether 4.16. Spirits of Turpentine 8.06. Sulphur 3.59. 


Volume, Pressure, and. Density of AAir at Various Tem¬ 
peratures. 

Volume and Atmospheric Pressure at 62° = 1. 


Temper¬ 

ature. 

Volume of 

1 lb. of air at 
atmospheric 
pressure of 
14:7 lbs. 

Pressure 
of a given 
weight of 
air. 

Density, or 
weight of one 
cube foot 
of air at 

14.7 lbs. 

Temper¬ 

ature. 

Volume of 

1 lb. of air at 
atmospheric 
pressure of 
14.7 lbs. 

Pressure 
of a given 
weight of 
air. 

Density, or 
weight of one 
cube foot 
of air at 

14.7 lbs. 

O 

Cube feet. 

Lbs. per 
Sq. Inch. 

Lbs. 

O 

Cube feet. 

Lbs. per 
Sq. Inch. 

Lbs. 

0 

11-58,3 

12.96. 

.086 331 

360 

20.63 

23.08 

.048 476 

32 

12.387 

13-86 

.080728 

380 

21.131 

23.64 

• 0 47 323 

40 

12.586 

14.08 

•079439 

400 

21.634 

24.2 

.046 223 

50 

12.84 

14.36 

.077 884 

425 

22.262 

24.9 

.04492 

62 

13.141 

14.7 

.076097 

450 

to 

N 

00 

VO 

25.61 

.043686 

70 

I 3-342 

14.92 

•074 95 

475 

23.518 

26.31 

.042 52 

80 

I 3 - 593 

15.21 

•073565 

500 

24.146 

27.01 

.041 414 

90 

I 3-845 

15-49 

.072 23 

525 

24-775 

27.71 

. 040 364 

100 

14.096 

15-77 

.070942 

55 o 

25.403 

28.42 

•039 365 

120 

14.592 

16.33 

.068 5 

575 

26.-031 

29.12 

•038415 

140 

I 5 - 1 

16.89 

.066 221 

600 

26.659 

29.82 

•037 51 

160 

15.603 

i 7-5 

.064 088 

650 

2 7 - 9 i 5 

' 3 i - 2 3 

.035822 

180 

16.106 

18.02 

.06209 

700 

29.171 

32-635 

.034 28 

200 

16.606 

18.58 

.06021 

75 o 

30.428 

34-°4 

.032 865 

210 

16.86 

18.86 

•059 313 

800 

31.684 

35-445 

.031 561 

212 

16.91 

18.92 

•059135 

850. 

32.941 

36.85 

•030358 

220 

17. in 

19.14 

.058 442 

900 

34-197 

38-255 

.029 242 

240 

17.612 

19.7 

•056774 

950 

35-454 

39-66 

.028 296 

260 

18.116 

2°. 27 

•0552 

1000 

36.811 

41.065 

.027 241 

280 

18.621 

20.83 

•05371 

1500 

49-375 

55-115 

.020 295 

3 °° 

19.121 

2I.39 

.052297 

2000 

61.94 

69.165 

.016 172 

320 

19.624 

21.95 

•050959 

2500 

74-565 

83.215 

.013441 

34 ° 

20.126 

22.51 

.049686 

, 3000 

87-13 

97.265 

.011499 


To Compute Volume of a Constant Weight of .A.ir or 
other Permanent Gras for any otlier Pressure and 
Temperature. 

When volume is knoivn at a given pressure and temperature. Rule.— Mul¬ 
tiply given volume by given pressure, and by new absolute temperature, 
and divide by new pressure, and by given absolute temperature. 

Example.—G iven volume 16.91 cube feet, pressure 13.86 lbs., and temperature 
32 0 ; what is volume at this temperature? 

Temperature for volume 16.91 ==212°. 

16.91 X 13.86 X 324-461-2-13.86 X 2124-461 = 12.39 cube feet. 

To Compute Pressure of a Constant Weight of .A_ir or 
other Permanent Gras for any other Volume and 
Temperature. 

When pressure is known for a given volume and temperature. Rule.— 
Multiply given pressure by new absolute temperature, and divide by given 
absolute temperature. * 

Note.— Absolute temperature is found by adding 461° to temperature. 

Example.— Given pressure 13.86 lbs., and temperature at this volume 32°; what 
is pressure at temperature of 212 0 ? 

13.86 X 2124-461-2-324-461 = 18.92 lbs. 

X x* 


















522 


HEAT. 


'.To Compute Volume of* a Constant Weight of Air or 
other Permanent Gras at any Temperature. 

When volume at a given temperature is known , pressure being constant. 
Rule. —Multiply given volume by new absolute temperature, and divide 
by given absolute temperature. 

Absolute zero-point by different thermometrical scales is: Fahrenheit —461.2 0 ; 
Reaumur —219.2 0 ; Centigrade —274 0 . 

Example.— Volume of 1 lb. air at 32° = 12.387 cube feet; what is its volume at 
212 ° ? 

12.387 X 212 -)- 461 -r- 32 -j- 461 = 16.91 cube feet. 


To Compute Increased. Volume of a Constant "W'eigh.t 

of Air. 

When initial volume at 62° = 1 under 1 atmosphere. Rule.— To given 
temperature add 461, and divide sum by 523 (32 -j- 461). 

Example. —Assume elements of preceding case. 

212 0 -(- 461 -f- 523 = 1.287 comparative volume to 1. 


To Compute Pressure of a Constant Weight of Air or 
other Gras at 62 °, and at 14r.?' IDs. Pressure per Sq. In., 
with Constant Volume, for a given Temperature. 

Rule. —Add 461 to given temperature, and divide sum by 35.58. 

Example.— Temperature is 212 0 ; what is pressure? 

212 -j- 461 -j- 35.58 = 18.92 lbs. 


To Compute Volume, Pressure, Temperature, and 
Density of Air. 


t -f- 461 
p 2.71 
V 

2.71 


:V; 


t -j- 461 


— V; 


t + 461 


—P\ V2.JOJ4P — \6i-t) and 


D. 


39-8 ’ V2.71 

f 4f)i - t representing temperature, p pressure in lbs. per sq. inch, V vol¬ 

ume in cube feet, and D weight of 1 cube foot at 14.7 lbs. per sq. inch. 

Product of volume and pressure of a constant weiglit of air, or any other 
permanent gas, is equal to product of absolute temperature and a coefficient, 
determined for each gas by its density. 

Or, \ p zzz C t -j- 461. 

Coefficients, as determined by volumes and consequent densities.* 

Hydrogen.1875 

Nitrogen.2.63 

Olefiant.2.67 


Air.2.71 

Carbonic acid.4.14 

Ether, vapor.7.02 


Oxygen. 2.99 

Mercury.18.88 

Steam.,. 1.68 


* See D. K. Clark, London, 1877, page 349. 


Decrease of Temperature by Altitudes. 


From 1 to a 000 feet 
1 “ 10000 “ . 
1 “ 20 000 ‘ ‘ , 


In clear sky. With cloudy sky. 

i° in 139 feet.i° in 222 feet. 

i° >4 288 i® u 331 

i° “ 365 “ .i° “ 468 “ 


To Compute Temperature to which a Substance of a 
given Length or Dimension must.be Submitted or 
Reduced, to give it a Greater or Less Length or Vol¬ 
ume by Expansion or Contraction. 


Lineal.— When Length is to be increased. -f f — T. L and l represent- 

ing lengths of increased and primitive substance in like denominations, T and t tem¬ 
peratures of L and l, and C expansion of substance for each degree of heat. 
























HEAT. 


523 


Illustration.— A copper rod at 32 0 is 100 feet in length; to what temperature 
must it be subjected to increase its length 1.1633 ins. ? 

Expansion for a length of 100 feet of copper for x° = .0115. 


100 X 12 +1.1633 — 100X12 . 1-1633 , 

-h 32 = - + 32 = 133.16 0 . 

.0115 J .0115 1 0 03 


L — l 


— T = «. 


When Length is to he reduced. 

Illustration.—T ake elements of preceding case. 
1201.1633 —1200 


.0115 


133. i 69 = xoi. 16 —133.16 = 32 0 . 


To Reduce Degrees of RaTirenlieit to Reaumur and Cen¬ 
tigrade, and Contrariwise. 

TTalirenlieit to Reaumur. If above zero. — Multiply difference 
between number of degrees and 32 by 4, and divide product by 9. 

Thus, 212 0 — 32 0 = 180 0 , and i8o° X 4-r- 9 = 8o°. 

If below zero. —Add 32 to number of degrees; multiply sum by 4, and 
divide product by 9. 

Thus, —40° -J- 32 0 —72 0 , and 72 0 X 4 -r- 9 = —32 0 . 

Reaumur to RaTirenlxeit. If above freezing-point. — Multiply 
number of degrees by 9, divide by 4, and add 32 to quotient. 

Thus, 8o° X 9-r-4 = i 2 o°, and 180 0 + 32 = 212°. 

If below freezing-point. —Multiply number of degrees by 9, divide by 4, 
and subtract 32 from product. 

Thus, —32 0 X 9 -r- 4 = 7 2 0 , and 72 0 — 32 = —40 0 . 

RaHrenlieit to Centigrade. If above zero. —Multiply difference 
between number of degrees and 32 by 5, and divide product by 9. 

Thus, 212 0 — 32 0 X 5 -7- 9 = 180 0 x 5 -4- 9 = ioo°. 

If below zero. —Add 32 to number of degrees, multiply sum by k, and 
divide product by 9. 

Thus, —40 0 + 32 0 X 5-4-9 = 72° X 5‘^"9 — — 4 °°- 

Centigrade to Ralirenheit. If above freezing-point. —Multiply 
number of degrees by 9, divide product by 5, and add 32 to quotient. 

Thus, ioo° x 9 —5 = 180 0 , and 180 0 -f- 32 = 212 0 . 

If below freezing-point. —Multiply number of degrees by 9, divide product 
by 5, and take difference between 32 and quotient. 

Thus, —io° X 9 - 4 - 5 = x8°, and i8° 32 = 14 0 . 

Reaumur to Centigrade. —Divide by 4, and add product. 

Thus, 8o° -T- 4 = 20 0 , and 20 0 -j- 8o° =: ioo°. 


Centigrade to Reanmnr.— Divide by 5, and subtract product. 
Thus, ioo° - 4 - 5 — 20 0 , and ioo° —>20 = 8o°. 

Corresponding Degrees upon the Three Scales. 


Fahr, 

Cent. 

Reaum. 

Fahr. 

Cent. 

Reaum. 

Fahr. 

Cent. 

I Reaum. 

212 

IOO 

80 

32 

O 

O 

—40 

—40 

1 — 32 . 


To Compute Expansion of TTnids in. Volume. 

Rule. —Proceed by preceding formulas for computing length of a sub¬ 
stance. Substitute Y and v for volume, instead of L and /, the lengths. 



















524 


HEAT, VENTILATION, BUILDINGS, ETC. 


Illustration.—A closed vessel contains 6 cube feet of water at a temperature of 
40°; to what height will a column of it rise in a pipe 1.152 ins. in area, when it is 
exposed to a temperature of 130° ? 

1.152 ins. = .008 sq.foot. C for water = .0002229. 


6 (1 -)- .000 222 9 (130 — 40)) = 6.125 95, and ——-— 15.744 lineal feet. 

. OOO 


Temperature by -Agitation.. 


Results of Experiments with Water enclosed in a Vessel and violently Agitated. 


Temperature of Air, 60.5°; of Water, 59.5 0 . 


Duration 
of Agitation. 

Increase 
of Temperature. 

Duration 
of Agitation. 

Increase 
of Temperature. 

Duration 
of Agitation. 

Increase 
of Temperature. 

Hour. 

0 

Hours. 

0 

Hours. 

O 

•5 

10 

2 

19-5 

5 

39-5 

I 

if 5 

3 

29-5 

6 

42-5 


VENTILATION. 

Biaildings, ^Apartixieiats, etc. 

Tn Ventilation of Apartments .—From 3.5 to 5 cube feet of air are required 
per minute in winter, and 5 to 10 feet in summer for each occupant. In 
Hospitals , this rate must be materially increased. 

Ventilation is attained by both natural draught and artificial means. In 
first case the ascensional force is measured by difference in weight of two 
columns of air of same height, the height being determined by total difference 
of level between entrance for warm air and its escape into the atmosphere. 
The difference of weight is ascertained from difference of temperatures of 
ascending warm air and the external atmosphere, as by Table, page 521, or 
by formula, page 522. 


Volumes of -Air Discharged through. a 'Ventilator One 
Foot Square of Opening, at Various Heights and. 
Temperatures. 


Height of 

Excess of Temperature of Apartment 

Height of 

Excess of Temperature of Apartment 

Ventilator 

above that of External Air. 

Ventilator 

above that of External Air. 

from 







from 







Base-line. 

5 ° 

IO° 

15 0 

20° 

25 ° 

3 °° 

Base-line. 

5 ° 

10 ° 

15 ° 

10 , 

O' 

O 

25 0 

3 °° 

Feet. 

C.ft. 

C. ft. 

C. ft. 

C.ft. 

C.ft. 

C.ft. 

Feet. 

C. ft. 

C.ft. 

C.ft. 

C. ft. 

C. ft. 

C. ft. 

IO 

Il6 

164 

200 

235 

260 

284 

35 

218 

306 

376 

436 

486 

53 i 

15 

142 

202 

245 

284 

3 l8 

34 S 

40 

235 

329 

403 

465 

518 

57 ° 

20 

1-64 

232 

285 

330 

368 

404 

45 

248 

34 8 

427 

493 

551 

605 

25 

184 

260 

3 l8 

368 

410 

450 

50 

260 

367 

450 

518 

579 

635 

3 ° 

201 

284 

347 

403 

450 

493 

55 

to 

VI 

0 

3 8 5 

472 

541 

605 

663 


Velocity of draft having been ascertained for any particular case, together with 
volume of air to be supplied per minute, sectional area of both air passages may be 
computed from these data. 


Heating by Hot "Water. 

One sq. foot of plate or pipe surface at 200° will heat from 40 to 100 cube 
feet of enclosed space to 70° where extreme depression of temperature is 

— io°. 

The range from 40 to 100 is to meet conditions of exposed or corner 
buildings, of buildings less exposed, as intermediate ones of a cluster or 
block, and of rooms intermediate between the front and rear. 

When the air is in constant course of change, as required for ventilation 
or occupation of space, these proportions are to be very materially increased 
as per following rules. 
































HEAT, VENTILATION, AND HEATING. 


525 


I11 determining length of pipe for any given space it is proper to include 
in the computation the character and occupancy of the space, Thus, a 
church, during hours of service, or a dwelling-room, will require less service 
of plate or length of pipe than a hallway or a public building. 

Reduction of Heat by Surfaces of Glass dr Metal. —In addition to the 
volume of air to be heated per minute for each occupant, 1.25 cube feet for 
each sq. foot of glass or metal the space is enclosed with must be added. 
The communicating power of the glass and metal being directly proportion¬ 
ate to difference of external and internal temperature of the air. Thus, 80 
feet of glass will reduce 100 feet of air per minute. 

When Pipes are laid in Trenches in the Earth. —The loss of heat is es¬ 
timated by Mr. Hood at from 5 to 7 per cent. 

Circulation of Water in Pipes. —In consequence of the complex forms of 
heating-pipes and the roughness of their internal surface, it is impracticable 
to apply a rule to determine the velocity of circulation, as consequent upon 
difference of weights of ascending and descending columns of the water. 

For a difference of temperature in the two columns of 30° (190° — 160 0 ) 
and a height of 20 feet, the velocity due to the height would be 3.74 feet. 
In practice not .3, and in some cases but .1, would be attained. 

In Churches and Large Public Rooms, with ordinary area of doors and windows 
and moderate ventilation, a large amount of heat is generated by the respiration 
of the persons assembled therein. 

In these cases it is not necessary to heat the air above 55°, and a rule that will 
meet the ordinary ranges of temperature from io 1 ? is to divide volume in cube 
feet by 200, and quotient will give area of plate in sq. feet or length of 4-inch pipe 
in lineal feet. 


Volume of Air required per Hour for each Occupant in an Enclosed Space. 

(General Morin.) 


Cube Feet. 

Hospitals.... 2100 to 3700 
Workshops .. 2100 “ 3500 


Cube Feet. 

Lecture-rooms 1000 to 2100 
Theatres:.1400 “ 1800 


Cube Feet. 

Prisons. 1800 

Schools.424 to 1060 


To Compute Length, of Iron IPipe required, to Heat Air 
in an Enclosed Space. 

By Hot Water. 


Rule. —Multiply volume of air to be heated per minute in cube feet by 
difference of temperatures in space and external air, divide product by differ¬ 
ence of temperatures of surface of pipe and space, multiply result by follow¬ 
ing coefficients, and product will give length of pipe in feet. 

For diameter of 4 ins. multiply by .5 to .55, for 3 ins. by .7 to .75. and for 
2 ins. by 1 to 1.1. 

A pipe 4 ins. in diameter, .375 inch thick, and 1 foot in length has an 
area of internal surface of 1.05 sq. feet. 


Example.— Volume of a room of a protected dwelling is 4000 cube feet; what 
length of 4 ins. pipe, at 200 0 , is necessary to maintain a temperature of 70 0 , when 
external air is at o° ? 

4qo,X _ ^H ° = ( 

200-70 

In computing length of pipe or surface of plate it is to be borne in mind 
that the coefficients here given and computation in following table are based 
upon a ventilation or change of air ordinarily of 3.5 to 5 cube feet per 
person, and from 5 to 10 cube feet in summer per minute. Hence, when 
the ventilation is restricted the coefficient may be correspondingly in¬ 
creased. 









526 


HEAT AND HEATING. 


Lengths of Fovir-Incli Eipe to Heat lOOO Cube Feet 
of -Air per NCinnte. (Chas. Hood.) 


Temperature 


Temperature of Pipe 200°. 

Temperature of Building. 


01 

External Air. 

45 ° 

O ' 

O 

IO 

55 ° 

6o° 

65° 

O 

O 

t'N 

75 ° 

8o° 

85° 

90 0 

O 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

10 

126 

150 

174 

200 

229 

259 

292 

328 

3 6 7 

4°9 

16 

105 

127 

151 

176 

204 

223 

265 

300 

337 

378 

20 

9 1 

112 

135 

160 

187 

216 

247 

281 

318 

358 

26 

69 

90 

112 

136 

162 

190 

220 

253 

288 

327 

30 

54 

75 

97 

120 

145 

173 

202 

234 

269 

307 

36 

32 

52 

73 

96 

120 

H 7 

175 

206 

239 

276 

40 

18 

37 

5 « 

80 

104 

129 

157 

187 

220 

255 

50 

— 

— 

19 

40 

62 

86 

112 

140 

171 

204 


Proper Temperatures of Enclosed. Spaces. 


Spaces. 

Temper¬ 

ature 

required. 

Work-rooms, manufactories, etc. 

O 

55 

Churches and like spaces. 

55 

Greenhouses. 

55 

Schools, lecture-rooms. 

58 

Halls, shops, waiting-rooms, etc. 

60 

Dwelling-rooms. 

65 


Spaces. 


Dwelling-rooms. 

Graperies. 

Hot-houses. 

Drying-rooms, when filled. 

“ “ for curing paper. 


Temper¬ 

ature 

required. 

0 

70 

70 

80 

80 

70 

120 


33 oiler. 


Boiler for steam-heating should be capable of evaporating as much water 
as the pipes or surfaces will condense in equal times. Mr. Hood recom¬ 
mends that 6 sq. feet of direct heating-surface of boiler should be provided 
to evaporate a cube foot per hour. Adopt mean weight of steam of 5 lbs. 
above pressure of atmosphere, or 20 lbs. absolute pressure, condensed per sq. 
foot of pipe per degree of difference of temperature per hour, viz., .002 35 lb. 
(as given by D. K. Clark), the quantity of pipe or plate surface that would 
form a cube foot of condensed water per hour, weight of like volume of 
water 62.4 lbs., would be, per i° difference of temperature, 

62.4 -=-.002 35 = 26550 sq. feet, and for differences of 168 0 , 158°, 148°, and 108 0 , 
required surface would be respectively (26550=168 = 158) 158, 168, 179, and 246 
sq. feet. 

Henoe, assuming, as previously stated, that 4 sq. feet of direct and effec¬ 
tive heating boiler-surface, or its equivalent flue or tube surface, will evap¬ 
orate 1 cube foot of water per hour, 158 sq. feet of steam-pipe or plate will 
require 4 sq. feet of direct surface, etc., for a temperature of 6o°, and cor¬ 
respondingly for other temperatures. 

Boiler-power. —One sq. foot of boiler-surface exposed to direct action of 
fire, or 3 sq. feet of flue-surface, will suffice, with good coal, for heating 50 
sq. feet of 4-inch, 66 of 3-inch, and 100 of 2-inch pipe. Mr. Hood assigns the 
proportion at 40 feet of 4-inch pipe for all purposes. Usual rate of com¬ 
bustion of coal is 10 or n lbs. per sq. foot of grate-surface, and at this rate, 
20 sq. ins. of grate suffice for heating 40 feet of 4-inch pipe. 

Four sq. feet of direct heating boiler-surface, or equivalent flue or tube 
surface, exposed to direct action of a good fire, are capable of evaporating 
1 cube foot of water per hour. 

According to M. Grouvelle, 1 sq. meter of pipe-surface (10.76 sq. feet), heated to 
6o° an ordinary room alike to a library or office, of from 90 to 100 cube meters 
(3178 to 3531 cube feet). 































HEAT, WARMING BUILDINGS, ETC. 527 


If a workshop to be heated to a high temperature, 1 sq. meter (10.76 sq. feet) of 
surface is assigned to 70 cube meters (2472 cube feet) = 4.35 sq. feet or 5.11 lineal 
feet of 4-inch pipe per 1000 cube feet. 

For heating workshops, having a transverse section of 260 sq. feet, with a window- 
surface of one sixth total surface, it is customary in France to assign 1.33 sq. feet 
of iron pipe surface per lineal foot of shop = 5.2 sq. feet per 1000 cube feet. 

Illustrations of extensive Heating by Steam. (R- Briggs , M. I. C. E.) 


1. Total number of rooms, including halls and vaults. 286 

“ Area of floor surface. 137 370 sq. feet. 

“ Volume of rooms.1 923 500 cube feet. 

“ Number of occupants. 650 

Maximum average of occupants at any time. 1300 

Volume per occupant, excluding vaults. 1443 cube feet. 


Boilers .—8 with 173 sq. feet of grate surface and 8000 sq. feet of heating surface. 
Furnishing steam in addition to the above, to operate the elevators and electric 
dynamos, elevating water, and supplying steam to heat a distant building, requiring 
one third of their capacity. 

By Steam. 

To Compute JLiengtli of Iron IPipe required to Heat A.ir 
in. an Enclosed. Space, -witli Steam at 5 IDs. per Sq. 
Inch above Pressure of Atmosphere. 

Rule.—M ultiply volume of air in cube feet to be heated per minute, by 
difference of temperature in space and external air, divide product by coeffi¬ 
cients in following table, and quotient will give length of 4-inch pipe in 
lineal feet, or area of plate-surface in sq. feet. 

Temperature of steam at 5 lbs.—)— pressure = 228°. Hence, if temperature of space 
required is 6 o°, 70 0 , 8 o°, or 120 0 , the differences will be 168 0 , 158°, 148°, and 108 0 , 
which for a coefficient of .5, as given in rule for hot water, would be 336, 316, 296, 
and 216, for a pipe 4 ins. in diameter, and for 



6o° 

7 °° 

• 8o° 

120° 

7-inch pipe_ 


237 

222 

162 

2 “ “ .... 


158 

148 

108 

T U <( 

.... 84 

79 

74 

54 


Illustration. —Volume of combined spaces of a factory is 50000 cube feet; what 
surface of wrought-iron plate at 200° is necessary to maintain a temperature of 50° 
when external air is at o° ? _ 

50 000 X 50 — 0 x 4 — 666 6 square feet. 

200 — 50 

Coal Consumed per Hour to Heat IOO Feet of l?ipe. 

(Chas. Hood.) 


Difference of Temperature of Pipe and Air in Space, in Degrees. 


jjiam. 01 
Pipe. 

150 

i45 

140 

i 35 

130 

125 

120 

ii 5 

no 

105 

IOO 

95 

90 

85 

80 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

I. I 

I. I 

I. I 

I 

I 

•9 

•9 

•9 

.8 

.8 

•7 

•7 

•7 

.6 

.6 

2 

2-3 

2.2 

2.2 

2.1 

2 

1.9 

1,8 

1.8 

i -7 

1.6 

i-5 

1.4 

1.4 

i -3 

1.2 

3 

3-5 

3-4 

3-3 

3 -i 

3 

2.9 

2.8 

2.7 

2.5 

2.4 

2-3 

2.2 

2. I 

2 

1.8 

4 

4-7 

4-5 

4.4 

4.2 

4.1 

3-9 

3-7 

3-6 

3-4 

3-2 

3- 1 

2.9 

2.8 

2.6 

2-5 


To warm a factory, according to M. Claudel, 43 feet in width by 10.5 high, a single 
line of hot-water pipe 6.25 ins. in diameter per foot of length of room, appears to be 
sufficient, temperature in pipe being from 170 0 to 180 0 . Also, water being at 180 , 
and air at 60°, making a difference of 120°, it is convenient to estimate from 1.5 
to 1 75 sq feet of water-heated surface as equivalent to one sq. foot of steam-heated 
surface, and to allow from 8 to 9 sq. feet of hot-water pipe-surface per 1000 cube 
feet of room. 

M. Grouvelle states that 4 sq. feet of cast-iron pipe-surface, whether heated by 
steam or by water at 176^ to 194 0 , will warm 1000 cube feet of workshop, main¬ 
taining a temperature of 6o°. Steam is condensed at rate of .328 lb. per sq. foot 
per hour. 








































528 


HEAT, WARMING BUILDINGS, ETC. 


2. (R. L. Greene .) Length of fronts of buildings.. lineal feet. 

Total volume of rooms.2 574084 cube feet. 

Radiating surfaces, direct, 10804. 1 ioo gq f eet 

indirect, 23 296. j ^4 

Boilers. —Grate-surface. 180 “ 

Heating surface.5863 “ 


Volume of‘ ,A.ir Heated, by Radiators ; Consumption of 
Coal; Areas of Grate and Heating-surface of Roiler. 

(Rob't Briggs.) 

Per xoo Sq. Feet of Warming-surface of Radiator. 

Pressure of steam per sq.. inch +) 

atmosphere in lbs . j 

Heat from radiators per minute) 

in units .j 

Volume of air heated i° per min-) 

ute in cube feet .j 

Efficiency of radiators in ratio .... 

Coal consumed per hour in lbs .... 

Area of grate consuming 8 lbs. 

coal per hour in sq. feet . 

do. 12 lbs. 

Heating surface of boiler ; coal) 
consumed per hour X2.8 in sq.feet j 

8 lbs. X 2.8. 

12 lbs. X 2. 8 . 


— 

3 

IO 

3 ° 

60 

456 

486 

537 

642 

74 i 

25 IIO 

26 772 

29 570 

35 352 

40803 

I 

1.066 

1.178 

1.408 

1.625 

3 C 4 

3-24 

3-58 

4.28 

4.94 

•38 

•405 

.448 

— 

— 

— 

— 

. 298 

•357 

.412 

8.512 

9.072 

10.02 

00 

ON 

13-83 

22.4 

22.4 

22.4 

— 

— 

— 

— 

33-6 

33-6 

33-6 


I iy Hot-Air Furnaces or Stoves. 


A square foot of heating surface in a hot-air furnace or stove is held to 
be equivalent to 7 sq. feet of hot water pipe. 

M. Peclet deduced that when the flue-pipe of a stove radiated its heat 
directly to air of a space, the heat radiated per sq. foot per hour, for i° 
difference of temperature, were, for: Cast iron, 3.65 units; Wrought iron, 1.45 
units, and Terra cotta .4 inch thick, 1.42 units. 

In ordinary practice, 1 sq. foot of cast iron is assigned to 328 cube feet 
of space. 

Open lEbLres. 


According to M. Claudel, the quantity of heat radiated into an apart¬ 
ment from an ordinary fireplace is .25 of total heat radiated by combustible. 

For wood the heat utilized is but from 6 to 7 per cent., and for coal 13 per 
cent. 


In combustion of wood, chimney of an ordinary open fireplace draws 
from 1000 to 1600 cube feet of air per pound of fuel, and a sectional area 
of from 50 to 60 sq, ins. is sufficient for an ordinary apartment. 

Proportions of fuel required to heat an apartment are: For ordinary fire¬ 
places, 100; metal stoves, 63; and open fires, 13 to 16. 

Furnaces. 

By D. K. Clark, from investigations of Mr. J. Lothian Bell. 

Cupola. —M. Peclet estimates that in melting pig-iron by combustion 
of 30 per cent, of its weight of coke, 14 per cent, only of the heat of combus¬ 
tion is utilized. 

Metallurgical.— According to Dr. Siemens, 1 ton of coal is consumed 
in heating 1.66 tons of wrought iron to welding-point of 2700°, and a ton 
of coal is capable of heating up 39 tons of iron; from which it appears that 
only 4.5 per cent, of whole heat is appropriated by the iron. Similarly, he 
estimates 1.5 per cent, of whole heat generated is utilized in melting pot 





















HEAT AND HEATING.-HYDRAULICS. 


529 


steel in ordinary furnaces, whilst, in his regenerative furnace, 1 ton of steel 
is melted by combustion of 1344 lbs. of small coal, showing that 6 per cent, 
of the heat is utilized. 

Blast-furnace. —Mr. Bell has formed detailed estimates of appro¬ 
priation of the heat of Durham coke in a blast-furnace; from which is de¬ 
duced following abstract: 

Durham coke consists of 92.5 per cent, of carbon, 2.5 of water, and 5 of 
ash and sulphur. To produce 1 ton of pig-iron, there are required 1232 lbs. 
of limestone, and 5388 lbs. of calcined iron-stone; the iron-stone consists of 
2083 lbs. of iron, 1008 lbs. of oxygen, and 2509 lbs. of earths. There is 
formed 813 lbs. of slag, of which 123 lbs. is formed with ash of the coke, 
and 690 lbs. with the limestone. There are 2397 lbs. of earths from the iron¬ 
stone, less 93 lbs. of bases taken up by the pig-iron and dissipated in fume, 
say 2314 lbs. Total of slag and earths, 3127 lbs. 

Mr. Bell assumes that 30.4 per cent, of the carbon of the fuel, which es¬ 
capes in a gaseous form, is carbonic acid; and that, therefore, only 51.27 
per cent, of heating power of fuel is developed, and remaining 48.73 per 
cent, leaves tunnel-head undeveloped. He adopts, as a unit of heat, the 
heat required to raise the temperature of 112 lbs. of water 33.8°. 


HYDRAULICS. 

Descending Fluids are actuated by same laws as Falling Bodies. 

A Fluid will fall through 1 foot in .25 of a second, 4 feet in .5 of a 
second, and through 9 feet in .75 of a second, and so on. 

Velocity of a fluid, flowing through an aperture in side of a vessel, 
reservoir, or bulkhead, is same that a heavy body would acquire by fall¬ 
ing freely from a height equal to that between surface of fluid and 
middle of aperture. 

Velocity of a fluid flowing out of an aperture is as square root of 
height of head of fluid. Theoretical velocity, therefore, in feet per sec¬ 
ond, is as square root of product of space fallen through in feet and 
64*333 = 3/2 gli\ consequently, for one foot it is V 64.333 ==: 8.02 feet. 
Mean velocity, however, of a number of experiments gives 5.4 feet, 
or .673. 

In short ajutages accurately rounded, and of form of contracted vein, 
(vena contracted ), coefficient of discharge = .974 of theoretical. 

Fluids subside to a natural level, or curve similar to Earth’s convexity; apparent 
level, or level taken by any instrument for that purpose, is only a tangent to Earth’s 
circumference; hence, in leveling for canals, etc., difference caused by Earth’s cur¬ 
vature must be deducted from apparent level, to obtain true level. 

IDedfictions from Experiments 011 Discharge of Flnids 
from Reservoirs. 

1. That volumes of a fluid discharged in equal times by same apertures 
from same head are nearly as areas of apertures. 

2. That volumes of a fluid discharged in equal times by same aperturSfe, 
under different heads, are nearly as square roots of corresponding heights 
of fluid above surface of apertures. 

3. That, on account of friction, small-lipped or thin orifices discharge pro¬ 
portionally more fluid than those which are larger and of similar figure, 
under same height of fluid. 

Y Y 




530 


HYDRAULICS. 


4. That in consequence of a slight augmentation which contraction of the 
fluid vein undergoes, in proportion as the height of a fluid increases, the flow 
is a little diminished. 

5. That if a cylindrical horizontal tube is of greater length than its di¬ 
ameter, discharge of a fluid is much increased, and may be increased with 
advantage, up to a length of tube of four times diameter of aperture. 

6. That discharge of a fluid by a vertical pipe is augmented, on the prin¬ 
ciple of gravitation of falling bodies; consequently, greater the length of a 
pipe, greater the discharge of the fluid. 

7. That discharge of a fluid is inversely as square root of its density. 

8. That velocity of a fluid line passing from a reservoir at any point is 
equal to ordinate of a parabola, of which twice the action of gravity (2 (]) 
is parameter, the distance of this point below surface of reservoir being the 
abscissa.* Or, velocity of a jet being ascertained, its curve is a parabola, 
parameter Of which = 4 h, due to velocity of projection.! 

9. Volume of water discharged through an aperture from a prismatic 
vessel which empties itself, is only half of what it would have been during 
the time of emptying, if flow had taken place constantly under same head 
and corresponding velocity as at commencement of discharge; consequently, 
the time in which such a vessel empties itself is double the time in which 
all its fluid would have run out if the head had remained uniform. 

10. Mean velocity of a fluid flowing from a rectangular slit in side of a 
reservoir is tw.o thirds of that due to velocity at sill or lowest point, or it is 
that due to a point four ninths of whole height from surface of reservoir. 

11. When a fluid issues through a short tube, the vein is less contracted 
than in preceding case, in proportion of 16 to 13; and if it issues through 
an aperture which is alike to frustum of a cone, base of which is the aper¬ 
ture, the height of frustum half diameter of aperture, and area of small end 
to area of large end as 10 to 16, there will be no contraction of the vein. 
Hence this form of aperture will give greatest attainable discharge of a fluid. 

12. Velocity of efflux increases as square root of pressure on surface of a 
fluid. 

13. In efflux under water, difference of levels between the surfaces must 
be taken as head of the flowing water. 

14. To attain greatest mechanical effect, or vis viva , of water flowing 
through an opening, it should flow through a circular aperture in a thin 
plate, as it has less frictional surface. 

From CoiicLnits or IPipes. ( Bossui .) 

1. Less diameter of pipe, the less is proportional discharge of fluid. 

2. Discharges made in equal times by horizontal pipes of different lengths, 
but of same diameter, and under same altitude of fluid, are to one another 
in inverse ratio of sq. roots of their lengths. 

3. In order to have a perceptible and continuous discharge of fluid, the 
altitude of it in a reservoir, above plane of conduit pipe, must not be less 
than .082 ins. for every 100 feet of length of pipe. 

4. In construction of hydraulic machines, it is not enough that elbows and 
contractions be avoided, but also any intermediate enlargements, the in¬ 
jurious effects of which are proportionate, as in following Table, for like 
volumes of fluid, under like heads in pipes, having a different number of 
enlarged parts. 


No. 

of Parts. 

Velocity. 

No. 

of Parts. 

Velocity. 

No. 

of Parts. 

Velocity. 

No. 

of Parts. 

Velocity 

O 

I 

I 

.741 

1 3 

•569 

5 

•454 


t Humber, page 57. 


* See D’Aubuisson, page 66. 


















HYDRAULICS. 


531 


Friction. 

Flowing of liquids through pipes or in natural channels is materially af¬ 
fected by friction. 

If equal volumes of water were to he discharged through pipes of equal 
diameters and leifgths, but of following figures: 



Figs. 1. 

The times would be as... 1, 

And velocities as.1, 



2. 3 - 

1. n, and 1.55. 
.72, and .64. 


D iscliarges from Compound or Diviclecl Reservoirs. 

Velocity in each may be considered as generated by difference of heights 
in contiguous reservoirs; consequently, square root of difference will rep¬ 
resent velocities, which, if there are several apertures, must be inversely as 
their respective areas. 

Note.—W hen water flows into a vacuum, 32.166 feet must be added to height of 
it; and when into a rarefied space only, height due to difference of external and 
internal pressure must be added. 

VELOCITY OF WATER OR OF FLUIDS. 

Coefficients of Discharge. 

Coefficient of Discharge or Efflux is product of coefficients of Contraction 
and Velocity. 

It is ascertained in practice that water issuing from a Circular Aperture 
in a thin plate contracts its section at a distance of .5 its diameter from 
aperture to very nearly .8 diameter of aperture, so as to reduce its area 
from 1 to about .61.* Velocity at this point is also ascertained to be about 
.974 times theoretical velocity due to a body falling from a height equal 
to head of water. Mean velocity in aperture is therefore .974, which, X 
.61 = .594, theoretical discharge; and in this case .594 becomes coefficient of 
discharge, which, if expressed generally by C, will give for discharge itself 

a\/-2 g k'XG V. a representing area of aperture, and V volume discharged per 

second. Or, 4.95 afh — Y. Or, 3.91 d 2 fh =. V. d representing diameter in feet. 

Hence, for cube feet per second, 4.95 afh, or 3.91 d 2 fh. 

Illustration. —Assume head of water 10 feet, diameter of opening 1.127 feet, 
area 1 sq. foot, and C = .62. 

Then 1 V2 g 10 X -62 = 15.72 cube feet. 4.97 X 1 X fio= 15.72 cube feet , and 
3.91 X i-i27 2 X V IO = I 5'7 cube feet. 

For square aperture it is .615, and for rectangular .621. 

Volume of water or a fluid discharged in a given time from an aperture 
of a given area depends on head, form of aperture, and nature of approaches. 

- y2 

64.333 h — v 2 , and -— h. h representing height to centre of opening in feet. 

6 4-333 

Note. —Head , or height, h, may be measured from surface of water to centre of 
aperture without practical error, for it has been proved by Mr. Neville that for cir¬ 
cular apertures, having their centre at the depth of their radius below the surface, 
and therefore circumference touching the surface, the error cannot exceed 4 per 
cent, in excess of the true theoretical discharge, and that for depths exceeding three 


* Bayer, .61. Observed discharges of water coincide nearer to unit of Bayer than that of all others. 
























532 


HYDRAULICS. 


times the diameter, the error is practically immaterial. For rectangular apertures 
it is also shown that, when their upper side is at surface of the water, as in notches, 
the extreme error cannot exceed 4.17 per cent, in excess; and when the upper is 
three times depth of aperture below the surface, the excess is inappreciable. 

For notches , weirs , slits , etc., however, it is usual to take full depth for head, when 
.666 only of above equation must be taken to ascertain the discharge. 

Experiments show that coefficient for similar apertures in thin plates, for 
small apertures and low velocities, is greater than for large apertures and 
high velocities, and that for elongated and small apertures it is greater than 
for apertures which have a regular form, and which approximate to the 
circle. 

When Discharge of a Fluid is tinder the Surface of another body of a 
like Fluid. —The difference of levels between the two surfaces must be taken 
as the head of the fluid. 

Or, V2 g (h — Id) = v. 

When Outer Side of opening of a discharging Vessel is pressed by a Force. 
—The difference of height of head of fluid and quotient of pressures on two 
sides of vessel, divided by density of fluid, must be taken as heads of fluid. 

Or, 2 g — v , s representing density of fluid. 

Illustration. —Assume head of water in open reservoir is 12 feet above water¬ 
line in boiler, and pressures of atmosphere and steam are 14.7 and 19.7 lbs. 

'/»4»-- » 7 ~£ 1 7X ^l = /6,333 X(. 2 -^] = ^6 feet. 

When Water flows into a rarefied Space , as into Condenser of a Steam- 
engine , and is either pressed upon or open to Atmosphere. —The height due to 
mean pressure of atmosphere within condenser, added to height of water 
above internal surface of it, must be taken as head of the water. 

Or, V2 g (h -J- h') = v. 

Illustration. —Assume head of water external to condenser of a steam-engine to 
be 3 feet, vacuum gauge to indicate a column of mercury of 26.467 ins. (= 13 lbs.), 
and a column of water of 13 lbs. = 29.9 feet. 

Then V 2 g (3 + 29.9) = V64.333 x 32.9 = f 2116.57 = 46 feet. 

Relative Velocity of Discharge of Water through, differ¬ 
ent Apertures and under lihe Heads. 

Velocity that would result from direct , unretarded action of the column of 


water which produces it, being a constant , or . 1 

Through a cylindrical aperture in a thin plate.625 

A tube from 2 to 3 diameters in length, projecting outward.8125 

A tube of the same length, projecting inward.6812 

A conical tube of form of contracted vein.974 

Wide opening, bottom of which is on a level with that of reservoir; 

sluice with walls in a line with orifice; or bridge with pointed piers.96 

Narrow opening, bottom of which is on a level with that of reservoir; 

abrupt projections and square piers of bridges.86 

Sluice without side walls..63 


Discharge or Efflux of Water for various Openings and 

Apertures. 

Rectangular "Weir. 

Weirs are designated Perfect when their sill is above surface of natural 
stream, and Imperfect , Submerged , or Drowned when it is below that surface. 




















HYDRAULICS. 


533 


Height measured from Surface of Water to Sill. ( Jas. B. Francis.) 


Mean Head. 

Length of Opening. 

Mean Discharge per Second. 

Mean Coefficient. 

.62 to 1.55 feet. 

10 feet. 

32.9 cube feet. 

.623 


Principal causes for variation in coefficients derived from most experi¬ 
ments giving discharge of water over weirs arises from, 

1. Depth being taken from only one part of surface, for it has been proved 
that heads on, at, and above a weir should be taken in order to determine 
true discharge. 

2. Nature of the approaches, including ratio of the water-way in channel 
above, to water-way on weir. 

When a weir extends from side to side of a channel, the contraction is 
less than when it forms a notch, or Poncelet weir, and coefficient sometimes 
rises as high as .667. 

When weir or notch extends only one fourth, or a less portion of width, 
coefficient has been found to vary from .584 to .6. 

When wing-boards are added at an angle of about 64°, coefficient is greater 
than even when head is less. 


Computation of* ‘V'olo.m.e of* Discharge. 

Mean velocity of a fluid issuing through a rectangular opening in 
side of a vessel is two thirds of that due to velocity at sill or lower 
edge of opening, or it is that due to a point four ninths of whole height 
from surface of fluid. 


Height measured from Surface of Head of Water to Sill of Opening. 


Rule.— Multiply square root of product of 64.333 and height or whole 
depth of the fluid ‘in feet, by area in feet, and by coefficient for opening, and 
two thirds of product will give volume in cube feet per second. 


Or, b h V 2 gh C = Y; 


—. t ; and 


i - b h V 2 gh C 

t representing time in seconds and Y volume in cube feet. 


C b h 




Example.— Sill of a weir is 1 foot below surface of water, and its breadth is 10 
feet; what volume of water will it discharge in one second? 

C = .623, V64.33 X 1 X kOO -- 80.2, and f 80.2 X .623 = 33.32 cube feet. 


Note._ Mean coefficient of discharge of weirs, breadth of which is no more than 

third part of breadth of stream, is two thirds of .6 = .4; and for weirs which extend 
whole width of stream it is two thirds of .666 = .444. 

Or, 214 Vh 3 = V in cube feet per minute. When h is in ins., put 5.15/or 214. 

Or, C b h V2 g h — V. C for a depth .1 of length — .417, and for .33 of length = .4. 

Or, by formula of Jas. B. Francis: 3.33 (L — .1 n H) H a = V. 

L representing length of weir and H depth of water in canal , sufficiently far from 
weir to be unaffected by depression caused by the current, both in feet, and n number 
of end contractions. 

Note.— When contraction exists at each end of weir. n = 2; and when weir is of 
w idth of canal or conduit, end contraction does not exist, and n~ o. 

This formula is applicable only to rectangular and horizontal weirs in side of a 
dam, vertical on water-side, with sharp edges to current; for if bevelled or rounded 
off in any perceptible degree, a material effect will be produced in the discharge^ 
it is essential also that the stream, after passing the edges, should in nowise be 
restricted in its flow and descent. 


Y Y* 













534 


HYDRAULICS. 


In cases in which depth exceeds one third of length of w r eir, this foimula is not 
applicable. In the observations from which it was deduced, the depth varied from 
7 to nearly 19 ins. 

With end contraction, a distance from side of canal to weir equal to depth on 
weir is least admissible, in order that formula may apply correctly. 

Depth of water in canal should not be less than three times that on weir for ac¬ 
curate computation of flow. 

Illustration. —If an overfull weir has a length of 7.94 feet and a depth of .986 
(as determined by a hook gauge), what volume will it discharge in 24 hours? 


3-33 (7.94 —.2 X .986) . 9863=3.33X7-94 —-1972 X-979°7 = 3-33 X 7-7428 X 
.97907 ±= 25.243 875, which X 60 X 60 X 24 = 2 181 061 cube feet. 

By Logarithms.—Log. 3.33 = .522444 

7.7428 = .888898 

3. _ 

.986 s = 1.993 877 
3 


2 ) 1.981 631 

1.990815 — 1.990815 
1.402 157 

Log. 24 hours = 86 400 seconds. 4-936 514 

6.338671 

Log. 6.33867 = 2181073 cube feet. C in this case = .615. 

Or, 214V1P and 5.15 v/Tf* = V, if stream above the sill is not in motion. H 
representing height of surface of water above sill in feet , h in inches ; and 
214 v/H 3 -(-.035 v 2 H3 = V, if in motion, v representing velocity of approach of 
water in feet per second , and V volume in cube feet discharged over each lineal foot 
of sill per minute. 

In gauging, waste-board must have a thin edge. Height measured to level of sur¬ 
face not affected by the current of overfall. (Molesworth.) 


To Compute Depth, of Flow over a, Sill that will Dis¬ 
charge a given. Volume of Water. 


( - 3 - ~ = + 

\2CbV2g 

fiows to the weir. 


3_\2 
7 c 2 |3 


V 2 

■ Jc — d. lc — — 
2 g 


representing height due to velocity (v) as it 


Note. —When back-water is raised considerably, say 2 feet, velocity of water ap¬ 
proaching weir ( 7 c) may be neglected. 


Rectangular Notches, or Vertical Apertures or Slits. 

A Notch is an opening, either vertical or oblique, in side of a vessel, reser¬ 
voir, etc., alike to a narrow and deep weir. 

Vertical Apertures or Slits are narrow notches or weirs, running to or 
near to bottom of vessel or reservoir. 

Coefficient for opening, 8 ins. by 5, mean .606 (Poncelet and Lesbros). 

Coefficient increases as depth decreases, or as ratio of length of notch to 
its depth increases. 

When sides and under edge of a notch increase in thickness, so as to be converted 
into a short open channel, coefficients reduce considerably, and to an extent beyond 
what increased resistance from friction, particularly for small depths, indicates. 

Poncelet and Lesbros found, for apertures 8X8 ins., that addition of a horizontal 
shoot 21 ins. long reduced coefficient from .604 to .601, with a head of about 4 feet; 
but for a head of 4.5 ins. coefficient fell from .572 to .483. 

For Rule and Formulas, see preceding page. 







HYDRAULICS. 


535 


Rectangular Openings or Sluices, or Horizontal Slits. 

Height measured from Surface of Head of Water to Upper Side and to Sill 

of Opening. 


Coefficient for 


Opening, i inch by i inch. Head, 7 to 23 feet. = .621. 


3 “3 " “ 7 “ 23 “ =.614. 

2 feet “ 1 foot. “ 1 “ 2 “ =.641. 

Poncelet and Lesbros deduced that coefficient of discharge increases with small 
and very oblong apertures as they approach the surface, and decreases with large 
and square apertures under like circumstances. 

Coefficients ranged, in square apertures of 8 by 8 ins., under a head of 6 ins. to 
rectangular apertures, 8 by 4 ins.; under a head of 10 feet, from .572 to .745. 

In a Thin Plate , C = .616 (Bossut) ; C = .61 ( Miclielotti). 

To Compute Discharge. 

Rule.—M ultiply square root of 64.333 and breadth of opening in feet, by 
coefficient for opening, and by difference of products of heights of water and 
their square roots, and two thirds of whole product will give discharge in 
cube feet per second. 

Or, — by/2 g (h ffh — It' f IT) C = V; - - ==;-—-— t- and 

3 | b V2 g (hfh — h'y/h') C 


—-— p= v. h and h' representing depth to sill arid opening in feet, and v velocity 

in feet per second. 

Example. —Sill of a rectangular sluice, 6 feet in width by 5 feet in depth, is 9 feet 
below surface of water; what is discharge in cube feet per second? 

C = .625, 9 — 5 = 4. and — V2 6 X -625 X (9V9 —4 X 3/4) — 380.95 cube feet. 

_ 3 

Or, V2 g d a C = V. d representing depth to centre of opening in feet. 


d = 9 —2.5 = 6.5, a —6 X 5 = 3 °, and V64.33 X 6.5 X 3° X -625 = 383.44 cube ft. 

Sluice 'W'eirs ox* Sluices. 

Discharge of w r ater by Sluices occurs under three forms—viz., Unimpeded , 
Impeded , or Partly Unimpeded. 

To Compute Discharge when. Unimpeded. 


C d b y/ 2 g h — V. d representing depth of opening and h taken from centre of 
opening to surface of water. 

If velocity, k, with which water flows to sluice is considered, 

V V 


w_v_y 

2 g \C d b) 


h- 


C b V 2 g h 


■ d: 


and 


d. 


h' representing height to which water is raised by dam above sill. 

Illustration. — How high must the gate of a sluice weir be raised, to discharge 
250 cube feet of water per second, its breadth being 24 feet and height, h', 5 feet? 

C by experiment = .6. d approximately = 1. 

2 5 ° 2 5 o 

— 1.0204 jee,t. 


.6X2,1/64.33(5-4) “MX. 7 - 0,4 

impute Discharg 
C dby/2 g h = V, and 


To Compute Discharge when Impeded. 

y 


:d. 


CbV2gh 

k representing difference of level between supply and back-water. 















536 


HYDRAULICS. 


To Compute Discharge when partly Impeded. 

Cby / 2g(^dh — — -j- d'-flij — V. d' representing depth of back-water above 
upper edge of sill. 

Illustration. — Dimensions of a sluice are 18 feet in breadth by .5 in depth; 
height of opening above surface of water .7 feet, and difference between levels of 
supply and surface water is 2 feet; what is discharge per second? 

.6 x 18 X 8.02 ^.7^/2-- -f- .5 — 86.62 X -896 + .707 = 138.85 cube feet. 

Coefficients of* Circular Openings or Slnices. 

Height measured from Surface of Head of Water to Centre of Opening. 

Contraction of section from 1 to .633, and reduction of velocity to .974; hence 
.633 X -974 = -617 [Neville). 

In a Thin Plate , C = .666 ( Bossut ); .631 ( Venturi ); .64 (Eytelwein). 

Cylindrical Ajutages , or Additional Tubes , give a greater discharge than 
apertures in a thin side, head and area of opening being the same; but it 
is necessary that the flowing water should entirely fill mouth of ajutage. 

Mean coefficient, as deduced by Castel , Bossut , and Eytelivein , is .82. 


Short Tubes, ALontli-pieces, and. Cylindrical Prolonga¬ 
tions or Ajutages. 


pig- 4. 


Fig- 5- 


If an aperture be placed in side of a 
vessel of from 1.5 to 2.5 diameters in 
thickness, it is converted thereby into a 
short tube, and coefficient, instead of being 
reduced by increased friction, is increased 
from mean value up to about .815, when 
opening is cylindrical, as in Fig. 4; and 
when junction is rounded, as in Fig. 5, to form of contracted vein, coefficient 
increases to .958, .959, and .975 for heads of 1,10, and 15 feet. 


Conically Convergent and Divergent Tubes. 

In conically divergent tube, Fig. 6, coeffi¬ 
cient of discharge is greater than for same 
tube placed convergent, fluid filling in both 
0 cases, and the smaller diameters, or those at 
same distance from centres, 0 O, being used 
in the computations. 

A tube, angle of convergence, O, of which 
is 5 0 nearly, with a head of from 1 to 10 
a feet, axial length of which is 3.5 ins., small 
diameter 1 inch, and large diameter 1.3 ins., 
b gives, when placed as at Fig. 6, .921 for co¬ 
efficient ; but when placed as at Fig. 7, co¬ 
efficient increases up to .948. Coefficient of velocity is, however, larger for 
Fig. 6 than for Fig. 7, and discharging jet has greater amplitude in falling. 
If a prismatic tube project beyond sides into a vessel, coefficient will be re¬ 
duced to .7x5 nearly. 

Form of tube which gives greatest discharge is that of a truncated cone, 
lesser base being fitted to reservoir, Fig. 7. Venturi concluded from his ex- 



Pig. 6. 

















HYDRAULICS. 


537 

periments that tube of greatest discharge has a length 9 times diameter of 
lesser opening base, and a diverging angle of 5 0 6'—discharge being 2.5 
greater than that through a thin plate, 1.9 times greater than through a 
short cylindrical tube, and 1.46 greater than theoretic discharge. 


Compound Moutli-pieces ancl Ajutages. 



Coefficients for HVIoutL - pieces, Sliort Tubes, and. Cyl¬ 
indrical Prolongations. 

Computed and reduced by Mr. Neville, from Venturi's Experiments. 


Description of Aperture, Mouth-piece, or Tube. 


1. An aperture 1.5 ins. diameter, in a thin plate. 

2. Tube 1.5 ins. diameter, and 4.5 ins. long, Fig. 4... 

3. Tube, Fig. 5, having junction rounded to form of contracted 

vein. 

4. Short conical convergent mouth piece, Fig. 6. 

5. Like tube divergent, with smaller diameter at junction with 

resei’voir; length 3.5 ins.,or = i in.,and ab = 1.3 ins. ... 

6. Double conical tube, a 0, S T, r 6, Fig. 9, when a6 = ST = i.5 

ins . x or — 1.21 ins., ao = . 92 in., and 0 S = 4.1 ins. 

7. Like tube when, as in Fig. 8, ao rb = o S T?-, and ao S — 

1.84 ins.. 

8. Like tube when S T = 1.46 ins., and 0 S = 2.17 ins. 

9. Like tube when S T = 3 ins., and o S = 9.5 ins. 

10. Like tube when 0 S =.6.5 ins., and ST — 1.92 ins. 

11. Like tube when S T = 2.25 ins., and oS = 12.125 ins. 

12. A tube, Fig. 10, when os — rt = 3 ins., or — st — x.'zi ins., 

and tube 0 S T r, as in No. 6, S T = 1.5 ins., and s S=4.1 ins. 


C. for 
iam. a b . 

C. for 
Diam. o r. 

.622 

•974 

.823 

.823 

.6ll 

•956 

.607 

•934 

.561 

.948 

.928 

00 

01 

H 

.823 

1.266 

.823 

1.266 

.911 

1.4 

1.02 

1.569 

1-215 

i -855 

• 895 

L 377 


Mean of various experiments with tubes of .5 to 3 ins. in diameter, and 
with a head of fluid of from 3 to 20 feet, gave a coefficient of .813; and as 
mean for circular apertures in a thin plate is .63, it follows that under 
similar circumstances, .813 .63=: 1.29 times as much fluid flows through 

a tube as through a like aperture in a thin plate. 

Preceding Table gives coefficients of discharge for figures given, and it 
will be found of great value, as coefficients are calculated for large as well 
as small diameters, and the necessity for taking into consideration form of 
junction of a pipe with a reservoir will be understood from the results. 


Circular Sluices, etc. 

To Compute Discharge. 

Height measured from Surface of Head of Water to Centre of Opening. 

Rule. —Multiply square root of product of 64.333 and depth of centre of 
opening from surface of water, by area of opening in square feet, and this 
product by coefficient for the opening, and whole product will give discharge 
in cube feet per second. 

Or, V2 g d, a C = V. a representing area in sq. feet, and d depth of surface of 
fluid from centre of opening in feet. 






























HYDRAULICS 


538 


Example. —Diameter of a circular sluice is 1 foot, and its centre is 1.5 feet below 
surface of the water; what is discharge in cube feet per second? 

Area of 1 foot = .7854; C = .64, and V 64.333 X 1.5 X -7854 X .64 = 4.938 cube feet. 


When Circumference reaches Surface of Water. V 2 g r, .9604 a C = V. 

r representing radius of circle in feet. 

Illustration. — In what time will 800 cube feet of water be discharged through a 
circular opening of .025 sq. foot, centre of which is 8 feet below surface of water? 

„ 800 800 

C = .61. -=-—- -— = 2230 . c;8 = 27 min. 10.0 sec. 

4 VTjjdX. 025 X .63 22.68 X -025 X -63 

Note. —For circular orifices, the formula f 2 g d aC — Y is sufficiently exact for 
all depths exceeding 3 times diameter; the finish of openings being of more effect 
than extreme accuracy in coefficient. 

Semicircular Sluices. 

When Diameter is either Upward or Downward, f 2 gd aC — Y. d repre¬ 
senting depth of centre of gravity offigure from surface. 

When Diameter as above is at Depth d, below Surface. V 2 gd 1.188 a C = V. 

Circular, Semicircular, Triangular, Trapezoidal, Pris¬ 
matic Wedges, Sluices, Slits, etc. 

See Neville , London , i860, pp. 51-63, and Weisbach , vol. i.p. 456. 

For greater number of apertures at any depth below surface of water, 
product of area, and velocity of depth of centre, or centre of gravity, 
if practicable to obtain it, will give discharge with sufficient accuracy. 

Discharge from Vessels not Receiving any Supply. 

For prismatic vessels the general law applies, that twice as much would 
be discharged from like apertures if the vessels were kept full during the 
time which is required for emptying them. 

_ _ 2 A fh 2 Ah 

To Compute Time. -— — ——- = t. < 

Caf 2 g v 

Illustration.— A rectangular cistern has a transverse horizontal section of 14 
feet, a depth of 4 feet, and a circular opening in its bottom of 2 ins. in diameter; in 
what time will it discharge its volume of water, when supply to it is cut off and 
cistern allowed to be emptied of its conteuts? 

A = 4 feet, a = 2 2 X .7854 = 144 = .0218, C = .6i3, and V 2 gh X a X C =.2143 

cube foot per second. Then 2 X 14 X 4 _ ^ 22 ^ seconds. 

.2143 

To Compute Time and Fall. 

Depression or subsidence of surface of water in a vessel, corresponding to 
a given time of efflux, is h — h! . hi representing lesser depth. 


2 A 


C af 2 g 


-— (fh — li') — t. 


Inversely, (fh- 


C a V 2 g 
2 A 


= h’. 


Illustration. — In what time will the water in cistern, as given in preceding 
case, subside 1.6 feet, and how much will it subside in that time ? 


14, C=6, a = . 021 

2 X 14 


h' — 4 — 1.6 = : 


(y/4 


.6 X .0218 X 8.02 
.6 X .0218 X 8.02 
2 X 14 


^2 g — 8.02, h = 4, 

. . 28 

X (v 4 — V 2-4) = _ ■ X (2 — 1.55) = 120. 1 seconds. 


.1049 


X I20.ij =2— .45 = 2.4 feet; hence, 4 — 2.4 = 1.6 feet. 


When Supply is maintained .—Divide result obtained as preceding by 2. 

















HYDRAULICS. 


539 


Discharge, when Form and. Dimensions of Vessel of 
Efflux are not known. 

Volume discharged may be estimated by observing heads of the water at 
equal intervals of time; and at end of half time of discharge, head of water 
will be .25 of whole height from surface to delivery. 

When t — such interval. For openings in bottom or side, C a t V2 g 
= V, for 1 depth ; C a £ V2 g ( ^ h + 4 aA i + V h z \ _ y y 0J . 2 g e pif lS . an( j 
C atVTg ( VA±Wh ^3 + = y f(jr + depths 

Note. —At end of half time of discharge, head of water will be . 25 of whole height 
from surface to delivery. 

■\Veirs or ^Notches. 

~ C b t V 2 g (\A 3 + 4 V h3 i + \/ft 3 2) = V. b representing breadth in feet. 

Illustration. —A prismatic reservoir 9 feet in depth is discharged through a 
notch 2.222 feet wide, surface subsiding 6.75 feet in 935 seconds; what is volume 
discharged? 


C = .6, /t x = 9 — 6.75 — 2.25 feet, 


and - 6 X 2.222 X 935 X 8.02 (VgS -f- 4 


V2.253-}- 3/o 3 ) = 2221.6 X 40.5 = 89974.8 cube feet. 

When there is an Influx and Ejflux. 

If a reservoir during an efflux from it has an influx into it, determination 
of time in which surface of water rises or falls a certain height becomes so 
complicated that an approximate determination is here alone essayed. 

A state of permanency or constant height occurs whenever head of water is in¬ 
creased or decreased b3 r — (c a) = 1 representing influx in cube feet per second. 

Ai u 

Time (t) in which variable head (x) increases by volume ( v ) = j_ C a > 

and time in which it sinks height, Jc, by Al 1 


__ . Time of efflux, in which 

C a V 2 gx — I 

subsiding surface falls from A to Ai, etc., and head of water from h to ft t , when 
k is represented by - — =: y / lc , is 


ft- 


C a V2 g 
A 4 Ai 


4 A : 


1-V*/ 


12 C a V1 g W'ft — y/ht — y/k y/h 2 — y/k y/h 3 — y/k y/h 4 

Illustration. —In what time will surface of water in a pond, as in a previous 
example, fall 6 feet, if there is an Influx into it of 3.0444 cube feet per second? 


y/k = 
20 —14 


3-044 


.537 X .8836 X 8.02 
600 000 


12 X -537 X .8836 X 8.02 
265 000 


X 




= .8. C = . 537 and a = .8836. 

4X495000 2X410000 4X325000 


472- 


8 


4.301 —.8 


4.123 


8 


3.937 —.8 


3 - 74 2 


\ —-— X 1 480 201 = 194 486 seconds — 54 ft., 1 min., 26 sec. 

— .8/ 45-665 


Prismatic Vessels. 
If vessel has a uniform transverse section, A. 


Then 


C a 3/2 g 
head of water flows from h to h t ; 


[v/< - v, + V‘x »» j 


= t = time in which 


























540 


HYDRAULICS. 


Illustration.—A reservoir has a surface of 500000 sq. feet, a depth of 20 feet; it 
is fed by a stream affording a supply of 3.0444 cube feet per second, and outlet has 
an area of .8836 sq. foot; in what time will it subside 6 feet? 

■fie, as before, =.8, C=:.537, and — * 500 °° g x [1/20 — A /i4 + -8 X hyp, log. 

C a 3 / .2 g L 


/ V 2 ° 

WM- 


-.8 


X 2-303j = 238414 seconds — 66 h. 13 min. 34 sec. 


To Compute Fall in a given Time. 

This is determining head hi at end of that time, and it should be sub¬ 
tracted from head h at commencement of discharge. Put into preceding 
equation several values of h x , until one is found to meet the condition. 


Illustration.— Take a prismatic pond having a surface of 38750 sq. feet, a depth 
to centre of opening of sluice of 10.5 feet, a supply of 33.6 cube feet, and a discharge 
of 40 cube feet per second. 

-fk —. 84. 

Putting these numerical values into the equation, and assuming different values 
for hi, a value which nearly satisfies the equation is 4. Consequently, 10.5 —4 = 
6.5 feet, fall. 


iff 


f C b 3/2 g 


hyp. log. 


1 = 


hif-y/hj^k-flc 

(V*i-V*) 2 ' 


if 12 arc ^tang. 


• 3/3*1 


2y/k-\- if hj 


= t; 


= k] arc (tang. = y, arc tangent of which = y, and I as preceding. 


According as k is ^ h, and influx of water, I^fC 1V2 gh 3 , there is a rise or fall 

of fluid surface, the condition of permanency occurring when h x —k, and time cor¬ 
responding becomes co. 


Illustration. —In what time will w~ater in a rectangular tank, 12 feet in length 
by 6 feet in breadth, rise from sill of a weir or notch, 6 inches broad, to 2 feet 
above it, when 5 cube feet of water flow into the tank per second? 

h z — 2 , h = zo , A:= 12 X 6 = 72, 1 — 5 , b -- 5 , G = . 6 . 


k = (— 

w. 


‘f .6 X .5 X 8.02 


) f = 


= ty 3 - II 7 2 = 2 -i 33 8 - 


72 x 2.1338 r, , ... 24-3/2X2.1338-1-2.1338 • / 

Then i- hyp. logarithm - — — —f -—-b 3/ 12 arc tang. — 

3X5 L (V 2 — V 2 - I 33^) V 

-) | = xo.2423 X hyp. log. 6-199 - 3-464! X arc (tang. ■ ^ = 

'■} J .CX32162 V 4-3356/ 


- V3 X 


2 V 2>I 33 8_ b 3 / 2 


.002 162 

10.2423 X [7-961 — (3.461 X arc, tangent of which = .56497, or 29 0 28' = 29.466, 


length of which = .5143) — 1.781] = 10.2423 — 7.961 — 1.781 = 10.2423 X 6.18 = 
63.297 seconds. 


Discharge of "Water under Wariahle Pressures. 
To Compute Time, Rise and Fall, and. Volume. 

2 gx — v. x representing variable head, A and a areas of transverse horizon¬ 
tal section of vessel and discharge, and v theoretical velocity of efflux. 

To Compute Volume. 

A y = V. y representing extent of fall, and V volume of water discharged, as 
h — h'. 

Illustration. —Assume elements of preceding case. 

A = 14. 2/ = 4 feet. Then 56 x 4 = 224 cube feet. 


















HYDRAULICS. 


541 


Discharge from 'V'essels of* Communication. 
When Reservoir of Supply is maintained at a uniform Height .— Fig. 11. 

To Compute Time. 2 ^ tA — t. 

C a V2 g 

Illustration i. —In what time will level of water in a receiving vessel having a 
section of 14 sq. feet attain height of that in supply, through a pipe 2 ins. in diam¬ 
eter, placed 4 feet below level of supply? 

2 x 14 X V 4 56 


C = -613. 
Fig. n. „ 


.613 X .0218 X 8.02 .1072 


522.3 seconds. 



2_Assume C, vessel, Fig. n, to be a cylinder 18 

ins. in diameter, head of water in A = 4 feet, at A' 
1 foot, and 2 feet below outlet 0; in what time will 
water in vessel run out and over at 0 through a pipe, 
a, 1.5 ins. diameter? 

h — h' = 4 — 1 — 2 — 1 foot. C —. 8. 

A 


<£)■— 


288 


6.424 


X 1.73 — 1 == 32.73 seconds. 


When Vessel of Supply has no Influx , and is not indefinitely great compared 

with Receiving Vessel. 

2 A A’ fh 


C ci (A -|— A') V2 g 


— t. A' representing section of receiving vessel , t time in which 


the two surfaces of water attain same level; and 


2 A A' (fh — fh') 


t, time within 


C a (A + A') fTg 

which level falls from h to It. 

Illustration. —Section of a cistern from which water is to be drawn is 10 sq. 
feet, and section of receiving cistern is 4 sq. feet; initial difference of level is 3 feet, 
and diameter of communicating pipe is 1 inch; in what time will surfaces of water 
in both vessels attain like levels? 

i = .7854. —- 2ALP - - = - = 276 seconds. 


C = .82. 


.82 X -7854 X X 8.02 

144 


.502 


Discharge from a. fSTotcla* in Side of a 'Vessel. 

3 A 


When it has no Influx. 


( —fir -~) —t.b breadth of notch in feet. 

W h V"/ 


Cb X V2 g 

Illustration. —If a reservoir of water, no feet in length by 40 in breadth, has a 
notch in end of 9 ins. in width; in what time will head of water of 15 ins. fall to 6? 

C = .6. 9" = .75 foot. h' — . 5. h = 1.25. 


3 X no X 40 


X 


(f 


13 200 


, . . , , . — , X 1.414 — .894 = 1901 seconds. 

•6 X-75X8.02 W .5 fi.zsJ 3 - 6 i y 

Note.—F or discharge of vessels in motion, see Weisbach, vol. i, pp. 394-396. 
Reservoirs or Cisterns. 

To Compute Time of Uhlling and of Emptying a Reser¬ 
voir under Operation of Dotla Supply and Discharge. 


- = T, and 


g p — — t. V representing volume of vessel, S supply of water , 

and D discharge of water , both per minute, and in cube feet. T time of filling vessel , 
and t time of discharging it , both in minutes. 


* When the notch extends to the bottom of the reservoir, etc., the time for the water to run out is 
indefinite, as A' = o. 

Z z 



































542 


HYDRAULICS. 


a 4 

fh\ 


^ — t ; li, h , 


Irz’egui.lar-Sh.aped. Vessels, as a Pond, Lake, etc. 
To Compute Time and. Volume Discharged. 
Operation. —Divide whole mass of water into four or six strata of equal 
depths. 

, li — hA / a , 4 oi , 2 ct2 4 a 3 

Then, for 4 Strata, -— X (—tt + -wtv + — r ,—h -777 ■ 

12 Ca 7 2 g W h V ,lZ V k2 V h3 
etc., representing depths of strata at a, a 1, etc., commencing at surface; ay a2, 

. , /i— A 4 

etc., fcei'n# areas of first, second, etc., transverse sections oj pond, etc. ; ana ——— 
Xa + 4« T + 2 a 2 -|- 4 a 3 -|-a 4 = :V. 

I2 ' A a h ~ Illustration. —In what time 

2 g i— 7,1 —^=-_=--=X/C vvill depth of w r ater in a lake, 

A 6 C, Fig. 12, subside 6 feet, sur¬ 
faces of its strata having follow¬ 
ing areas, outline of sluice being 
a semicircle, 18 ins. wide, 9 deep, 
and 60 feet in length? 



a 

at 

20 feet 

(h ) 

depth of water 

= area of 600 000 sq. feet. 

a 1 

u 

M 

0° 

Ln 

(hi) 

U tt 

_ tt 

495000 “ 

a 2 

tt 

17 “ 

(A 2 ) 

tt tt 

_ (t 

410000 “ 

a 3 

tt 

15-5 “ 

m 

tt tt 

_ tt 

325000 “ 

a 4 

tt 

14 “ 

(A 4 ) 

it tt 

_ tt 

265000 “ 



a = area of 

18 -r- 2 = .8836 

sq. feet; 

C = - 537 - 


Then 


14 


12 X -537 X .8836 X 8.02 
265 ooo\ 6 


X 


/600 000 

V A u 


.472 


4 X 495 000 
4.301 


2 X 410000 
4.123 


4 X325000 
3-937 


3-742 


45-665 


X 1194 431 = 156 938 sec. =43 h., 35 min. 38 sec. 


And discharge = —• X (600 000 -f- 4 X 495 000 4-2X410 000 + 4 X 325 000 -f- 265 000) 
12 

= .5X4 965 000 = 2482 500 cube feet. 

For 6 Strata, put 204, instead of a 4 , and 4 a 5 and a6 additional, and divide by 
18 instead of 12. 

Flow of Water in Beds. 

Flow of water in beds is either Uniform or Variable. It is uniform when 
mean velocity at all transverse sections is the same, and consequently when 
areas of sections are equal; it is variable when mean velocities, and there¬ 
fore areas of sections, vary. 

To Compute Dali of Flow. 

C X — — h. C representing coefficient offriction , l length of flow, p perimeter 

of sides and bottom of bed, and hfall in feet. 

Illustration. —A canal 2600 feet in length has breadths of 3 and 7 feet, a depth 
of 3 feet, with a flow of 40 cube feet per second; what is its fall? 

C = as per table below .007 565 ; p — Vs 2 -\- 2 2 X 2 -f 3 = 10.2; 


15; and 


: 40 ~ r ~ 15 — 2.66. Hence .007 565 x 


2600 x 10.2 

15 


2.66 2 

X 2-= I -47 fict. 

64-33 


To Compute Velocity of Flow. . / —-— ■zqli — v. 

VC Xl P 

Illustration. —A canal 5800 feet in length has breadths of 4 and 12 feet, a depth 
of 5, and a fall of 3; w’hat is velocity and volume of flow? 

P — V5 J -\-a 2 X 2 + 4 = 16.8, and a — 40. 

,oo 7 565X^00X16.8 X 64 ' 33 X 3 = v/ -° 54a x = 3- 2 3 fi*. Hence 
volume ~ 40 x 3.23 = 129.2 cube feet. 


Then 


7 



































HYDRAULICS. 


543 


Coefficients of Friction of Flow of Water in Beds, as 
in Rivers, Canals, Streams, etc. 


In Feet per Second. 


Velocity. 

C. 

A T elocity. 

C. 

Velocity. 

C. 

Velocity. 

c. 

• 3 

.008 15 

•7 

•°°7 73 

1-5 

.007 59 

5 

.00745 

•4 

.007 97 

.8 

.007 69 

2 

.007 52 

8 

.007 44 

•5 

.007 85 

•9 

.007 66 

2-5 

.007 51 

IO 

.00743 

.6 

.007 78 

I 

.007 63 

3 

.007 49 

12 

.OO7 42 


Forms of Transverse Sections of Canals, etc. 

Resistance or friction which bed of a stream, etc., opposes to flow of water, 
in consequence of its adhesion or viscosity, increases with surface of contact 
between bed and water, and therefore with the perimeter of Avater profile, or 
of that portion of transverse section which comprises the bed. 

Friction of flow of water in a bed is inversely as area of it. 

Of all regular figures, that which has greatest number of sides has for 
same area least perimeter; hence, for enclosed conduits, nearer its trans¬ 
verse profile approaches to a regular figure, less the coefficient of its friction; 
consequently, a circle has the profile which presents minimum of friction. 

When a canal is cut in earth or sand and not Availed up, the slope of its 
sides should not exceed 45°. 

~V r ai*ia"ble lYtotion. 

Variable motion of water in beds of rivers or streams may be reduced to 
rules of uniform motion when resistance of friction for an observed length 
of river can be taken as constant. 


To Compute Volixme of* Water flowing in a Fiver. 



V 2 gh 


= V. 



a 1 + a\a^a>) 


A and A r representing areas of upper 
and lower transverse sections of 
flow. 


Illustration. —A stream having a mean perimeter of water profile of 40 feet for 
a length of 300 feet has a fall of 9.6 ins.; area of its upper section is 70 sq. feet, and 
of its lower 60; what is volume of its discharge? 


To obtain C for velocity due to this case, 92.35 
coefficient for which, see Table above, = .007 44. 


-\ / 7 °+ 6 ° x if 

40 X 300 


S. 59 feet, 


V64.33 X (9.6-4-12) 


7W4 


/j_ l 

v 70 2 60 


300 x 4° 
6o* + ' K>744 -p+to 


( — - + — ^ 
\70 2 6o 2 / 


v: 


394.6 cube feet; 


000 330 < 


and mean velocity = 394 '^ = 6.07 fleet, C for which is .007 45. 

70 -)- 60 


FRICTION IN PIPES AND SEWERS. 

Friction in flow of Avater through pipes, etc., of a uniform diameter is in¬ 
dependent of pressure, and increases directly as length, very nearly as square 
of velocity of flow, and inversely as diameter of pipe. 

With wooden pipes friction is 1.75 times greater than in metallic. 

Time occupied in flowing of an equal quantity of water through Pipes or 
SeAvers of equal lengths, and with equal heads, is proportionally as follows: 

In a Right Line as 90, in a True Curve as 100, and in a Right Angle as 140. 





























544 


HYDRAULICS. 


To Compute Head, necessary to overcome Friction of* 

3 ?ipe. ( Weisbach.) 


( 0144 4 - l 0 1 7 4^ \ l_ _ y j t , representing head to overcome friction of 
V V v J d 5-4 

flo w in pipe, l length of pipe, and v velocity of water per second, both in feet , and d 
internal diameter of pipe in ins. 

Illustration. —Length of a conduit-pipe is iooo feet, its diameter 3 ins., and the 
required velocity of its discharge 4 feet per second; what is required head of water 
to overcome friction of flow in pipe? 

( .017 a6\ iooo 16 , „ - , 

.0144 -\ - y-) X - X-= -023 13 X 333-333 X 2.963 = 22.845 feel. 

V 4 / 3 5-4 

Head here deduced is height necessary to overcome friction of water in 
pipe alone. 

Whole or entire head or fall includes, in addition to above, height between 
surface of supply and centre of opening of pipe at its upper end. Conse¬ 
quently, it is whole height or vertical distance between supply and centre 
of outlet. 

To Compute whole Head, or Height from Surface of 
Supply to Centre of Discharge. 

< Cx “ + i-5>X^ = fc 


1.5 is taken as a mean, and is coefficient of friction for interior orifice, or that of 
upper portion of pipe. 


To obtain C or coefficient. 


( , -017 46\ 



For facilitating computation, following Table of coefficients of resistance 
is introduced, being a reduction of preceding formula: 

Coefficients of Friction of Water. 

In IPipes at Different Velocities. 


V. 

c. 

v. 

C. 

V. 

C. 

V. 

c. 

V. 

C 

Ft. Ins. 


Ft. Ins. 


Ft. Ins. 


Ft. Ins. 

- 

Ft. Ins. 


4 

•0443 

2 8 

.025 

5 

.0221 

7 4 

.0208 

ii 6 

.0195 

8 

•0356 

3 

.0244 

5 4 

.0219 

7 8 

.0206 

12 

.0194 

I 

•0317 

3 4 

.0239 

5 8 

.0217 

8 

.0205 

12 6 

.0193 

1 4 

.0294 

3 8 

.0234 

6 

.0215 

8 6 

.0204 

13 

.OI9I 

i 8 

.0278 

4 

.0231 

6 4 

.0213 

9 

.0202 

14 

.0189 

2 

.0266 

4 4 

.0227 

6 8 

.0211 

10 

.0199 

15 

.0188 

2 4 

.0257 

4 8 

.0224 

7 

.0209 

11 

.0196 

l6 

.0187 


Illustration i.— Coefficient due to a velocity of 4 feet per second is .0231. 


2.—Take elements of preceding case. 

iooo X 12 , . 4 2 16 

(.0231 X---h i- 5 ) X --= 93-9 X 2 -= 23.35 feet. 

3 64.33 64.33 

Note.— In preceding formula l was taken in feet, as the multiplier of 12 for ins. 
was cancelled by taking 5.4 for 2 g, but jn above formula it is necessary to restore 
this multiplier. 

Raclii of Curvatures. 


When Pipes branch off from Mains, or when they are deflected at right 
angles, radius of curvature should be proportionate to their diameter. Thus, 


Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Diameter. 2 to 3 

3 to 4 

6 

8 

IO 

Radius.' 18 

20 

30 

42 

60 









































HYDRAULICS. 


545 


Curves and. Bends. 


Resistance or loss of head due to curves and bends, alike to that of friction, 
increases as square of velocity; when, however, curves have a long- radius 
and bends are obtuse, the loss is small. 

Curved Circular Pipe . ( Weisbach). —- x I.131 +1.847 ( — \ ¥ 1 v— — h 

180 L \2 rj J 2 g 

a representing angle of curve , d diameter of pipe , r radius of curve, and h height 
due to friction or resistance of curve, all in feet. 

For facility of computations, following values of .131 -f- 1.847 ( — ) 8 
duced. 

Coefficients of Resistance. 

In Curved Pipes with. Section of a Circle. 
d 


are intro- 


2 r 


I 

• 131 

•25 

• M5 

• 4 

. 206 

.6 

•44 

•75 

.806 

•9 

15 

• 133 

• 3 

.158 

•45 

.244 

•65 

•54 

.8 

•977 

•95 

2 

.138 

•35 

.178 

•5 

.294 

•7 

.661 

•85 

!.l 77 

1 


1.408 

1-674 

1.978 

Illustration. — If in a pipe 18 ins. in diameter and 1 mile in length there is a 
right-angled curve of 5 feet radius, what additional head of flow should be given to 
attain velocity due to a head of 20 feet? 

a — 90 0 , v for such a pipe and head = 4 feet per second; 18 = 1.5 and — 1-5 
= . 15, and ■ 15 by table . 133. 


HenCe ’ ^ X 133 X 

180 64.33 


= -5 X .133 X 


l6 


64-33 


•016 53 foot. 


Note.— If angle is greater than 90 0 , head should be proportionately increased. 

Bent or Angular Circular Pipes. 

Coefficient for angle of bend = .9457 sin. 2 x -j-2.047 sin.4 x. Hence, 


X 

IO° 

20° 

3 °° 

O 

O 

45 ° 

La 

O 

O 

55 ° 

6o° 

65° 

^4 

O 

O 

C 

.046 

•139 

•364 

•74 

.984 

1.26 

i -556 

1.861 

2.158 

2.431 


and -— x C = h. x representing half angle of bend. 

2 g 


Illustration. — Assume v = 4 feet , and angle = 90 0 ; x = -— = 45 0 . 

2 

Then —‘-— x -984 = .2447 foot additional head required. 

64-33 


In Valve Grates or Slide Valves. 
In Rectangular Pipes. 


r 

I 

•9 

.8 

•7 

.6 

•5 

•4 

•3 

.2 

.1 

C 

.O 

.09 

•39 

•95 

2.08 

4.02 

8.12 

00 

44-5 

x 93 


r = ratio of cross section. 

In Cylindrical Pipes. 


h 

O 

.125 

•25 

•375 

•5 

• 625 

•75 

•875 

r 

I 

.948 

.856 

•74 

.609 

.466 

• 3 i 5 

•i 59 

C 

.O 

.07 

.26 

.81 

2.06 

5-52 

x 7 

97.8 


h — relative height of opening. 


In a Throttle Valve. In Cylindrical Pipes. 


A 

5° 

IO° 

*5° 

20° 

25° 

3 °° 

35° 

4 ^ 

O 

° 

45° 

O 

0 

10 

6o° 

O 

O 

r 

•9 X 3 

.826 

•74 1 

.658 

•577 

•5 

.426 

• 357 

•293 

• 234 

•i34 

.06 

C 

.24 

•52 

•9 

x -54 

2 - 5 x 

3-9 1 

6.22 

10.8 

w 

OC 

^4 

32.6 

Il8 

75 i 


A = angle of position. 

Z z* 
































































































546 


HYDRAULICS. 


In a Claclw or Trap Valve. 


Angle of opening. 

15 ° 

20° 

25 ° 

O 

0 

cn 

35 ° 

O 

0 

45 ° 

50 0 

55 ° 

6o° 

65° 

O 

O 

C 

90 

62 

42 

30 

20 

14 

9-5 

6.6 

4.6 

3 - 2 

2 -3 

i -7 


In a Cock. In Cylindrical Pipes. 


A 

5 ° 

IO° 

i 5 ° 

20° 

25° 

30 0 

35 ° 

0 

0 

45 ° 

50 0 

55 ° 

6o° 

65° 

r 

.926 

•85 

.772 

.692 

.613 

•535 

•458 

•385 

• 3 V 

•25 

.19 

•137 

.O9I 

c 

•°5 

.29 

•75 

I.56 

3 -i 

5-47 

9.68 

I 7 , 3 

31.2 

52.6 

106 

206 

486 


In a Conical Valve. ^1.645 ^ = C. a and a' = areas of pipe 

and opening. 

/ CL \ 2 

In Imperfect Contractions, (fTf '— I ) = 0 . C —a factor, rang¬ 
ing from .624 for — .1 to 1 for —, = j, being greater the greater the ratio. 

d Qj 

Illustration.— If a slide valve is set in a cylindrical pipe 3 ins. in diameter and 
500 feet in length, is opened to .375 of diameter of pipe (hence, .625 diameter closed), 
what volume of water will it discharge under a head of 100 feet, coefficient of en¬ 
trance of pipe assumed at .5 ? 


C, by table, p. 545, pipe being .625 closed = 5.52. 


V2 g y/h 


= u 


\/(i‘ 5 H" c -f- c 

C = from table, p. 544 ,for an assumed velocity of n feet 6 ins. — .0195. 

8.03 X 10 


Then 


V 64.33 X V100 


80.3 

V(7-° 2 + 39) _ 6.78 


—11.85 fat. 


it , , 500 X I2\ 

^i .5 + 5-52 + -OI 95 - 2 - J 

Hence, area of 3 ins. = 7.07, and 7.07 X 12 X 11-85 =3 1005.4 cube feet per second. 

Valves. ( Conical , Spherical, or Flap.) 

Conical or Splierical Valve Puppet. 

v 2 

Height due to resistance or loss of head of water = 11 —. v representing 

2 g 

velocity of water in full diameter of pipe or vessel. 

— i j = C. A and A' representing transverse areas of vessel and of valve 

( A \ 2 

1.645 — — 1J = C of contraction in general. 

Illustration.— If A' = .5 of vessel, C = ^1.645 X -y ——2.292 = 5.24. 

Clack or Trap Valve .—C decreases with diameter of vessel. 

Illustration. —If a single-acting force-pump, 6 ins. in diameter, delivers at each 
stroke 5 cube feet of water in 4 seconds, diameter of valve seat 3.5 ins., and of valve 
4.5; what resistance has water in its passage, and what is loss of mechanical effect? 

a — . 196. =.34 ratio of transverse area of opening. 1 — =-44 ratio 

of annular contraction to transverse area of vessel. 

Hence, '-———-=.39 mean ratio, and coefficient of resistance corresponding 
2 


thereto 


/1.645 \ 2 2 

~ — ij = 3.22 2 = 10.37. 


4 X .196 


= 6.37 velocity per second. 































































HYDRAULICS. 


547 


6 . ^ 7 2 

-—— = .63 height due to velocity. Consequently, 10.37 X -63 = 6.53 height due to 

64-33 

resistance of valve, and - X 62.5 X 6.53 = 510.15 lbs. mechanical effect lost. 

4 


Discharge of Water in Pipes. 


For any Length, and Head, and for Diameters from 
1 Inch to IO Feet. In Cube Feet per Minute. (Beardmore.) 


Diam. 

Tab. No. 

Diam. 

Tab. No. 

Diam. 

Tab. No. 

Diam. 

Tab. No. 

Diam. 

Tab. No. 

Ins. 


Ft. Ins. 


Ft. Ins. 


Ft. Ins. 


Ft. Ins. 


I 

4.71 


9 

1 147.6 

I 

II 

n 983 

3 

I 

39 329 

4 

9 

115854 

1.25 

8.48 


IO 

M93-5 

2 


13328 

3 

2 

42 O4O 

5 


131 703 

i-5 

13.°2 


II 

1 894.9 

2 

I 

14758 

3 

3 

44 863 

5 

3 

148 79 1 

I -75 

ICMS 

I 


2 356 

2 

2 

16 278 

3 

4 

47 794 

5 

6 

167 139 

2 

26.69 

I 

I 

2 876.7 

2 

3 

17 889 

3 

5 

50835 

5 

9 

186 786 

2.5 

46.67 

I 

2 

3 463-3 

2 

4 

19 592 

3 

6 

53 995 

6 


207 754 

3 

73-5 

I 

3 

4II5-9 

2 

5 

21 390 

3 

7 

57265 

6 

6 

253781 

3-5 

108.14 

I 

4 

4836.9 

2 

6 

23 282 

3 

8 

60 648 

7 


305 437 

4 

151.02 

I 

5 

5628.5 

2 

7 

25 270 

3 

9 

64 156 

7 

6 

362 935 

4-5 

194.84 

I 

6 

6 493.1 

2 

8 

27 358 

3 

IO 

67 782 

8 


426 481 

5 

263.87 

I 

7 

7 433 

2 

9 

29 547 

3 

II 

71 526 

8 

6 

496 275 

6 

416.54 

I 

8 

8449 

2 

IO 

3i 834 

4 


75 392 

9 


572 508 

7 

612.32 

I 

9 

9 544 

2 

II 

34228 

4 

3 

87730 

9 

6 

655 369 

8 

854.99 

I 

IO 

10 722 

3 


36 725 

4 

61 

IOI 207 

IO 


745 038 


This Table is applicable to Sewers and Drains by talcing same proportion 
of tabular numbers that area of cross-section of water in sewer or drain 
bears to whole area of sewer or drain. 


Formula upon which the table is constructed is, 3356 ,^/y x d 5 = V in 

cube feel per minute, and 39.27 y X d 5 = V in cube feet per second, h represent¬ 

ing height of fall of water and d diameter of pipe and l length, all in feet. 

To Compute Discharge. 

_ , , . , Id* h „ , 5AV 2 „ , , ,, , ... 

(Eytelwexn .) w ——- 4. 71 = V, and f / —- — . 538 = d. d = diameter of pipe in 

ins., I length of pipe and h head of water, both in feet. 


T T , * /G 2 1 1 . . /(15 dp h _ 

(Hawksley.) —--- = d, and —[ -= G- 

gallons per hour, and l length of pipes in yards. 


G = number of Imperial 


(Neville .) 140 Vr s — 11 vr s = v in feet per second, r — hydraulic mean depth 
in feet, and s sine of the inclination or total fall divided by total length. 

v 47.124 d 2 — V, and v 293.7286 d~ = Imperial gallons per minute, d = diameter 
of pipe in feet. 

To Compute Volume discharged. 

When Length of Pipe, Height or Fall, and Diameter are given. Rule. 
— Divide tabular number, opposite to diameter of tube, by square root of 
rate of inclination, and quotient will give volume required in cube foet per 
minute. 

Example.— A pipe has a diameter of 9 ins., and a length of 4750 feet; what is 
its discharge per minute under a head of 17.5 feet? 

Tab. No. 9 ins. =1x47.6, and -—— — — 69.67 cube feet. 

/ 475 o 16.47 
























548 


HYDRAULICS. 


To Compute Diameter. 

When Length , Head , and Volume are given. Rule.—M ultiply discharge 
per minute by square root of ratio of inclination ; take nearest corresponding 
number in Table, and opposite to it is diameter required. 

Example. —Take elements of preceding case. 


69.67 X 


= 1147.61, and opposite to this is 9 ins. 


0T '</\ 


v l 


= d in feet, v representing velocity in feet per second and l length 


1542 h 

in feet. 

To Compute Head. 

When Length, Discharge, and Diameter are given. Rule. — Divide 
tabular number for diameter by discharge per minute, square quotient, and 
divide length of pipe by it; quotient will give head necessary to force given 
volume of water through pipe in one minute. 

Example. —Take elements of preceding cases. 


1147.61 

69.67 


= 16.47; i6.47 2 = 27i.3; 4750 = 271.2 = 17.5 feet. 


To Compute whole Head necessarj’- to furnish, requisite 

Discharge. 

See Formula and Illustration, page 544. 

To Compute "Velocity. 

When Volume and Diameter alone are given. Rule. — Divide volume 
when in feet per minute by area in feet, and quotient, divided by 60, will 
give velocity in feet per second. 

Example. —Take elements of preceding case. 

— ^'67 .. = go = 2.63 feet. 

•75 2 X. 7 854 

When Volume is not given. Rule. — Multiply square root of product of 
height of pipe by diameter in feet, divided by length in feet, by 50, and 
product will give velocity in feet per second. ( Beardmore .) 

To Compute Inclination of a Dipe. 

( V \ 2 i h 
2356/ df ^~ T ' 

Illustration. —Take elements of preceding case. 


X —- = .000874 X 4.214: 
75 ‘ 


I7.5 

:.003 68, and --=. 003 68, or 4750 x • 003 68 


/ 69.6 7 y 

V2356/ 

—17.49 feet head. 

To Compute Elements of Dong Pipes. 


4 V 


3.1416 X d 2 


r. 27 3=p = »; (. +c + ci) AI =A; 


V2 gh 


\/ I + C + C l 


V Y 2 

(1.505 Xd-\-cl) — = d ir, 


This latter formula will only give an approximate dimension in consequence Of 

unknown element d, and also of C, as v = - - 

’ 3.1416 xd 2 

For Illustration, see Miscellaneous Illustration, page 556. 










HYDRAULICS. 


549 


To Compute Vertical Height of a Stream projected, from 
3 ?ipe of a, IFire-engine or 3 ?nmp. 

Rule. —Ascertain velocity of stream by computing volume of water 
running or forced through opening in a second; then, by Rule in Gravita¬ 
tion, page 488, ascertair height to which stream would be elevated if wholly 
unobstructed, which multiply by a coefficient for particular case. 

Example. —If a fire-engine discharges 14 cube feet of water through a pipe .75 
inch in diameter in one minute, how high will the water be projected, the pipe be¬ 
ing directed vertically? 

14X1728-^.4417 area of pipe, -=-12 ins. in a foot,-1-60 seconds — 76.07 feet ve¬ 
locity ; and as velocity of a stream of water from a vessel is but two thirds that due 

to its head, then 76.07 x — = 114.1 feet. 

2 

According to elements furnished by observation, coefficient in this case 
would be .57; hence, 114.1 x .57 = 65.04 feet. 

In great heights and with small apertures, coefficients should be reduced. 
In consequence of the varying elements and conditions of operation of fire- 
engines, it is difficult to assign a coefficient for them. Difference between 
actual discharge and that as computed by capacity and stroke of cylinder, 
as ascertained by Mr. Larned, 1859, was 18 per cent. = a coefficient of .82. 

A steam fire-engine of the Portland Company, discharging a stream 1.125 
ins. in diameter, through 100 feet 2.5 inch hose, gave a theoretical head, 
computed from actual discharge, of 225 feet, and stream vertically projected 
was 200 feet ; hence coefficient in this case was .88. 

Experiments by Preston Water Company in England gave for a discharge 
of 14.4 cube feet per minute, through a pipe .75 inch in diameter, a height 
of jet of 64 feet, from an actual head of no feet, a coefficient of .58, or from 
theoretical head, as computed in preceding example, .57. 

Cylindrical Ajutage. 

Mean coefficient as determined by Mariotte and Bossut = .003 066 square 
of effective head for cylindrical ajutages; hence, for conical, alike to that of 
an engine pipe, coefficient ranges from .72 to .9, or a mean of .81. 

By formula of D’Aubuisson, .003 047 h 2 = h'. 

Effective head, or h, in preceding example = 1 14. 1. Then 114.1—.003047 X 
114.1 2 = 114.1 — 39.67 = 74-43 .feet height of jet. 

Hence, fora conical or engine pipe, 74.43 X .81 = 60.29 feel, or a coefficient of .535. 

To Compute Distance a Jet of* Water will be projected, 
from a Vessel tlirongh. an Opening in its Side. 

B C, Fig. 13, is equal to twice square root of A 0 X 0 B. 

If s is 4 times as deep below A as a is, s will discharge 
twice volume of water that will flow from a in same time, 
as 2 is y/ of A s and 1 is yf of A a. 

Note.— Water will spout farthest when 0 is equidistant 
from A and B; and if vessel is raised above a plane, B must 
_ be taken upon plane. 

C T? Volumes of water passing through equal apertures in 

same time are as square roots of their depths from surface. 

Rule. —Multiply square root of product of distance of opening from sur¬ 
face of water, and its height from plane upon which water flows, in feet by 
2, and product will give distance in feet. 

Example.— A vessel 20 feet deep is raised 5 feet above a plane; how far will a jet 
reach that is 5 feet from bottom of vessel ? 

20 — sXs + S^ I 5°; and V I 5° X 2 = 24.495 feet. 








550 


HYDRAULICS. 


Velocity of a jet of water flowing from a cylindrical tube is determined to 
be .974 to .98 of actual to theoretic velocity, or = .82 of that due to height 
of reservoir. Hence volume of discharge through a cylindrical opening 
= .82 a V2 g k. 

Fig. 14. Jets d’Eau. (Fig. 14.) 


That a jet may ascend to greatest practicable height, 
communication with supply should be perfectly free. 

Short tubes shaped alike to contracted fluid vein, and 
conically convergent pipes, are those which give greatest 
velocities of efflux. Hence, to attain greatest effect, as in 
fire-engines, long and slightly conically convergent tubes 
or pipes should be applied. 

In order to diminish resistance of descending water, a 
jet must be directed with a slight inclination from vertical. 

Effect of combined causes which diminish height of a jet from that due 
to elevation of its supply can only be determined by experiments. Great 
jets rise higher than small ones. 

With cylindrical tubes, velocity being reduced in ratio of 1 to .82, and as 
heights of jets are as squares of these coefficients or ratios, or as 1 to .67, 
height of a jet through a cylindrical tube is two thirds that of head of 
water from which it flows. 



HC = h. H representing head of water , C coefficient , and h height of jet. 
worth .) 


When d = H - 4 - 

C L U __ ii _L_ 

it u ___ u » 

11 it_ tt » 

ti tt _ i t « 


300, C = .96. 
450, “ = .93. 
600, = .9. 

800, “ = .87. 
1000, “ = .83. 


When d = 

ii ti _ 

it it __ 
ti it ___ 
it it_ 


H -f- 1500, 0 =. 8. 
“ -T- 1800, “ = .7. 
“ -7- 2800, “ = . 6 . 
“ -f-3500, “ = .5. 
-^45oo, ‘‘ = .25. 


(Moles- 


FLOW OF WATER IN RIVERS, CANALS, AND STREAMS. 

Running Water .—Water flows either in a natural or artificial bed 
or course. In first case it forms Streams, Brooks, and Rivers ; in 
second, Drains, Cuts, and Canals. 

Bed of a water-course is formed of a Bottom and two Banks or Shores. 

Transverse Section is a vertical plane at right angles to course of the 
flowing water; Perimeter is length of this section in its bed. 

Longitudinal Section or Profile is a vertical plane in the course or thread 
o f current of flowing water. 

Slope or Declivity is the mean angle of inclination of surface of the water 
to the horizon. 

Fall is vertical distance of the two extreme points of a defined length of 
the flowing course, measured upon a horizontal plane, and this fall assigns 
angle for defined length of the course. 

Line or Thread of Current is the point where flowing water attains its 
maximum velocity. 

Mid-channel is deepest point of the bed in thread of current. Velocity is 
greatest at surface and in middle of current; and surface of flowing water 
is highest in current, and lowest at banks or shore. 

A River, Canal, etc., is in a state of permanency when an equal quantity 
of water flows through each of its transverse sections in an equal time, or 
when V, product of area of section , and mean velocity through whole extent 
of the stream , is a constant number. 



















HYDRAULICS. 


551 


To Compute Mean Deptli of Flowing Water. 

Rule. —Set off breadth of the stream, etc., into any convenient number of 
divisions; ascertain mean depths of these divisions; then divide their sum 
by number of divisions, and quotient is the mean depth. 

To Compute fYIean Area of Flowing Water. 

Rule i.— Multiply breadth or breadths of the stream, etc., by the mean 
depth or depths, and product is the area. 

2.—Divide the volume flowing in cube feet per second by mean velocity 
in feet per second, and quotient is area in sq. feet. 

To Compute Volume of Flowing Water. 

Rule. —Multiply area of the stream, etc., in sq. feet, by the mean velocity 
of its flow in feet, and product is volume in cube feet. 

To Compute HVlean "Velocity of Flowing "Water. 

Rule. —Divide surface velocity of flow in feet per second by area of the 
stream, etc., and quotient, multiplied by coefficient of velocity, will give 
mean velocity in feet. 

Mean velocity at half depth of a stream has been ascertained to be as .915 to 1, 
and at bottom of it as .83 to 1, compared with velocity at surface. Again, the ve¬ 
locity diminishes from line of current toward banks, and, to obtain mean superficial 
velocity, Vl _|_ U2 _i_ v 

— 1 - ! —- = .9151;; hence, 

n v j > * 

To Compute fVIea.ii "Velocity in -wliole Profile of a, !N"avi- 

galile River, etc., 

V -j- 1 —2 fV = velocity at bottom , and V-f- .5 — y/V = mean velocity. 

In rivers of low velocities multiply mean velocity by .8. 

Obstruction in Rivers. ( Molesworth.) 

^2 /A\ ^ 

-^ - - -f- .05 X y—J — 1 — R. v representing velocity in ins. per second previous 

to obstruction , A and a areas of river unobstructed and at obstruction in sq.feet, and 
R rise in feet. 

Illustration. —Velocity of obstructed flow of a river is 6 feet per second, and 
areas of section before and after obstruction are 100 and 90 sq. feet; what would 
be rise in feet? 

^2 /joo\^ 

1 = .664 X .232 .154 feet. 


58.6 1 '° 5 V 90 / 

Flow of Water in Lined. Clirinnels. 
/CD „ 1 


(Bazin .) 






C. D representing mean hydraulic depth in feet, F 
fall, or length of channel to fall of 1, x and 
y as per table, and C as per table p. 543. 



X 

y 


X 

Plastered. 


10.16 

Rubble Masonry... 


Cut Stone .... 


4-354 

Earth. 



y 

1.219 
.214 


For Sections of Uniform Area , as Canals, Sewers , etc. 2 D — v. A — 

area of flow in sq. feet, P wet perimeter of section, and D fall of stream per mile 
in feet. 

Illustration.— Area of transverse section of a sewer is 50 sq. feet, its wet perim¬ 
eter 20 feet, and its fall 5 feet per mile. 

x 2 x sj = V 25 = 5 feet. For Sections of Rivers. 12 D ^ = v. 

Illustration. —Assume area 500 sq. feet, wet perimeter 200, and fall 5 feet per mile. 

















552 


HYDRAULICS. 


Hydraulic Radius or Mean Depth is obtained by dividing area of trans¬ 
verse section by wet perimeter, both in feet. 

To Compute Fall per IMile for a req.ri.ired. ZVIean Velocity-. 


tV X 12 


2 r — D. r representing hydraulic radius in ins. 


Upper surface of flowing water is not exactly horizontal, as water at its surface 
flows with different velocities with respect to each other, and consequently exert 
on each other different pressures. 

If v and Vi are velocities at line of current and bank of a stream, the difference 


of the two levels is 


»-*— Vi - 

2 9 


— h. 


Illustration. —If v = 5 feet, and Vi 9 -u; then 


■• 9 X 5 4-75 


2? 


64-33 


.0738 foot. 


A velocity of 7 to 8 ins. per second is necessary to prevent deposit of slime and 
growth of grass, and 15 ins. is necessary to prevent deposit of sand. 

Maximum velocity of water in a canal should depend on character of bed of the 
channel. 


Thus, Mean Velocity should not exceed per second over 


Fine clay.6 ins. 

A slimy bed.8 “ 


Common clay.6 “ 


River sand... 
Small gravel. 
Large shingle. 


1 ft. Broken stones. 

1 “ Stones. 

3“ Loose rocks... 


4 ft. 
6 “ 
r r\ ^ 


To Compute Velocity- of Flow or Discharge of Water in 
Streams, Pipes, Canals, etc. 


1. When Volume discharged per Minute is given in Cube Feet , and Area of 
Canal, etc., in Sq. Feet. Rule. —Divide volume by area, and quotient, di¬ 
vided by 60, will give velocity in feet per second. 

2. When Volume is given in Cube Feet , and Area in Sq. Ins. Rule. —Di¬ 
vide volume by area; multiply quotient by 144, and divide product by 60. 

3. When Volume is given in Cube Ins ., and Area in Sq. Ins. Rule.— Di¬ 
vide volume by area, and again by 12 and by 60. 

To Compute Flow or Volume of Discharge. 

1. When Area is given in Sq. Feet. Rule. —Multiply area of flow by its 
velocity in feet per second, and product, multiplied by 60, will give volume 
in cube feet per minute. 

2. When Area is given in Sq. Ins. Rule. —Multiply area by its velocity, 
and again by 60, and divide product by 144. 

Note i.— Velocities and discharges here deduced are theoretical, actual results de¬ 
pending upon coefficient of efflux used. Mean velocity, however, as before given, 

page 529, may be taken at f 1 g .673 = 5.4 feet, instead of 8.02 feet. 

2.—As a rule, with large bodies, as vessels, etc., their floating velocity is some¬ 
what greater than that of flow of water, not only because in floating they descend 
an inclined plane, formed by surface of the water, but because they are but slightly 
affected by the irregular intimate motion of water: the variation for small bodies 
is so slight that it may be neglected. 


To Compute FCeiglit of Head of Flowing Water. 


When Volume and Area of Flow are given in Feet. Rule. —Divide vol¬ 
ume in feet per second by product of area, and f- coefficient for opening, and 
square of quotient, divided by 64.33, will give height in feet. 


Example. —Assume volume 266.48 cube feet, area 40 sq. feet, and C = .623. 


Then 


1 266.48 

vt° X f-623 




64-33 


257-28 

64-33 


-- 4 feet. 

















HYDRAULICS. 


5S3 


Submerged or Drowned Orifices and. Weirs. 

When, wholly submerged (Fig. 15). —Available pressure at any point in depth 

of nrifir.fi is ennal t.o difference nf pressure nn 

Fig, 15-_ 

Whence, C V2 g h — v, and C a \/ 2 gh = \. 
a representing area of sluice in sq.feet. 
Illustration. — Assume opening 3 feet by 5, 



Then, 5 X 3 X 5 V64.33 X 4 = 7-5 X 16.04 = 
120.3 cube feet per second. 


When partly submerged (Fig. 16). h' — h = d = submerged depth, and h — 
Pig j6 h" = d' = remaining portion of depth; whence 



d' -\-d — entire depth, and 

C ifiVg (d y/h + f h y/h — h" fill") = V. 
Illustration. — Assume opening as above, h — 
4 feet , h' = 6, h" — 3, and C = . 5. Then d — 6 — 
—^ 4 ~ 3 " 2 feet. 


Then .5 X 5 X 8.02 (2 y /4 + f X 4 V 4 — 3 V3) 
— 20.05 X 5-869 = 117.67 cube feet per second. 


Fig. 17. 





Tl.1 


When drowned (Fig. 17). 

C l V2 g h (d -J- •§■ h) = Y. 

ifjzffgljg Illustration. — Assume opening as above, 
h—^feet^ d — 2, and C = .52. 

Then, .52 X 5 X V64.33 X 4 X (2 + f 4) = 2.6 
X 16.04 X 4-66 = 194.34 cube feet per second. 




CANAL LOCKS. 
Single Locks. 


When a fluid passes from one level or reservoir to another, through an 
aperture covered by the fluid in the latter, effective head on each point of 
aperture, and consequently head due to velocity of efflux at each instant, is 
the difference of levels of the two reservoirs at that instant. 


Hence C a fi -2. gh' — V per second, h' representing difference of levels. 


To Compute Time of Willing and Discharging a Single 

Lock.—Wig. IS. 


When Sluice in Upper Gate is entirely under Water , and above Lower Level. 


Ah' 


: time of filling up to centre of sluice. 


Fig. 18. 


C ay/zgh 

h representing height of centre of sluice, in, upper 
gate from surface of canal or reservoir , and h' height 
of centre of sluice in upper gate from lower sur- 
face, or water in the lode or river , all in feet y and 

2 Ah time of filing the remaining space , 




Upper 

level 


. z^=k 4JSluice Xower level [ 






!m 


C « V2 gh 

where a gradual diminution of head of ivater occurs. 






Consequently, + 2 — t time of filling a single loch. 

C a fi 2 gh 


When Aperture or Sluice in Loioer Gate is entirely under Water , and above 

Loiver Level. 2 A + A _ time of emptying or discharging it. a' representing 
C a' fi^Ug 

area of lower sluice. 


T A 


























































554 


HYDRAULICS. 


Illustration. —Mean dimensions of a lock, Fig. 18, are 200 feet in length by 24 
in breadth; height of centre of aperture of sluice from upper and lower surfaces is 
5 feet; breadth of both upper and lower sluices is 2.5 feet; height of upper is 4 feet, 
and of lower—entirely under water—5 feet; required the times of filling and dis¬ 
charging. 

h — s, h' = 5, A = 200 X 24 = 4800, C = .545, a = 4 X 2.5 = 10, a' = 5 X 2.5 = 12.5. 
4800 X 5 24 000 


= 245.59 seconds— time of filling lock up to centre of 
48 000 


.545 X 10 X f 2 gh 97 - 7 2 

, . . 2 X 4800 X 5 4° „ _ ... . 

sluice ; and -- - -- = --= 491.18 seconds = time of filling remain- 

.545X10X^2^/1 97 - 7 2 

ing space , or Lock above centre of sluice , and 245.59 -f- 491.18 = 736.77 seconds , whole 
time. 


(5 -f- 2 X 5) X 4800 72 000 


0r ’ / 
.545X 10 x V zgh 

30358.08 


== 736.77 sec. = time of filling. 


54-7 


97.72 

= 554.9 seconds = time of discharging. 


2 X 4800 V 5 + 5 
•545 X 12.5 x Vvg 


When Aperture or Sluice in Upper Gate is entirely under Water and beloio 
2 A Vh — h' 


Lower Level. 


CaVz g 


- — time of filling lock. 


When Sluice in the Lower Gate is in part above Surface of Lower Ljevel 
and in part below it. —- 2 . A ( -— time of dis- 


C b V2 g [dfh 4- h' — - -f- d' Vh -f- h'^j 


charging, d and d' representing distances of part of aperture above and of below 
surface of lower water , b breadth of aperture, and h and h’ as before. 

Illustration.— Assume sluice in preceding example to be 1 foot above lower 
level of water, or that of lower canal; what is time of discharge of lock, distance 
of part of aperture 1 foot and of that below surface of water 4 feet? 

2 X 4800 (5 -f 5) 96 000 


•545 X 2.5 X 8.02 [1 X V5 + 5 — (i-i-2) + 4 X V5 + 5] 

96 000 


10.93 X (3.082-f-12.65) 


= 558.3 seconds. 


i7i-95 

Double Lock. (J. D. Van Burcn , Jr.) 

A double lock is not a duplication of a single lock in its operation, for in 
lower chamber supply of water 

is from upper one, having no ” ” * '£• I 9 - 

influx, instead of a uniform sup¬ 
ply flowing directly from sur¬ 
face level of canal or feeder. 

Operation, therefore, of a 
double lock is complex, addition 
to formula for a single lock be¬ 
ing that of discharging of water 
in upper lock to till lower, the 
head of water gradually decreas- 
ing in the chamber, which is 
closed from upper reach during discharge into lower. 

To Compute Time required for 'Water to Fall from 
XJpper to Uniform Water Level. 

1. c (V/+ V 2 h — V2/1 — 2d) = t. A representing horizontal area of lock, 

and a area of sluice opening , both in sq. feet, C coefficient of discharge ~ . 545 for 
openings with square arrises , g acceleration of gravity , f depth of centre of sluice 






































HYDRAULICS. 


555 


below uniform level , h depth of centre sluice opening below upper water level , and d 
height of centre of sluice above lower water level , alt in feet, and t time for water to 
fall from upper to uniform water level , in seconds. 

Illustration.—A = 2000 sq. feet; € = .545; a = 5; /= 6; 7 i = 14; andd = 
2 feet. (Fig. 19.) 


Then, 


■545 X 5 X 5-67 15-45 


X 7.74 — 4.9 = 367.6 seconds. 


. A V/ 

2 . Ifd = o; -- y—t 

C afg 


2000 x V s 


5660 


366.34 seconds. 


-545X5X5-67 15-45 

Note.— /is never greater than Z (lift in feet); it is equal to l when d — o; / 2 is 
equal to l when f t =0, never greater. In each case it is the unbalanced head above 
sluice, however far below the lowest water level the sluice is. 

To Fill XJpper Lock or Empty Lower. 

To fill upper lock or empty lower, when the sluice is below the lowest water-line, 
in either case, takes the same time; for the head diminishes at the same rate, one 
from the upper surface, the other from the bottom. 

A'v/a/ _ ^ Here, / being below lowest water level of lock = 8 feet, as d = o, 
(2a fg 

and f— whole lift 


2000 V 2 X 8 8000 

•545 X 5 X 5-67 ~ 15-45 


= 517.8 seconds. 


To Discharge a like Volume under a Constant Head. 


A yff 


C aV2 g G 2 g 


= t. 


•545 X 5 V 64-33 




8 


258.9 seconds , 


Or, one half the time given by preceding case. 


The times deduced by preceding formulas are in the following proportions in 

, / V 2 / 1 

order, as 1 : v 2 : — , or 1 : v 2 *- —7- • 

2 V 2 

If sluice of upper lock, through which it is filled, is above lowest water level, 
then, by combining formulas 3 and 4, the time is thus deduced. 

To Jill from Lowest Water Level of said Lock to Level of Centre of Sluice. 

A f f 

5. - v — — t'. f representing height of centre of sluice above said lowest water 

C a V 2 g level. 

To fill remaining Portion of Lock above Sluice. 

6. 2 ^ ^^ = t". f" representing depth below upper water level of centre of 
C a V 2 g 

sluice or remaining portion of lift. Hence, t' -j- t" = 




C af2g 

To fill Lower Lock tinder Constant Head from Upper Caned Level. 
A Vh ^ 


7 - ~ 


d 2 Vh —f 


C a V 2 g ■ 


H 


h V h 

8. If both lifts are the same, li — f=l, and — A ^ (24-^- — 2 a /— \ = t. 

CaVig' h V> 2 / 

If lower lock is filled from upper one under a constant head, when latter is drawn 
down to lowest level, formula 7 will apply by making h —f and 

- A — (2 V /+ which is identical with 7, for/=/ 2 and d —f the cases 

C a V 2 g ' VJ J 

being the same. 


















556 


HYDRAULICS. 


MISCELLANEOUS ILLUSTRATIONS. 

1. If external height of fresh water, at 6o° above injection opening in condenser 
of a steam-engine, is 3 feet, and the indicated vacuum at 23 ins., velocity of water 
ilowing into condenser is thus determined. (Formulapage 532.) 

v V 2 g (h -f- /«.'). h' representing height of a column of water equivalent to press¬ 
ure of atmosphere within condenser. 

Assuming mean pressure of atmosphere == 14.7 lbs. per sq. inch, height of a column 
of fresh water equivalent thereto — 33.95 feet. 

Then, if 1 inch = .4912 lbs., 23 ins. = 11.3 lbs.; and if 14.7 lbs. — 33.95 feet, 11.3 
lbs. == 26.1 feet. 

Hence v = Vz g (3 + 26.1)1=43.27 feet, less retardation due to coefficient of both 
influx and efflux. 

2. What breadth must be given to a rectangular weir, to admit of a flow of 6 cube 
feet of water, under a head of 8 ins. ? (Formula page 533.) 

—- 6 ■ - =-— = 2.21 feet. 

^X-(>2sV2g 66 * 4*7 X 6.55 

3. It being required to ascertain volume of water flowing in a stream, a tem¬ 
porary dam is raised across it. with a notch in it 2 feet in breadth by 1 in depth, 
which so arrests flow that it raises to a head of 1.75 feet above sill of notch; what 
is volume of flow r per second.? (Formula gage 533.) 


C = .635. - X -635 X 2 X i-75 V2 g X j.75 — 1.481 X 10.6: 

3 


: 15.7 cube feet. 


4. A rectangular sluice 6 feet in breadth by 5 in depth, has a depth of 9 feet of 
water over its sill, and discharges, as per example page 535, 380.95 cube feet per 
second; what is velocity of flow? (Formula page 535.) 

380.95 380.95 

-—-— — ; =- = 12.7 feet. 

6 X (9 — 4) 30 ' 

2 , — fh 3 \/h' 3 

If volume was not given: — CV2 gX ~— : - 7-, — -~v. C = .62?. 

3 h — h 

Then — X .625 X 8.02 X ■ V ^ 729 -^^=3.341 X 3-8 = 12.7 feet. 

3 9 — 4 

5. If a river has an inclination of 1.5 feet per mile, is 40 feet in breadth with nearly 
vertical banks, and 3 feet depth; what is volume of its discharge ? (Formula p. 542.) 

Perimeter 40 + 2 X 3 = 46 feet; hydraulic mean depth -= 2.61 feet; 

46 

a = 120 feet; C per table , page 543 .for assumed velocity of 2. 5 feet = .0075. 


Then 


V: 


x r - 

~ 75 x 528q x 4 g X 64.33 X 1.5 = V.0659 X 96.-5 = 2.52 feet velocity. 
Hence 120 x 2.52: 


302.4 cube feet. 

6. WhfR is head of w T ater necessary to give a discharge of 25 cube feet of water 
per minute, through a pipe 5 ins. in diam. and 150 feet in length? (Formulap. 548.) 

Tabular number for diameter 5 ins., page 547, — 263.87. 

- 2 

Then 263.87 = 25 = 111.3, and 150 = 111.3 = 1.35 feet. 

If this pipe had 2 rectangular knees or bends, what then would be head of water 
required? (Formulapage 545.) 

C, page 545, /or- — = .984, area of 5 ins. — .13 6 feet, and --—^ = 60 = 3.06 feet 
2 -136 

*3 06^ 

velocity. Then ~—• X -984 X 2 = .2863, which, added to 1.35 = 1.64 feet. 

04-33 

By formulas foot of page 548, C = .o24, and c .505 velocity = 3.06 feet; head = 
1.49 feet, and volume 26.38 cube feet. 

7. If a stream of water has a mean velocity of 2.25 feet per second at a breadth 
of 560 feet, and a mean depth of 9 feet, what will be its mean velocity when it has 
a breadth of 320 feet, and a mean depth of 7.5 feet? (Rulepage 548.) 

560X9 X 2.25 1134° ^ * 

— =4.725 feet. 


320 X 7-5 


2400 

















HYDRAULICS. 


557 


8. What volume will a pipe 48 feet in length and 2 ins. in diameter, under a head 
of 5 feet, deliver per second? {Formulapage 547.) 

Tabular number for diameter 2 ins., page 547, = 26.69. 

3.1. Then —-^ = 8.61, which -7-60 = .143 cube feet. 


V 


48 

5 


3-i 


d 


If this pipe had 5 curves of 90° with radii —— - —.5; what would be its dis- 

2 r 4 

charge per second ? 


V = .i43; a = 2-4-144 = .0139; C per table =•—: 

2 r 

_ 90 0 10.29” 

Then .294X^5 X-^- 


.294; 


•i43 


139 


10.29 f ee t- 


33 


.147 X 1.64 = 241, which X S for 5 curves = 1.2 = 


height due to resistance of curves, h = 5 — 1.2 = 3.8. 

Hence, if V2 # 5 =. 143; V2 # 3.8 = . 125 cube feet. 

9. If a slide stop valve, set in a cylindrical conduit 500 feet in length and 3 ins. in 
diameter, is raised so as to close .625 of conduit; what volume will it discharge 
under a head of 4 feet? {Formulapage 546.) 

C for conduit —. .5, for friction .025, and for slide valve .375 open, table, page 545, 
5.52, d — . 25, and a == 7.07 sq. ins. 

2 g h 16.06 

i ben- :: ■ - . - = —- r— = 2.13 feet velocity, and 


\J ( I +*5+S.Sa+.o 35 -fg) 

2.13 X 12 X 7.07 = 180.71 cube ins. 


V( 7-02 + 50) 


10. If a single lock chamber is 200 feet in length by 24 in breadth, with a depth 
of 10 feet, centre of upper gate, which is 4 feet in depth by 2.5 in breadth, is at 
middle of depth of chamber, lower gate, 5 feet in depth by 2.5 in breadth and wholly 
immersed; what is time required for filling and discharging it? {Formulap. 553.) 

C = .6i5, 7t = 5, = 5, A = 200 X 24 = 4800, (£ = 4X2.5 = 10, and o'= 5 

X 2.5 = 12.5 


(2X5 + 5) 4 8o ° 


72 000 


.615X10^/64.33x5 no.27 

2 X 4800 X V 5 + 5 30 336 - 


652.8 seconds time of filling. 


491.4 seconds time of emptying. 


, / — — 61.73 

.615 x 12.5 v 2 g 

11. In a moderately direct and uniform course of a river, the depths and velocities 
are as follows; what is the volume of its flow' and what its mean velocity ? {p. 551.) 

Area of profiles = 5X3 + 
12X6 + 20X11 + 15X8 + 
7 X 4 = 455 sq.feet. 

15 X 1.9 + 72 X 2.3 + 220 X 2.8 + 120 X 2,4 + 28 X 2.1 = 1156.9 cube feet volume, 
ji 56-9 — 54 feet velocity. 



Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Distances. 


12 

20 

15 

7 

Depths. 

••• 3 

6 

II 

8 

4 

Mean velocity... 

... 1.9 

2-3 

2.8 

2.4 

2. I 


and 


455 


IVIirier’s Incli. 

A u Miner’s inch ” is a measure for flow of water, and is an opening one 
inch square through a plank two inches in thickness, under a head of six 
inches of water to upper edge of opening. 

It will discharge 11.625 U. S. gallons water in one minute. 

Theoretical IP under different Heads. 

Heads in feet J100 190 I80 170 |5o 1 50 I40 130 1 20 I.15 110 I 51 3| 1 

Ins. per IP... | 3.25) 3.611 4.061 4.641 5.41I 6.51 8.12I10.8I16.2I21.6I32.5I65I108I325 

Water Inch (Pouce d'eau ).—Circular opening of 1 inch in a thin plate is 
equal to a discharge of 19.1953 cube meters per 24 hours. 

3 A* 



















558 


HYDRODYNAMICS. 


HYDRODYNAMICS. 


Hydrodynamics treats of the force of action of Liquids or Inelastic 
Fluids, and it embraces Hydraulics and Hydrostatics: the former of 
which treats of liquids in motion, as flow of water in pipes, etc., and 
latter of pressure, weight, and equilibrium of liquids in a state of rest. 


Fluids are of two kinds, aeriform and liquid, or elastic and inelastic, 
and they press equally in all directions, and any pressure communicated 
to a fluid at rest is equally transmitted throughout the whole fluid. 

Pressure of a fluid at any depth is as depth or vertical height, and 
pressure upon bottom of a containing vessel is as base and perpendicu¬ 
lar height, whatever may be the figure of vessel. Pressure, therefore, 
of a fluid, upon any surface , whether Vertical , Oblique, or Horizontal , is 
equal to weight of a column of the fluid, base of which is equal to sur¬ 
face pressed, and height equal to distance of centre of gravity of sur¬ 
face pressed, below surface of the fluid. 

Side of any vessel sustains a pressure equal to its area, multiplied by 
half depth of fluid, and whole pressure upon bottom and against sides 
of a vessel is equal to three times weight of fluid. 

Pressure upon a number of surfaces is ascertained by multiplying 
sum of surfaces into depth of their common centre of gravity, below 
surface of fluid. 

When a body is partly or wholly immersed in a fluid, vertical press¬ 
ure of the fluid tends to raise the body with a force equal to weight of 
fluid displaced; hence weight of any quantity of a fluid displaced by a 
buoyant body equals weight of that body. 

Centre of Pressure is that point of a surface against which any fluid 
presses, to which, if a force equal to whole pressure were applied, it 
would keep surface at rest. Hence distance of centre of pressure of 
any given surface from surface of fluid is same as Centre of Percussion. 


Centres of Pressure. 

Parallelogram , Side , Base, Tangent , or Vertex of Figure at Surface of Fluid , is at 
.66 of line (measuring downward) that joins centres of two horizontal sides. 

Triangle, Base uppermost, \s at centre of a line raised vertically from lower apex, 
and joining it with centre of base; and Vertex uppermost, it is at .75 of a line let 
fall perpendicularly from vertex, and joining it with centre of base. 

Right-angled Triangle, Base uppermost , is at intersection of a line extended from 
centre of base to extremity of triangle by a line running horizontally from centre 
of side of triangle. Vertex or Extremity uppermost, is at intersection of a line ex¬ 
tended from the centre of the base to the vertex, by a line running horizontally from 
• 375 of side of triangle, measured from base. 


Trapezoid, either of parallel Sides at Surface , - , X a = d. b and b' renre- 

2 0 4 0 r 

senting breadths of figure, d distance from surface of fluid, and a length of line join¬ 
ing opposite sides. 


Circle*, at 1.25 of its radius, measured from upper edge. 


'X 7) T 

Semicircle, Diameter at Surface of Fluid, - - = d. 

16 


r representing radius of circle 


and p = 3.1416. 


„ , 15 pr — 22 r 

Diam. downward, — - -— = d. 

i2 p — 16 



HYDRODYNAMICS. 


559 


Side, Base, or Tangent of Figure below Surface of 

Fluid. 


Rectangle or Parallelog'm. - X 

3 ft 


h'3 — /i3 


.d ; or. 


3 m o -f- m 2 


, ?r- 


d: and — = d". 
3 o 


ft" ' 3 0 

h and h' representing depths of upper and lower surfaces of figure and d depth, 
both from surface of fluid, m half depth of figure, o depth of centre of gravity of 
figure from surface of fluid, d' distance from upper side of figure, and d" distance 
from centre of gravity. 

I 2 

■— d . Base Uppermost. 


I 2 4 - 18 o 2 

Triangle. — Vertex Uppermost. —- -_ g • 


18 o ’ 18 o 

Z 2 +i8o 2 

-— —— d. I representing depth of figure, d distance from surface of fluid upon 

a line from vertex to centre of base, and d' distance from centre of gravity of figure. 


Circle. 


4 o 2 -\-r 2 
4 o 


d, or —-== distance from centre of circle. 


1 2 j5 £2 

Semicircle. — Diam. Horizontal and Upward or Downward. - \-o = d: 

4 o q p o 

3 pi — \l , \l , l 2 16 l 2 

-= a ; -= a , and-, — = c. d representing distance from 

3 P 3 P 4 o 9 p o 

surface of fluid, d' distance of centre of gravity from centre of arc, d" distance of 
centre of gravity from diameter when it is uppermost, and c centre of pressure. 


Pressure. 

To Compute Pressure of a Fluid upon Bottom of its 

Containing Vessel. 

Rule. —Multiply area of base by height of fluid in feet, and product by 
weight of a cube foot of fluid. 


To Compute Pressure of a Fluid upon a ''Vertical, In¬ 
clined, Curved, or any Surface. 

Rule.— Multiply area of surface by height of centre of gravity of fluid 
in feet, and product by weight of a cube foot of fluid. 

Example i. —What is pressure upon a sloping side of a pond of fresh water io feet 
setuare and 8 feet in depth'? 

Centre of gravity, 84-2 = 4 fei from surface. Then io 2 X 4 X 62.5 = 25 000 lbs. 

2. —What is pressure upon staves of a cylindrical reservoir when filled with fresh 
water, depth being 6 feet, and diameter of base 5 feet? 

5X3-1416 — 15.708 feet curved surface of reservoir, which is considered as a plane. 

15.708 X 6 X 6 - 4 - 2 = 282.744, which X 62.5 = 17 671.5 lbs. 

3. —A rectangular flood-gate in fresh water is 25 feet in length by 12 feet deep; 
what is pressure upon it? 

25 X 12 X 12-f- 2 = 1800, which X 62.5 = 112 500 lbs. 

When water presses against both sides of a plane surface, there arises from 
resultant forces, corresponding to the two sides, a new resultant, which is 
obtained by subtraction of former, as they are opposed to each other. 

Illustration. —Depth of water in a canal is 7 feet; in its adjoining lock it is 4 
feet, and breadth of gates is 15 feet; what mean pressure have they to sustain, and 
what is depth of point of its application below surface? 

7 X 15 = 105, and 4 X 15 = 60 sq.feet. (105 X - —60 x 2) X 62.5 = 1546.875 lbs., 

2 

mean pressure. 

Then 1546.875 = 62.5 = 247.5 =cube feet pressing upon gates upon high side, and 
247. 54-15X7 = 2.35 feet =,.depth of centre of gravity of mean pressure. 

To Compute Pressure on. a Sluice. 

Awd = P, and CP = P'. A representing area of sluice in sq.feet, w weight of 
water per cube foot, d mean depth of sluice below surface, in feet, P pressure on sluice, 
and P' power required to operate it, both in lbs. 

C = .68 when sluice is of wood, and .31 when of iron. 















560 


HYDRODYNAMICS. 


Example. —What is pressure on a sluice-gate 3 feet square, its centre of gravity 
being 30 feet below surface of a pond of fresh water ? 

3 X 3 X 30 = 270, which X 62.5 = 16 875 lbs. 

To Compute Pressure of a Column of a Fluid, per 

Sep Inch. 

Rule. —Multiply height of column in feet by weight of a cube foot of 
fluid, and divide product by 144; quotient will give weight or pressure per 
sq. inch in lbs. 

Note. —When height is given in ins., omit division by 144. 

PIPES. 

To Compute required Thickness of a JPipe. 

Rule. —Multiply pressure in lbs. per sq. inch by diameter of pipe in ins., 
and divide product by twice assumed tensile resistance or value of a sq. 
inch of material of which pipe is constructed. 

By experiment, it has been found that a cast-iron pipe 15 ins. in diameter, and 
.75 of an inch thick, will support a head of water of 600 feet; and that one of oak, 
of same diameter, and 2 ins. thick, will support a head of 180 feet? 

Example i. —Pressure upon a cast-iron pipe 15 ins. in diameter is 300 lbs. per sq. 
inch; what is required thickness of metal? 

300 X 15 = 4500, which -T- 3000 X 2 = .75 inch. 

Note. —Here 3000 is taken as value of tensile strength of cast iron in ordinary 
small water-pipes. This is in consequence of liability of such castings to be im¬ 
perfect from honey-combs, springing of core, etc. 

2.—Pressure upon a lead pipe 1 inch in diameter is 150 lbs. per sq. inch; what is 
required thickness of metal? 

Here 500 is taken as value of tensile strength. 

150 X 1 — 150, wh ich - 4 - 500 X 2 =. 15 inch. 

Cast-iron Pipes. 

To Compute Thickness, etc., of Flanged Pipes. 



For 

75 

lbs. 

Pressure. 


For 100 lbs. Pressure. 


.025 D - 


•25 


T 

•03 

D + 

•3 = 

:T 

•°3 

D - 


3 


-. t 

.035 D -f 

•45 = 

t 

•05 

D - 

- I 

i 5 


— l 

•05 

1 ) + 

1.15 : 

l 

•03 

D h 


35 


= / 

.04 

D + 

.6 r= 

■f 

1.05 

D -| 

- 4 

•25 

d + 1.25 

: 0 

I. I 

D + 

5 X d -f- 1.5 = 

0 

1.05 

D-f 2 

X 

d- f 1 

- O' 

I. I 

D + 

"2.5X^+1.4 

o' 


.7 D 

+ 

2.2 

= n; 

A x p- 4 -n , 

--- — a, and 

J 

“ + C — d. 






4COO 


V 

7854 



D representing diam. of pipe, T thickness of metal, t thickness and l length of boss, 
f thickness of flange, o diam. of flange. 0' diam. of centres at bolt holes, and d diam. 
of bolts, all in ins.; A area of pipe and a area of bolt at base of its thread, in sq. ins., 
p pressure in lbs. per sq. inch, and C a coefficient due to diam. of bolt. 

Thus, diam. .125+ .032, .25+ .064, .5+ .107, 1 + .16, 1.5+ .214, and 2-}- .285. 

Illustration. —What should be dimensions of a flanged pipe, 10 ins. in diameter, 
for a pressure of 100 lbs. per sq. inch? 

.7 X 10 -)- 2.2 = 9-2 = 10 number of bolts, and diam. 10 ins. = 78.54 ins. area = A. 


78.54 X 100 4 -10 
4000 


.,9635, 


' C = \/.25 = .5 ; hence, .5-f. 107 = 


.607 = .625 lbs. diameter of bolls ; .03 x 10-f-. 3 — .6 — thickness of metal; .035 X 10 
.45 = .8 = thickness of flange ; .05 x 10 -f -1.15 — 1.65 = length of boss ; .04 X 10 
+ .6 = 1 = thickness o f flange; 1. 1 X 10 + 5 X .625 + 1.5 = 15.625 = diameter of 
flange ; and 1.1 X 10 + 2.5 X -625 + 1.4 = 13.9625 = diameter of bolt holes. 


For Tables of Cast-iron Pipes, see page 132 . 















HYDRODYNAMICS, 


561 


To Compute Elements of "Water-pipes. 

.000124 5 P d -}- C = t ; or, .000054 H d -f- C = <; .4336 H — P; and 

D 2 — d 2 X 2.45 = W. P representing pressure of water in lbs. per sq. inch, D and d 
external and internal diameters of pipe, and t thickness of metal, all in ins., C coeffi¬ 
cient for diameter of pipe, and H head of water in feet. 

C = .37 for pipes less than 12 ins. in diameter, .5 from 12 to 30, and .6 from 30 to 50. 

To Compute "W r eiglxt of Pipes. 

To Diameter add thickness of metal, multiply sum by 10 times thickness, 
and product will give weight in lbs. per foot of length. 

Weight of Faucet end is equal to 8 ins. of length of pipe. 


Hydrostatic Press. 

To Compute Elements of a Hydrostatic Press. 


PZA 
l' a 


= W; 


W l'a 
P l 


— A; 


Wl'fl 
l A 



PAZ 

\VZ' 


= a. 


P representing power or press¬ 


ure applied , W weight or resistance in lbs., I and V lengths of lever and fulcrum in 
ins. or feet, and A and a areas of ram and piston in sq. ins. 

Illustration. —Areas of a ram and piston are 86.6 and 1 sq. ins., lengths of lever 
and fulcrum 4 feet and 9 ins., and power applied 20 lbs.; what is weight that may¬ 
be sustained? 


20 X 4 X 12 x 86.6 83 136 

9X1 ~ 9 


9237.3 lbs. 


To Compute Thickness of IVEetal to Resist a given 

Pressure. 

Rule. —Multiply pressure per sq. inch in lbs. by diameter of cylinder in 
ins., and divide product by twice estimated tensile resistance or value of 
metal in lbs. per sq. inch, and quotient will give thickness of metal required. 

Example.— Pressure required is 9000 lbs. per sq. inch, and diameter of cylinder is 
5.3 ins.; w T hat is required thickness of metal of cast iron? 

Value of metal is taken at 6000. 9 °°° 5 - 3 __ 47 7 °° _ t - ns . 

6000X2 12000 J 


Values of Different Metals in Tons. ( Molesworth .) 

Cast iron.41 | Gun metal.22 | Wrought iron.. .14 | Steel.06 

Hydraulic Ram. 

Useful effect of an Hydraulic Ram, as determined by Eytelwein, varied 
from .9 to .18 of power expended. When height to which water is raised 
compared to fall is low, effect is greater than with any other machine; but 
it diminishes as height increases. 

Length of supply pipe should not be less than .75 of height to which 
water is to be raised, or 5 times height of supply; it may be much longer. 


To Compute Elements. 


.00113 Vfc = H>; — = Y ; 1.45 v^ = D; .75 and|x^ = 

efficiency. V and v representing volumes expended and raised, in cube feet per 
minute , h and h' heights from which water is drawn and elevated in feet , D and d 
diameters of supply and discharging pipes in ins., and IP effective horsepower. 

Illustration.— Heights of a hill and of elevation are 10 and 26.3 feet, and vol¬ 
umes expended and raised per minute are 1.71 and .543 cube feet. 


.00113 X 1.71 X 10 = .0193 IP ; 


81 X.0193 


= 1.71 cube feet; 1.45 ffi.pi — id 


f ^ f J O ^ 26 Q 

ins.; .75 \/i.7 1 —-975 ins.; and — x--— — .696 efficiency. 

o i • i X 10 










HYDRODYNAMICS. 


562 


Results of Operations of Hydraulic Rams. 


Strokes 

Fall. 

Eleva- 

Water 

Useful 

Strokes 

Fall. 

Eleva- 

Water 

Useful 

per M. 

tion. 

Expen’d. 

Raised. 

Effect. 

per M. 

tion. 

Expen’d. 

Raised. 

Effect. 

No. 

Feet. 

Feet. 

C. Ft. 

C. Ft. 


No. 

Feet. 

Feet. 

C. Ft. 

C. Ft. 


66 

10.06 

26.3 

1.71 

•543 

•9 

15 

3.22 

38.6 

1.98 

.058 

•35 

50 

9-93 

38.6 

1-93 

.421 

•85 

IO 

1-97 

38.6 

1.58 

.014 

.18 

36 

6.05 

38.6 

i -43 

. 169 

•75 

— 

22.8 

196.8 

•38 

.029 

.67 

3 i 

5.06 

38.6 

I. 29 

•113 

.67 

— 

8.5 

52-7 

2 

.186 

•57 


Note. — Volume of air vessel = volume of delivery pipe. One seventh of water 
may be raised to about 4 times head of fall, or one fourteenth 8 times, or one twenty- 
eighth 16 times. 


WATER POWER. 

Water acts as a moving power, either by its weight or by its vis viva , and 
in latter case it acts either by Pressure or by Impact. 

Natural Effect or Poicer of a fall of water is equal to weight of its volume 
and vertical height of its fall. 

If water is made to impinge upon a machine, the velocity with which it 
impinges may be estimated in the effect of the machine. Result or effect, 
however, is in nowise altered; for in first case P =zY w h, and in latter = 

v 2 

— Y w. V representing volume in cube feet w weight in lbs., and v velocity 
of flow in feet per second. 

62.5 V h = P, and 3.2* a flli = V. P representing pressure in lbs ., a area of open¬ 
ing in sq. feet , and h height of flow in feet per second. 


To Compute Power of a Fall of Water. 

Rule. —Multiply volume of flowing water in cube feet per minute by 
62.5, and this product by vertical height of fall in feet. 

Note.— When Flow is over a Weir or Notch , height is measured from surface of 
tail-race to a point four ninths of height of weir, or to centre of velocity or pressure 
of opening of flow. 


When Flow is through a Sluice or Horizontal Slit, height is measured from sur¬ 
face of tail-race to centre of pressure of opening. 

Example.—W hat is power of a stream of water when flowing over a weir 5 feet 
in breadth by 1 in depth, and having a fall of 20 feet from centre of pressure of flow? 

By Rule, page 533, — 5XiV2^fiX -625 = 16.68 cube feet per second. 

3 

16.68 X 60 X 62.5 X 20 = 1 251 000 lbs., which -4- 33 000 = 37.91 horses' power. 

Or, .1135 V h — theoretical IP. h representing height from race in feet. 

Illustration. —If flow of a stream is 17.9 cube feet per second, to what height 
and area of flow of 1 foot in depth should it be dammed to attain a power of 10 
horses. 


33 °°° " X ' I ° = 5500 lbs. per second, and ^°- = 88 cube feet per second. 
60 62.5 

4.92 feet height. Hence, - .6 V2 g x 1 = 3.2, and 17.9 - 4 - 3.2 = 5.59 sq. feet. 


88 

17-9 


Water sometimes acts by its weight and vis viva simultaneously, by com¬ 
bining effect of an acquired velocity with fall through which it flows upon 
wheel or instrument. 


In this case 



X 62.5 — mechanical effect. 


* As determined by — C. 


























HYDRODYNAMICS. 


563 


WATER-WHEELS. 

Water-wheels are divided into two classes, Vertical and Horizontal. 
Vertical comprises Overshot, Breast, and Undershot; and Horizontal, 
Turbine, Impact, or Reaction wheels. 


Vertical wheels are limited by construction to falls of less than 60 feet. 
Turbines are applicable to falls of any height from 1 foot upward. 

Vertical wheels applied to a fall of from 20 to 40 feet give a greater 
elfect than a Turbine, and for very low falls Turbines give a greater effect. 


Slviices.— Methods of admitting water to an Overshot or Breast 
Wheel are various, consisting of Overfall, Guide-bucket, and Penstock. 

An Overfall Sluice is a saddle-beam with a curved surface, so as to‘direct the 
current of water tangentially to buckets ; a Guide-bucket is an apron by which 
water is guided in a course tangential to buckets; and a Penstock is sluice-board or 
gate, placed as close to wheel as practicable, and of such thickness at its lower edge 
as to avoid a contraction of current. Bottom surface of penstock is formed with a 
parabolic lip. 

Shrouding of a wheel consists of plates at its periphery, which 
form the sides of the bucket. 

Height of fall of a water-wheel is measured between surfaces of water in penstock 
and in tail-race , and, ordinarily, two thirds of height between level of reservoir and 
point at which water strikes a wheel is lost for all effective operation. 

Velocity of a wheel at centre of percussion of fluid should be from .5 to .6 that 
of flow of the water. 

Total effect in a fall of water is expressed by product of its weight 
and height of its fall. 


Ratio of* Effective Power of Water INLotors. 


O'b/east, 1 ^ from " 68 to - 6 to 1 

Turbine. “ .6 to .8 to 1 

Breast. “ .45 to .65 to 1 

Hydraulic Ram. “ .6 to 1 


Undershot, Poncelet’s, from .6 to.4 to 1 

Undershot. “ .27 to.45 to 1 

Impact and Reac -1 ,, . , 

tion.....} -3 t0 -5 to 1 

Water-pressure engine “ .8 to x 


Overshot-wheel, 


Overshot-wheel.— The flow of water acts in some degree by impact, 
but chiefly by its weight. 

Lower the speed of wheel at its circumference, the greater will be mechan¬ 
ical effect of the water, in some cases rising to 80 per cent.; with velocities 
of from 3 to 6.5 feet, efficiency ranges from 70 to 75 per cent. Proper ve¬ 
locity is about 5 feet per second. 

Number of buckets should be as great, and should retain water as long, as 
practicable. Maximum effect is attained when the buckets are so numerous 
and close that water surface in the bucket commencing to be emptied should 
come in contact with the under side of the bucket next above it. Moles- 
worth gives 12 ins. apart. 

Curved buckets give greatest effect, and Radial give but .78 of effect of 
Elbow buckets. Wheel 40 feet in diameter should have 152 buckets. 

Small wheels give a less effect than large, in consequence of their greater 
centrifugal action, and discharging water from the buckets at an earlier 
period than with larger wheels, or when their velocity is lower. 

When head of water bears to fall or height of wheel a proportion as great 
as 1 to 4 or 5, ratio of effect to power is reduced. The general law there¬ 
fore is, that ratio of effect to power decreases as proportion of head to total 
head and fall increases. 









564 


HYDRODYNAMICS. 


Wheel with shallow Shrouding acts more efficiently than one where it is 
deep, and depth is usually made 10 or 12 ins., but in some cases it has been 
increased to 15. 

Breadth of a wheel depends upon capacity necessary to give the buckets 
to receive required volume of water. 

Form of Buckets.—Radial buckets—that is, when the bottom is a right line—in¬ 
volve so great a loss of mechanical effect as to render their use incompatible with 
economy; and when a bucket is formed of two pieces, lower or inner piece is 
termed bottom ox floor, and outer piece arm or wrist. Former is usually placed in 
a line with radius of wheel. 

Line of a circle passing through elbow , made by junction of floor and arm , is 
termed division circle, or bucket pitch, and it is usual to put this at one half depth 

of shrouding. „ n 

360° 

When arm of a bucket is included in division angle of buckets, that is, n 

representing number of buckets, the cells are not sufficiently covered, except for very 
shallow shrouding; hence it is best to extend arm of a bucket over 1.2 of division 
angle, so as to cover or overlap elbow of bucket next in advance of it. 

Construction of Buckets (Fig. 1).—Capacity of bucket should be 3 times volume 

of water. 

Fairbairn gives area of opening of a bucket in a 
wheel of great diameter, compared to the volume of it, 
as 5 to 24. 

Buckets having a bottom of two planes, that is, with 
two bottoms, and two division circles or bucket pitches 
and an arm, give a greater effect than with one bottom. 

When an opening is made in base of buckets, so as 
to afford an escape of air contained within, without a 
loss of w r ater admitted, the buckets are termed ven¬ 
tilated, and effective power of wheel is much greater 
than with closed buckets. 

D z= distance apart at periphery — d,d depth of 
shrouding, s length of radial starts.33 d, l length of 
bucket curve= 1.25 d in large wheels, and 1 in wheels 
under 25 feet, a angle of radius of curve of bucket, 
with radial line of wheel at points of bucket = 15°. 
(. Molesworth .) 


Fig. x. 



To Compute Radius and. Revolutions of an Overshot- 
wheel, and Height of Fall of Water. 


When whole Fall and Velocity of Flow , etc., are given. 


h — h' 


h c 


1 -j- cos. a 


3.1416 r 


— 1.1= ft', and 3 ’ I4I ^ n - = c. h representing height of whole 
2 g 30 


fall , h' height between the centre of gravity of discharge and half depth of bucket 
upon which water flows, v velocity of flow in feet per second, a angle which point of 
entrance of water into a bucket makes with summit of wheel , n number of revolutions 
per minute , c velocity of wheel at its circumference per second , and r its radius. 

Note.— Height of whole fall is distance between surface of water in flume and 
point at which lower buckets are emptied of water, and as a proportion of velocity 
of flow is lost, it is proper to assume height h' as above given. 

Illustration. —A fall of water is 30 feet, velocity of its flow is 16 feet per second, 
angle of its impact upon buckets is 12 0 , and required velocity of wheel is 8 feet per 
second; what is required radius, number of revolutions, and height of fall upon 
wheel ? 


16 2 
‘2 ~9 

>X 8 


X 1.1=4.38 feet; cos. 12° = .978; 


X 8 -—■ 240 = 3. q. revolutions. 

X 12.95 40.68 


30 — 4.38 _ 25.62 
1 -f.978 — 1.978 


= 12.95 feet radius ; 















HYDRODYNAMICS. 


565 


When Number of Revolutions and Ratio between Velocities of Flow and at 
Circumference of Wheel are given. 

3.1416 n r 


'curnj erence of 

V .000772 (x n)~ A —(1 -f- cos. a) 2 — 1 -f- cos. a) 

.000386 (xn) 2 c 30 

Illustration. —If number of revolutions are 5, x = 2, and fall, etc., as in previous 
case; what is radius of wheel, velocity of flow, and height of fall? 


= r,x= - , and 
c 


V .ooo772 (2 X 5 )~ X 30+ (1.978 ) 2 — 1.978 .518 


.000386 (2 X 5 ) 2 


.0386 


= 13.41 feet. 


3.1416 X 5 X X 3 - 4 1 
30 

X 1.1 — 3.37 feet. 


— 7.03 feet. Hence 7.03 X 2 — 14.06 velocity of flow, and 


14.06 2 

64-33 


To Compute “Wicltli of an Overshot-wheel. 


C V 


s c 


- = w. C representing a coefficient = 3, when buckets are filed to an excess, and 

5 when they are deficiently filled, Y volume of water in cube feet per second, s depth 
of shrouding , w width of buckets, both in feet, and c' velocity of wheel at centre of 
shrouding, in feet per second. 

Illustration. —A wheel is to be 31 feet in diameter, with a depth of shrouding of 
1 foot, and is required to make 5 revolutions per minute under a discharge of 10 
cube feet of water per second; what should be width of buckets? 


Assume C = 4, and c' 


31 —1 X 31416 X 5 

60 


: 7.854. Then— 4 ** 0 5.09 feet. 

' M 1X7-854 


7 + d- 12 = d, and 


To Oompnte nNTnmDer of Bncluets. 
Dps 


d 


D representing diameter of wheel, d dis¬ 


tance between centres of buckets, in feet, and n number of buckets. 

Illustration. —Take elements of preceding case. 

/ , 1 \ „ ,31 — 1X3.1416 X 1 

|^i -f- -g-y —7 X 2.2-4-12 = 1. 283, and 

360° 


Then 7 
buckets ; hence 


1 283 


= 73.4, say 72 


72 


5°, angle of subdivision of buckets. 


To Compute Effect of an Overshot-wheel. 

V Id iv — — V w 

--y-—-— = P. w representing weight of cube foot of water in lbs., 

v’ velocity of it discharged at tail of wheel, in feet per second , V volume of flow in 
cube feet, and ffriction of wheel in lbs. 

Illustration. — A volume of 12 cube feet per second has a fall of 10 feet,, wheel 
using but 8.5 feet of it, and velocity of water discharged is 9 feet per second; what 
is effect of fall ? 

Friction of wheel is assumed to be 750 lbs. 


12 X 8.5 X 62.5 ■ 


f—r — X 12 X 62.5 750^ 

\ 64 - 33 / 


6-375 — (1.26 X "Mo A- 750 4680 


12X10X62.5 7500 7500 

.624 = ratio of effect to power ; and 4680 X 60 seconds -r- 33 000 — 8.51 IP. 

To Compute Power of an Overshot-wlieel. 

Rule. — Multiply weight of water in lbs. discharged upon wheel in one 
minute by height or distance in feet from centre of opening in gate to sur¬ 
face of tail-race; divide product by 33 000, and multiply quotient by as¬ 
sumed or determined ratio of effect to power. Or, for general purposes, 
divide product by 50 000, and quotient is IP. 


Or, .0852 V ft = IP, and 


-1 7 IP 


V per second ; or, 

3 B 


771 IP 
h 


: V per minute. 




















5 66 


HYDRODYNAMICS. 


Mechanical Effect of water is product of its weight into height from which 
it falls. 

Example. —Volume of water discharged upon an overshot-wheel is 640 cube feet 
per minute, and effective height of fall is 22 feet; what is H?.? 

^ 4 ° _ X - 62 ' 5 X 22 _ 2 g w hj C p, x -75 = assumed ratio of effect to power = 20 IP. 

33 000 

Useful iCfFect of an Overshot-wheel. 

With a large wheel running in most advantageous manner, .84 of power 
may be taken for effect. 

Velocity of a wheel bears a constant ratio, for maximum effects, to that 
of the flowing water, and this ratio is at a mean .55. 

Ratio of effect to power with radial-buckets is .78 that of elbow-buckets. 
Ratio of effect decreases as proportion of head to total head and fall increases. 
Thus, a wheel 10 feet in diameter gave, with heads of water above gate, 
ranging from .25 to 3.75 feet, a ratio of effect decreasing from .82 to .67 of 
power. 

Higher an overshot-wheel is, in proportion to whole descent of water, 
greater will be its effect. Effect is as product of volume of water and its 
perpendicular height. 

Weight of arch of loaded buckets in lbs. is ascertained by multiplying 
.444 of their number by number of cube feet in each, and that product by 40. 

TJ n cl e rs la o t-w h eel. 

Undershot-wheel is usually set in a curb, with as little clearance for 
escape of water as practicable; hence a curb concentric to this wheel is more 
effective than one set straight or tangential to it. 

Computations for an undershot-wheel and rules for construction are near¬ 
ly identical with those for a breast-wheel. 

Buckets are usually set radially, but they may be inclined upward, so as 
to be more effectively relieved of water upon their return side, and they are 
usually tilled from .5 to .6 of their volume. Depth of shrouding should be 
from 15 to 18 ins., in order to prevent overflow of water within the wheel, 
which would retard it. 

Velocity of periphery should equal theoretical velocity due to head of 
water X .57. 

Note. —When constructed without shrouding, as in a current-wheel, etc., buckets 
become blades. 

Sluice-gate should be set at an inclination to plane of curb, or tangential 
to wheel, in order that its aperture may be as close to wheel as practicable; 
and in order to prevent partial contraction of flow of water, lower edge of 
sluice should be rounded. 

Effect of an undershot-wheel is less than that of a breast-wheel, as the 
fall available as weight is less than with latter. 

To Compute Power of an Undershot-wheel. 

Proceed as per rule for an overshot-wheel, using 93 750 for 50 000, and .4 
for .75. 

j — j 1 1 

Or, V li .coo66 — HP; or, ——— — V. V representing volume of water in cube 
feet per minute, and h head of water in feet. 




HYDRODYNAMICS. 


5 67 


ZPoneelet’s Wlieel. 

Poncelet’s Wheel. —Buckets are curved, so that flow of water is in 
course of their concave side, pressing upon them without impact; and effect 
is greater than when water impinges at nearly right angles to a plane sur¬ 
face or blade. 

This wheel is advantageous for application to falls under 6 feet, as its 
effect is greater than that of other undershot wheels with a curb, and for 
falls from 3 to 6 feet its effect is equal to that of a Turbine. 

For falls of 4 feet and less, efficiency is 65 per cent., for 4.25 to 5 feet, 60 
per cent., and from 6 to 6.5 feet, 55 to 50 per cent. 

In its arrangement, aperture of sluice should be brought close to face of 
wheel. First part of course should be inclined from 4 0 to 6° ; remainder of 
course, which should cover or embrace at least three buckets, should be car¬ 
ried concentric to wheel, and at end of it a quick fall of 6 ins. made, to guard 
against effect of back-water. Sluice should not be opened over 1 foot in any 
case, and 6 ins. is a suitable height for falls of 5 and 6 feet. 

Distance between two buckets should not exceed 8 or i<p ins., and radius 
of wheel should not be less than 40 ins., or more than 8 feet. 

Plane of stream or head of water should meet periphery of wheel at an 
angle of from 24 0 to 30°. Space between wheel and its curb should not ex¬ 
ceed .4 of an inch. 

Depth of shrouding should be at least .25 depth of head of water, or such 
as to prevent water from flowing through it and over the buckets, and width 
of wheel should be equal to that of stream of impinging water. 

Effect of this wheel increases with depth of water flow, and, therefore, 
other elements being equal, as filling of buckets, to obtain maximum effect, 
water should flow to buckets without impact, and velocity of wheel should 
be only a little less than half that of velocity of water flowing upon wheel. 


To Compute IProportions of a, IPoncelet 'Wlieel. 


Note. — As it is impracticable to arrive at the results by a direct formula, they 
must be obtained by gradual approximation. 

Example.— Height of fall is 4.5 feet; volume of water 40 cube feet per second; 
radius of wheel = 2 h, or 9 feet; depth of the stream =. 75 feet; and C assumed at .9. 

Y representing volume of water in cube feet per second, h height of fall, d depth of 


shrouding — — . - -1- d': d' opening of and e width of sluice, r radius of curva- 

4 2 g 

ture of buckets = —— , and a of wheel, all in feet; n number of revolutions — — - 

COS. Z J ) ct 

per minute ; c velocity of circumference of wheel and v velocity of water, both in feet 
per second; C coefficient of resistance of flow of water ; x angle between plane of 

x 

flowing water and that of circumference of wheel at point of contact, sin. of - — 

2 

1/cos. z ; z angle made by circumference of wheel with end of buckets — 2 tang, y; 

pc Id 


and y angle of direction of water from circumference of wheel = 


0 c / 

“Vi 


1 c 


Then v — .9 M'S. — .9 x 16.29 = 14.66 feet velocity of wheel, being 





5 68 


HYDRODYNAMICS. 


less than half velocity of water ; 


- — 14 * 66 *66 -. 

o — — / jccv) 


.25, angle corresponding to which = 14 0 30'; n = 


■/ 


d = -X 
4 


1.58 


r 

9 

30 x 7 


= x.222 X V 0403 — 


32.166 -j— 

9 


2 = 2 tang, y — 2 X -258 62 = .517 24 2 = 27 0 20 ; 


3.1416 X 9 

-,,cw. - 40 


e = 


7.43 revolutions ; 
2=3.63 feet; 


1.58 


1.58 


.75 X 14- 66 


__ =1.78 feet; x — sin. - = \/cos. 2 = Vcos.27 0 20' = .943 

cos. 27 0 20' .88835 ' J ’ 2 y 


= sin. of 70 0 34' /. x — 141 0 8'. Effect is a maximum when c = .5 v cos. y. 

Construction of Buckets (Fig. 2). (Molesworth.) 

From point of bucket, a, draw a line, a b, at an angle of 26° 
with radial line, point b, where this line cuts an imaginary cir¬ 
cle, drawn at a distance of s X 1-17 from periphery of wheel, is 
centre from which bucket is struck with radius, b a. Radius of 
wheel should not be less than 7, or more than 16 feet. 

Curb should fit wheel accurately for 18 or 20 ins., measured 
back from perpendicular line which passes through axis of 
wheel, the breast should then incline 1 in 10, or 1 in 15 towards 
sluice. 

After passing axis of wheel in tail-race, curb should make a 
sudden dip of 6 ins. 

To Compute Power of a. IPoncelet "Wheel. 

880 IP 

Vh.ooi 13=; IP, and —-— = V. V = velocity of theoretical periphery = .55.* 
h 

Number of buckets 1.6 D -)-1.6, D = diameter of wheel in feet. Shrouding .33 to 
.5 depth of head of water, and D ■= 2 h, and not less than 7 or more than 16 feet. 

DBreast-wlieel. 


Fig. 2. 



Breast-wheel is designed for falls of water varying from 5 to 15 feet, 
and for flows of from 5 to 80 cube feet per second. It is constructed with 
either ordinary buckets or with blades confined by a Curb. 

Enclosure within which water flows to a breast-wheel as it leaves the sluice 
is termed a Curb or Mantle. 

When blades are enclosed in a curb, they are not required to hold -water; 
hence they may be set radial , and they should be numerous, as the loss of 
w r ater escaping between the wdieel and the curb is less the greater their num¬ 
ber ; and that they may not lift or carry up water with them from tail-race, 
it is proper to give them such a plane that it may leave the water as nearly 
vertical as may be practicable. 

Distance between two buckets or blades should be from r.3 to 1.5 times 
head over gate for low velocity of wheel and more for a high velocity, or 
equal to depth of shrouding, or at from 10 to 15 ins. 

It is essential that there should be air-holes in floor of buckets, to prevent 
air from impeding flow of water into them, as the water admitted is nearly 
as deep as the interval between them ; and velocity of wheel should be such 
that buckets should be filled to .5 or .625 of their volume. 

When wheels are constructed of iron, and are accurately set in masonry, 
a clearance of .5 of an inch is sufficient. 


2 g h in feet per second. 



















HYDRODYNAMICS. 


569 


High Breast-wheel is used when level of water in tail-race and penstock 
or forehay are subject to variation of heights, as wheel revolves in direction 
in which water flows from blades, and back-ioater is therefore less disad¬ 
vantageous, added to which, penstocks can be so constructed as to admit of 
an adjustable point of opening for the water to flow upon the wheel. 

Effect of this wheel is equal to that of the overshot, and in some instances, 
from the advantageous manner in which water is admitted to it, it is greater 
when both wheels have same general proportions. 

Under circumstances of a variable supply of water, Breast-wheel is better 
designed for effective duty than Overshot, as it can be made of a greater 
diameter; whereby it affords an increased facility for reception of water 
into its buckets, also for its discharge at bottom; and further, its buckets 
more easily overcome retardation of back-water, enabling it to be worked 
for a longer period in back-water consequent upon a flood. 


In a well-constructed wheel an efficiency of 93 per cent, was observed by M. 
Morin, and Sir Wm. Fairbairn gives, at a velocity of circumference of wheel of 
5 feet, an efficiency of 75 per cent. Velocity usually adopted by him was from 4 
to 6 feet per second, both for high and low falls; a minimum of 3.5 feet for a fall of 
40 and a maximum of 7 feet for a fall of 5 to 6 feet. 

When water flows at from io° to 12 0 above horizontal centre of wheel, Fairbairn 
gives area of opening of buckets, compared with their volume, as 8 to 24. 

The capacity between two buckets or blades should be very nearly double that of 
volume of water expended. 


To Compute Proportions and Effect of a Breast-wheel. 


Illustration. —Flow of water is 15 cube feet per second; height of fall, measured 
from centre ofpressure of opening to tail-race, is 8.5 feet; velocity of circumference 
of wheel 5 feet per second; and depth of buckets or blades 1 foot, filled to .5 of their 
volume. 


Width of wheel = —, d representing depth , and v velocity of buckets; 


15 


s d 


3 ; 


1 X 5 

and as buckets are but .5 filled, 3 -=- .5 = 6 feet. Assume water is to flow with double 
velocity of circumference of wheel; 11 = 5 x 2 —10 feet; and fall required to gen¬ 


erate this velocity = — X 
2 9 


c. 1 — h' — 


64-33 


X 1.1 = 1.71 feet. 


Deducting this height from total fall, there remains for height of curb or shroud¬ 
ing, or fall during which weight of water alone acts, h — h' — 8.5 —1.71 = 6.79 feet. 

Making radius of wheel 12 feet, and radius of bucket circle n feet, whole mechan¬ 
ical effect of flow of water = 15 X 62.5 X 8.5 = 7968.75 lbs., from which is to be de¬ 
ducted from 10 to 15 per cent, for loss of water by escape. 

Theoretical efTect, as determined by M. Morin, velocity of circumference about 
.5 of that of water, and within velocities of 1.66 to 6 feet. 


^(e_cos._a._ v ^ v h'^j v 62.5. a representing angle of direction of velocity with 

which water flows to wheel at centre of thread of flow and direction of velocity of 
wheel at this line , and h" h — h' in feet. 

a is here assumed at 20 0 . See Weisbach, London, 1848, vol. ii. page 197, and for 
the necessarily small value of a, its cosine may be taken at 1. Cos. 20° = . 94. 


Then / ( IOx -94 5 ) 5 _|_ 6 \ x l5 x 62, 5 = 7 . 474 X 15 X 62.5 = 7006.9 Tbs., which 

V 32.16 / 

is to be reduced by a coefficient of .77 for a penstock sluice, and .8 for an overfall 
sluice. 

Theoretical effect, as determined by Weisbach, 7273 lbs., from which are 
to be deducted losses, which he computes as follows: 


Loss by escape of water between wheel and curb.= 916 

Loss by escape at sides of wheel and curb. = 180 

Friction and resistance of water = 2.5 per cent. = 160 

1256 lbs. 









570 


HYDRODYNAMICS. 


Friction of wheel as per formula, page 571, = Wr nC .0086; a — . 048 

.048 y 


16 500 


= 4.36 ins.; and n 


5 X 60 


: 4 revolutions. 


2 12 X 2 X 3-1416 

4.36 - 5 - 2 = 2.18. Then 16 500 X 2.18 X 4 X .08 X .0086 = 98.99 lbs. 

7006.9 —1256 + 9.9 


C = .o8. 


Whence, 

7968.75 

puted by Weisbach. 


.72 efficiency, upon assumption of losses as com- 


To Compute Power of a, Breast-wlaeel. 

Rule. —Proceed as per rule for an overshot-wheel, using 55 000 and .65 
with a high breast, and 62 500 and .6 for a low breast. 

13 3 IP 


Or, High breast, .0612 V h = IP, 

14.5 EP 


and 


— V; and Low breast .0546 V h — 


IP, and 


h 


V. 


Illustration. —Assume elements of preceding case. Then 
= 14.49, w hich X .7 = 10.14 horses. 

7006.9 — 1256 + 102.6 X 60 


15 X 62.5 X 8.5 X 60 


33000 


Or, 


33 °o° 


10.27 horses. 


Openings of Buckets or Blades.—High Breast , .33 sq. foot, and Low Breast , .2 sq. 
foot for each cube foot of their volume, or generally 6 to 8 in opening in a high 
breast and 9 to 12 in a low breast. 

Forms of Buckets.—Two Fart, d — T), s-=. 5 d , l 1.25 d in large wheels, and = d 
in wheels less than 25 feet in diameter. 

Three Part Buckets. — d divided into 3 equal parts; ^ = .25 d, d — D, s = . 33 d, l = 
d in large wheels, and .75 d in wheels less than 25 feet in diameter. 

Ventilating Buckets (Fairbairn's). Spaces are about 1 inch in width. 

Notes.—A Committee of the Franklin Institute ascertained that, with a high 
breast-wheel 20 feet in diameter, water admitted under a head of 9 ins., and at 17 
feet above bottom of wheel, elbow-buckets gave a ratio of effect to power of .731 at 
a maximum, and radial-blades .653. With water admitted at a height of 33 feet 
8 ins., elbow-buckets gave .658, and radial blades .628. 

At 10.96 feet above bottom of wheel, with a head of 4.29 feet, elbow-buckets gave 
. 544, and blades .329. 

At 7 feet above bottom of wheel, and a head of 2 feet, a loiv breast gave for 
elbow-buckets .62, and for blades .531. 

At 3 feet 8 ins. above bottom of wheel, and a head of 1 foot, elbow-buckets gave 
.555, and blades .533. 

Current-wheel. 

Current-wheel. —D. Iv. Clark assigns the most suitable ratio of veloc¬ 
ity of blades to that of current as 40 per cent. 

Depth of blades should be from .25 to .2 of radius; it should not be less 
than 12 or 14 ins. Diameter is usually from 13 to 16.5 feet, with 12 blades; 
but it is thought that there might be an advantage in applying 18 or even 
24. The blades should be completely submerged at lower side, but not more 
than 2 ins. under water, and not less than 2 at one time. 


a s , 
—- (v- 
150 


-s)2 —ip. a representing area of vertical section of immersed blades in 
sq. feet , s velocity of wheel at circumference , and v of stream , both in feet per second. 


Or, .38 -— V 62.5 = useful effect. Hence, efficiency = .38. 
2 g 











HYDRODYNAMICS. 


571 


IPluitter-wlieel. 

Flutter or Saio-mill Wheel —Is a small, low breast-wheel operating under 
a high head of water; the design of its construction, water being plenty, is 
the attainment of a simple application to high-speed connections, as a gang 
or circular saw. In effect it is from .6 to .7 that of an overshot-wlieel of 
like head of fall. 


V <; 

— (w —= 

150 


v and s as preceding. 


IT ruction, of’ Journals or G-udgeons. 

A very considerable portion of mechanical effect of a wheel is lost in ef¬ 
fect absorbed by friction of its gudgeons. 

To Compute Friction of Journals or Grndgeons of a, 

\V ater-wheel. 

Wi'iiC .0086 —f. W representing weight of wheel in lbs., r radius of gudgeon in 
ins., and n number of revolutions of wheel per minute. 

For well-turned surfaces and good bearings, C = .c>75 with oil or tallow; when 
best of oil is well supplied = .054; and, as in ordinary circumstances, when a black- 
lead unguent is alone applied = .n. 

Illustration. —A wheel weighing 25000 lbs. has gudgeons 6 ins. in diameter, and 
makes 6 revolutions per minute; what is loss of effectV 


Assume C = .o8. 


Then 25000 X - X 6 X -08 x 






Weights.— Iron wheels of 18 to 20 feet in diameter will weigh from 800 to 
1000 lbs. per IP 

Wood wheels of 30 feet in diameter, 2000 to 2500 lbs. per IP. 


To Compute Diameter and. Journals of a Shaft, Stress 
laid, uniformly- along its Length. 

i/WJ /IP 

Cast Iron, -—- — d. Wood, 6.12 3 / — = d. W representing weight or load in 

9 - ^ > 4 

lbs., I length of shaft between journals in feet, and d diameter of shaft in its body 
in ins. 


/W 

Journals or Gudgeons. — Cast Iron, .048 / — — d. 


When Shaft has to resist both Lateral and Torsional Stress. —Ascertain 
the diameter for each stress, and cube root of sum of their cubes will give 
diameter. 

To Compute Dimensions of Arms. 
cL 

Cast Iron 1 ' 7 — = w. d representing diameter of shaft, and 10 width of arm, both 
V n 

w 

in ins., n number of arms, — — t, and t thickness of arm. 

5 

When Arm is of Oak, w should be 1.4 times that of iron, and thickness .7 that 
of width. 

Memoranda. 


A volume of water of 17.5 cube feet per second, with a fall of 25 feet, applied to an 
undershot-wheel, will drive a hammer of 1500 lbs. in weight from 100 to 120 blows 
per minute, with a lift of from 1 to 1.5 feet.* 

A volume of water of 21.5 cube feet per second, with a fall of 12.5 feet, applied to 
a wheel having a great height of water above its summit, being 7.75 feet in diame¬ 
ter, will drive a hammer of 500 lbs. in weight 100 blows per minute, with a lift of 2 
feet 10 ins. Estimate of power 31.5 horses. 


* Volume of water required for a hammer increases in a much greater ratio than velocity to be given 
to it, it being nearly as cube of velocity. 





572 


HYDRODYNAMICS. 


A Stream and Overshot Wheel of following dimensions—viz., height of head to 
centre of opening, 24.875 ins.; opening. 1.75 by 80 ins.; wheel, 22 feet in diameter 
by 8 feet face; 52 buckets, each 1 foot in depth, making 3.5 revolutions per minute 
—drove 3 run of 4.5 feet stones 130 revolutions per minute, with all attendant ma¬ 
chinery, and ground and dressed 25 bushels of wheat per hour. 

4.5 bushels Southern and 5 bushels Northern wheat are required to make 1 bar¬ 
rel of flour. 

A Breast-wheel and Stream of following dimensions—viz., head, 20 feet; height 
of water upon wheel, 16 feet ; opening, 18 feet, by 2 ins.; diameter of wheel, 26 feet 
4 ins.; face of wheel, 20 feet 9 ins.; depth of buckets, 15.75 ins.; number of buck¬ 
ets, 70; revolutions, 4.5 per minute — drove 6144 self-acting mule spindles; 160 
looms, weaving printing-cloths 27 ins. wide of No. 33 yarn (33 hanks to a lb.), and 
producing 24000 hanks in a day of n hours. 

Horizontal "Wheels. 

In horizontal water - wheels , water produces its effect either by Impact , 
Pressure , or Reaction , but never directly by its weight. 

These wheels are therefore classed as Impact, Pressure, and Reaction, but 
are now designated by the generic term of Turbine. 

Txarhines. 

Turbines, being operated at a higher number of revolutions than Ver¬ 
tical Wheels, are more generally applicable to mechanical purposes ; but 
in operations requiring low velocities, Vertical Wheel is preferred. 

For variable resistances, as rolling-mills, etc., Vertical Wheel is far 
preferable, as its mass serves to regulate motion better than a small 
wheel. 

In economy of construction there is no essential difference between 
a Vertical Wheel and a Turbine. When, however, fall of water and 
volume of it are great, the Turbine is least expensive. Variations in 
supply of water affect vertical wheels less than Turbines. 

Durability of a Turbine is less than that of a Vertical Wheel; and it is 
indispensable to its operation that the water should be free from sand, silt, 
branches, leaves, etc. 

With Overshot and Breast Wheels, when only a small quantity of water is 
available, or when it is required or becomes necessary to produce only a por¬ 
tion of the power of th efall, their efficiency is relatively increased, from the 
blades being but proportionately filled; but with Turbines the effect is con¬ 
trary, as when the sluice is lowered or supply decreased water enters the 
wheel under circumstances involving greater loss of effect. To produce 
maximum effect of a stream of water upon a wheel, it must flow without im¬ 
pact upon it, and leave it without velocity; and distance between point at 
which the water flows upon a wheel and level of water in reservoir should 
be as short as practicable. 

Small wheels give less effect than large, in consequence of their making a 
greater number of revolutions and having a smaller water arc. 

In High-pressure Turbines reservoir (of wheel) is enclosed at top, and water 
is admitted through a pipe at its side. In Low-pressure , water flows into res¬ 
ervoir, which is open. 

In Turbines working under water, height is measured from surface of 
water in supply to surface of discharged water or race; and when they work 
in air, height is measured from surface in supply to centre of wheel. * 

In order to obtain maximum effect from water, velocity of it, when leav¬ 
ing a Turbine, should be the least practicable. 


HYDKODYNAMICS. 


573 


Efficiency is greater when sluice or supply is wide open, and it is less af¬ 
fected by head than by variations in supply of water. It varies but little 
with velocity, as it was ascertained by experiment that when 35 revolutions 
gave an effect of .64, 55 gave but .66. 

When Turbines operate under water, the flow is always full through them ; 
hence they become Reaction-tvheels , which are the most efficient. 

Experiments of Morin gave efficiency of Turbines as high as .75 of power. 

Angle of plane of water entering a Turbine, with inner periphery of it, 
should be greater than go°, and angle which plane of water leaving reservoir 
makes with inner circumference of Turbine should be less than 90°. 

When Turbines are constructed without a guide curve *, angle of plane of 
flowing water and inner circumference of wheel = 90°. 

Great curvature involves greater resistance to efflux of water; and hence 
it is advisable to make angle of plane of entering water rather obtuse than 
acute, say ioo°; angle of plane of water leaving, then, should be 50°, if in¬ 
ternal pressure is to balance the external; and if wheel operates free of 
water, it may be reduced to 25° and 30°. 

If blades are given increased length, and formed to such a hollow curve 
that the water leaves wheel in nearly a horizontal direction, water then both 
impinges on blades and exerts a pressure upon them; therefore effect is 
greater than with an impact-wheel alone. 

Turbines are of three descriptions : Outward, Downward, and Inward flow. 

Outward-flow Turbines. 

Fourneyron Turbine, as recently constructed, may be considered as one 
of the most perfect of horizontal wheels; it operates both in and out of 
back-water, is applicable to high or low falls, and is either a high or low 
pressure turbine. 

In high-pressure, the reservoir is closed at top and the water is led to it 
through a pipe. In low-pressure, the water flows directly into an open res¬ 
ervoir. Pressure upon the step is confined to weight of wheel alone. 

Fourneyron makes angle of plane of water entering =90°, and angle of 
plane of water leaving = 30°. 

Efficiency is reduced in proportion as sluice is lowered, for action of water 
on wheel is less favorably exerted. M. Morin tested a Fourneyron turbine 
6.56 feet in diameter, and he found that efficiency varied from a minimum 
of 24, to 79 per cent., when supply of water was reduced to .25 of full supply. 
In practice, radial length of blades of wheel is .25 of radius, for falls not ex¬ 
ceeding 6.5 feet, .3 for falls of from 6.5 to 19 feet, and .66 for higher falls. 

To Compute Elements and JR.esvi.lts. 

V V 1.77 V IP 

High Pressure, 6 . 6 fh=zv: - = A ; - - — =Df; 12.6 — = V ; and 

v fh h 

.079 Y h = IP. h representing head of water, v velocity of turbine at periphery per 
minute, and D internal diameter of turbine, all in feet, V volume of water in cube feet 
per second, A sum of area of orifices in sq. feet, and IP effective horse-power. 

1.2 D = external diameter of turbine in feet, when it is more than 6 feet, and 1.4 
when it is less than 6 feet. Number of guides = number of blades t when less than 
24, and number A- 3 when greater than 24. Area of section of supply-pipe = .4 Y. 

For construction of blades and guides, see Molesworth, London, 1882, page 540. 


* Guide curves are plates upon centre body of a Turbine, which give direction to flowing water, 
or to blades of wheel which surround them. 

+ In extreme cases of very high falls diameter given by this formula may be increased. 

% Fourneyron’s rule for the number of blades is constant number 36, irrespective of size of turbine. 




HYDKODYNAMICS. 


574 


Operation of High-Pressure Turbines. 


h 

3 ° 

40 

5 ° 

60 

70 

80 

90 

100 

120 

140 

160 

180 

200 

V 

4.2 

3 -i 

2-5 

2 . I 

1.8 

1.6 

1.4 

1.27 

1.05 

•9 

.8 

■7 

•63 

V 

36 

42 

47 

51 

55 

59 

63 

66 

73 

78 

84 

89 

94 


h — head of water in feet, V volume of water in cube feet required for each io IP, 
and v velocity of periphery of turbine in feet per second. 

Boyden Turbine. — Mr.-Boyden, of Massachusetts, designed an 

outward-flow turbine of 75 IP, which realized an efficiency of 88 per cent. 
Peculiar features, as compared with a Fourneyron turbine, are, 1st, and most 
important, the conduction of the water to turbine through a vertical trun¬ 
cated cone, concentric with the shaft. The water, as it descends, acquires a 
gradually increasing velocity, together with a spiral movement in direction 
of motion of wheel. The spiral movement is, in fact, a continuation of the 
motion of the water as it enters cone.—2d. Guide-plates at base are inclined, 
so as to meet tangentially the approaching water.—3d. A “ diffuser,” or annu¬ 
lar chamber surrounding wheel, into which water from wheel is discharged. 
This chamber expands outwardly, and, thus escaping velocity of water, is 
eased off and reduced to a fourth when outside of diffuser is reached. Effect 
of diffuser is to accelerate velocity of water through machine; and gain of 
efficiency is 3 per cent. Diffuser must be entirely submerged. [D. K. Clarlc.) 

Poncelet Turbine. —This wheel is alike to one of his undershot-wheels 
set horizontally, and it is the most simple of all horizontal wheels. 

To Compute Elements of Greneral Proportion and. 

Results. (Lt. F. A. Mahan, U. S. A.) 

.0425 D 2 h y/k = IP; 4- 8 5 — - 0 1 *5 B 2 y/h — V ; .1 D = H; 4.49 fh = v, 

3 (D —j— 10) = N ; 5 = t£ = W; D ~~ = d) .5 N to . 75 -N = » ; ~=w'; 

V' 

and C coefficient for V' in terms of V = —. D and d representing exterior and in¬ 
terior diameters of wheel, H and h heights of orifices of discharge at outer circum¬ 
ference and of fall acting on wheel, w and w' shortest distances between two adjacent 
blades and two adjacent guides, all in feet, V, V', and v velocities due to fall of water 
passing through narrowest section of wheel , and of interior circumference of wheel, 
all in feet per second, N and n numbers of blades and guides, and IP actual horse¬ 
power. 

For falls of from 5 feet to 40, and diameters not less than 2 feet, n w should be 
equal to diameter of wheel. H equal to . 1 D, n w' = d, and 4 w = width of crown. 
For falls exceeding this, H should be smaller, in proportion to diameter of wheel. 

Downward-flow Turbines. 

In turbines with downward flow, wheel is placed below an annular series 
of guide-blades, by which water is conducted to wheel. The water strikes 
curved blades, and falls vertically, or nearly so, into tail-race; consequently, 
centrifugal action is avoided, and downward flow is more compact. 

Fontaine Turbine yields an efficiency of 70 per cent., when fully 
charged. When supply of water is shut off to .75, by sluice, efficiency is 
57 per cent. Best velocity at mean circumference of wheel is equal to 55 
per cent, of that due to height of fall. It may vary .25 of this either way, 
without materially affecting efficiency. 

In operation the water in race is in immediate contact with wheel, and its 
efficiency is greatest when sluice is fully opened. Its efficiency, also, is less 
affected by variations of head of flow than in volume of water supplied; 
hence they are adapted for Tide-mills. 














HYDRODYNAMICS. 


575 


Jonval Turbine. —This wheel is essentially alike in its principal propor¬ 
tions to Fontaine’s, and in principle of operation it is the same. Water in 
race must be at a certain depth below wheel. 

For convenience, it is placed at some height above level of tail-race, within 
an air-tight cylinder, or “ draft-tube,” so that a partial vacuum or reduction 
of pressure is induced under wheel, and effect of wheel is by so much in¬ 
creased. Resulting efficiency is same as if wheel was placed at level of tail- 
race ; and thus, while it may be placed at any level, advantage is taken of 
whole height of fall, and its efficiency decreases as volume of water is di¬ 
minished or as sluice is contracted. 


To Compute Elements and. Results. 

how Pressure. —For falls of 30 feet and less. 

V f 1.77 V BP 

6 y/h — v; =- = A; -~ h — = D*; 12.7 — = V; and .079 Y ft = IP. 

ft representing head of water, v velocity of turbine at periphery per minute, and D 
internal diameter of turbine, all in feet, V volume of water in cube feet per second , 
A sum of area of orifices in sq.feet, and IP effective horse power. 

1.2 D = external diameter of turbine in feet, when it is more than 6 feet, and 1.4 
when it is less than 6 feet. Number of guides := number of blades t when less than 
24, and numbers- 3 when greater than 24. Area of section of supply-pipe = .4 V. 

For construction of blades and guides, see Molesworth, London, 1882. page 540. 


Low-Pressure Turbines. (Molesworth.) 


-o 


5 IP 

10 

IP 

15 IP 

20 BP 

30 

IP 

40 

IP 

50 IP 

<D 

K 

V 

V 

R 

V 

R 

V 

R 

Y 

R 

V 

R 

V 

R 

Y 

R 

2-5 

9.48 

25 

34 

50 

24 

75 

20 

IOO 

J 7 

— 

— 

— 

— 

— 

— 

5 

L 3 - 3 8 

12.5 

81 

25 

57 

3 « 

47 

50 

4 i 

75 

33 

IOO 

28 

126 

26 

7-5 

16.38 

8-5 

136 

W 

97 

25 

79 

33 

68 

5 i 

56 

68 

48 

85 

43 

IO 

18.96 

6-3 

180 

12.6 

128 

*9 

105 

25 

90 

38 

75 

5 o 

64 

63 

58 

i 5 

23.22 

4.2 

319 

8.4 

226 

12.6 

185 

17 

160 

25 

r 3 i 

33 

113 

42 

IOO 

20 

26.82 

— 

— 

6-3 

329 

9-3 

273 

12.6 

232 

18.9 

I 94 

25 

164 

31 

148 

25 

3 ° 

— 

— 

— 

— 

7-5 

358 

IO 

310 

i 5 

253 

20 

220 

25 

196 

30 

32.88 

— 

— 

— 

— 

— 

— 

8.4 

380 

12.6 

310 

w 

268 

21 

240 


v representing velocity of centre of blades in feet and V volume of water, in cube 
feet, both per second, R revolutions per minute, and IP effective horse power. 


Vertical Shaft. 


3/230EP 

V R 


= diameter of shaft in ins. 


Inward-flow TiirToine. 


Inward-flow Turbine. — Inward-flow or vortex wheel is made with 
radiating blades, and is surrounded by an annular case, closed externally, 
and open internally to wheel, having its inner circumference fitted with four 
curved guide-passages. The water is admitted by one or more pipes to the 
case, and it issues centripetally through the guide-passages upon circum¬ 
ference of wheel. The water acting against the curved blades, wheel is 
driven at a velocity dependent on height of fall, and water having expended 
its force, passes out at centre. This wheel has realized an efficiency as high 
as 77.5 per cent. It was originally designed by Prof. James Thomson. 

Swain Turbine. —Combines an inward and a downward discharge. Re¬ 
ceiving edges of buckets of wheel are vertical opposite guide-blades, and 
lower portions of the edges are bent into form of a quadrant. Each bucket 
thus forms, with the surface of adjoining bucket, an outlet which combines 
an inward and a downward discharge. One, 72 ins. in diameter, was tested 


* In extreme cases of very high falls diameter given by this formula may be increased, 
t Fourneyron’s rule for the number of blades is constant number 36, irrespective of size of turbine. 



































HYDRODYNAMICS. 


576 


by Mr. J. B. Francis, for several heights of gate or sluice, from 2 to 13.08 
ins., and circumferential velocities of wheel ranging from 60 to 80 per cent, 
of respective velocities due to heads acting on wheel. 

For a velocity of 60 per cent., and for heights of gate varying within limits al¬ 
ready stated, efficiency ranged from 47.5 to 76.5 per cent., and for a velocity of 80 
per cent, it ranged from 37.5 to 83 per cent. Maximum efficiency attained was 84 
per cent., with a 12-inch gate and a velocity-ratio of 76 per cent.; but from 9-inch 
to 13-inch gate, or from .66 gate to full gate, maximum efficiency varied within 
very narrow limits—from 83 to 84 per cent.,—velocity-ratios being 72 per cent, for 
9-inch gate, and 76.5 per cent, for full gate. At half-gate, maximum efficiency was 
78 per cent., when velocity-ratio was 68 per cent. At quarter-gate, maximum effi¬ 
ciency was 61 per cent., and velocity-ratio 66 per cent. 


T remont Turbine, as observed by Mr. Francis, in his experiments at 
Lowell, Mass., gave a ratio of effect to power as .793 to 1. 


Victor Turbine is alleged to have given an effect of .88 per cent, under 
a head of 18.34 feet, with a discharge of 977 euhe feet of water per minute, 
and with 343.5 revolutions. 

Tangential Wheel. 

Wheels to which water is applied at a portion only of the circumference 
are termed tangential. They are suited for very high falls, where diameter 
and high tangential velocity may be combined with moderate revolutions. 
The Girarcl turbine belongs to this class. It is employed at Goeschenen 
station for St. Gothard tunnel, it operates under a head of 279 feet. The 
wheels are 7 feet 10.5 ins. in diam., having 80 blades, and their speed is 160 
revolutions per minute, with a maximum charge of water of 67 gallons per 
second. An efficiency of 87 per cent, is claimed for them at the Paris 
water-works; ordinarily it is from 75 to 80 per cent. {D. K. Clark.) 

Impact and Reaction "Wheel. 

Impact-wheel. —Impact Turbine is most simple but least efficient form 
of impact-wheel. It consists of a series of rectangular buckets or blades, 
set upon a wheel at an angle of 50° to 70° to horizon; the water flows to 
blades through a pyramidal trough set at an angle of 20° to 40°, so that 
the water impinges nearly at right angles to blades. Effect is .5 entire me¬ 
chanical effect, which is increased by enclosing blades in a border or frame. 

If buckets are given increased length, and formed to such a hollow curve 
that the water leaves wheel in nearly a horizontal direction, the water then 
impinges on buckets and exerts a pressure upon them; effect therefore is 
greater than with the force of impact alone. 

By deductions of Weisbach it appears that effect of impact is only half 
available effect under most favorable circumstances. 


Reaction-wheel.— Reaction of water issuing from an orifice of less 
capacity than section of vessel of supply, is equal to weight o f a column of 
water , basis of which is area of orifice or of stream, and height of which is 
twice height due to velocity of water discharged. 

V 2 

Hence, the expression is 2. — a w = R. iv representing weight of a cube foot of 
2 Q 

water in lbs., and a area of opening in sq.feet. 


Whitelaw’s is a modification of Barker’s*, the arms taper from centre 
towards circumference and are curved in such a manner as to enable the 
water to pass from central openings to orifices in a line nearly right and 
radial, when instrument is operating at a proper velocity; in order that very 
little centrifugal force may be imparted to the water by the revolution of 
the arms, and consequently a minimum of frictional resistance is opposed 
to course of the water. 


HYDRODYNAMICS. 


577 


A Turbine 9.55 feet in diameter, with orifices 4.944 ins. in diameter, oper¬ 
ated by a fall of 25 feet, gave an efficiency of 75 per cent., including friction 
of gearing of an inclined plane. 

When a reaction wheel Is loaded, so that height due to velocity, corresponding to 

velocity of rotation «, is equal to fall, or : A, or w =3 V2 0 A, there is a loss of 17 

2 a 

, U 2 

per cent, of available effect: and when 2 A, there is a loss of but 10 per cent.: 

20 ’ A ’ 

and when - — 4 A, there is a loss of but 6 per cent. Consequently, for moderate 

falls, and when a velocity of rotation exceeding velocity due to height of fall rnay 
be adopted, this wheel works very effectively. 

Efficiency of wheel is but one half that of an undershot-wheel. 

When sluice is lowered, so that only a portion of w heel is opened, efficiency 
of a iieaetion-wheel is less than that of a Pressure Turbine. 


Ratio of 'Effect to Pouter of several Turbinas is as follows: 

Bakkkb’s Mill. —Effect ef this mill is considerably greater than that 
which same quantity of water would produce if' applied to an undershot- 
wheel, hut less than that which it would produce if properly applied to an 
overs hot-wheel. 

For a description of it, see Oner's Mechanics' Calculator, page 234; and for its 
formulas, see London Artisan , 1845, page 229. 


IMPULSE AND RESISTANCE OP FLUIDS. 

Impulse and Itewintanee of Water— Water or any other fluid, 
when flowing against a body, imparts a force to it by which its condition of 
motion is altered. Resistance which a fluid opposes to motion of a body 
does not essentially differ from Impulse. 

Impulse of one and same mass of fluid under otherwise similar circum¬ 
stances is proportional to relative velocities c vp v of fluid. 

For an equal transverse section of a stream, the impulse against, a surface 
at. rest increases as square of velocity of w'ater. 


Tmpv.Ua agafourl /‘lane Ha fares. —The impulse of a stream of water de¬ 
pends principally upon angle under which, after impulse, it leaves the water; 
it is nothing if the angle is o, and a maximum if it is deflected hack in a 

line parallel to that of its flow, or 180% 2 ~ - V w — P*. 


c T v 

Whan Surface of Resistance is a Plane , and = 90°, then - —~ Vw = P, and 
for a surface at rest , 2 a h w = P. a representing area of opening in sg.feeL 


p 2 A A w ; c «w.d w representing velocities of water and of surface upon which it 
im fringes in feet per second, w weight offluid, per cube foot in lbs:, A transverse section 
of stream in sg. ins., and cppv relative motions of water and surface. 


Normal impulse of water against a plane surface is equivalent to weight 
ef a column which has for its base transverse section of stream, and for 

c 2 

altitude twice height due to its velocity, 2 A = 2 -. 

2 0 


Resistance of a fluid to a body in motion is same as impulse of a fluid 
moving with same velocity against a body at rest. 


Welabach, New York, 1870, vol. i. page 1008. 

3C 








HYDRODYNAMICS. 


578 


Maximum Effect of Impulse. — Effect of impulse depends principally on 
velocity v of impinged surface. It is, for example, o, both when v — c and 
v=.o] hence there is a velocity for which effect of impulse is a maximum 

— (c — v) v\ that is, v — — , and maximum effect of impulse of water is ob¬ 
tained when surface impinged moves from it with half velocity of water. 


Illustration. —A stream of water having a transverse section of 40 sq. ins., dis¬ 
charges 5 cube feet per second against a plane surface, and flows off with a velocity 

C ^ V c ^ j ( i 

of 12 feet per second; effect of its impulse, then, is -Vw = P; c — -- = 18; 


g — 32.16; w — 6 2.5; ——^ X 5 X 62.5 = 58.28 lbs. 

32.16 

Hence mechanical effect upon surface = P»= 58.28 X 12 = 699.36 lbs. 

^ j ^ ^ j j j^2 

Maximum effect would be v = - = - x -— = 9 feet, and - X •— X 5 X 62.5 

2 2 40 2 2 g 


= - X 5-036 X 312.5 = 786.87 lbs.; and hydraulic pressure _ 87.44 lbs. 

2 9 


When Surface is a Plane and at an Angle , then (1 — cos. a) — V w = P. 


Illustration. —A stream of water, having a transverse section of 64 sq. ins., dis¬ 
charges 17.778 cube feet per second against a fixed cone, having an angle of con¬ 
vergence from flow of stream of 50 0 , hydraulic pressure in direction of stream; 
17.778 AO 

thenc = -—;-= 4°; cos. 50° = .64279. (1 —.64279) -- X 17.778 x 62.5 = 

64=144 32.16 

.357 21 X 1382.2 = 494.26 lbs. 


When Surface of Resistance is a Plane at 90°, and has Borders added to 
its Perimeter , effect will be greater, depending upon height of border and 
ratio of transverse section between stream and part confined. 

Oblique Impulse .—In oblique impulse against a plane, the stream may flow 
in one, two, or in all directions over plane. 

£ _ 1) 

When Stream is confined at Three Sides, (1 cos. a) -- V w = P. 

c — v 

When Stream is confined at Two Sides, — — sin. a 2 V w = P. 

Normal impulse of a stream increases as sine of angle of incidence; par¬ 
allel impulse as square of sine of angle; and lateral'impulse as double the 
angle. 


When an Inclined Surface is not Bordered , then stream can spread over 
it in all directions, and impulse is greater, because of all the angles by 
which the water is deflected, a is least; hence each particle that does not 
move in normal plane exerts a greater pressure than particle in that plane, 
2 sin. a 2 


and 


1 sin. a- 


X 


9 


V io = P. 


Impulse and Resistance against Surfaces. 

Coefficient of resistance, C, or number with which height due to velocity is to be 
multiplied, to obtain height of a column of water measuring this hydraulic press¬ 
ure, varies for bodies of different figures, and only for surfaces which are at right 
angles to direction of motion is it nearly a definite quantity. 

According to experiments of Du Buat and Thibault, 0 = 1.85 for impulse of air 
or water against a plane surface at rest, and for resistance of air or water against a 
surface in motion, C = 1.4. In each case about .66 of effect is expended upon front 
surface, and .34 upon rear. 











HYDRODYNAMICS. 


579 


Comparison between Turbines and. otlier Water-wheels. 

Turbines are applicable to falls of water at any height, from i to 500 feet. 

Their efficiency for very high falls is less than for smaller, in consequence 
of the hydraulic resistances involved, and which increase as the square of 
the velocity of the water. They can only be operated in clear water. 

With Fourneyron’s, the stress and pressure on the step is that of the wheel 
in motion; with Fontaine’s, the whole weight of the water is added to that 
of the wheel; they are well adapted, however, for tide-mills. Experiments 
on Jouval’s gave equal results with Fontaine’s. 

Vertical Water-wheels are limited in their application to falls under 60 
feet in height. 

For falls of from 40 to 20 feet they give a greater effect than any turbine; 
for falls of from 20 to 10 feet, they are equal to them; and for very low 
falls, they have much less efficiency. 

Variations in the supply of water effect them less than turbines. 


Water-pressure Engine. 

By experiments of M. Jordan, he ascertained that a mean useful effect of 
.84 was attainable. 

Weisback, London, 1848, vol. ii. page 349. 

PERCUSSION OF FLUIDS. 

When a stream strikes a plane perpendicular to its action, force with 
which it strikes is estimated by product of area of plane, density of fluid, 
and square of its velocity. 

Or, A d v 2 = P. A representing area in sq. feet, d weight of fluid in lbs., and v 
velocity in feet per seeond. 

If plane is itself in motion, then force becomes A d (v — v') 2 — P. v' representing 
velocity of plane. 


If C represent a coefficient to be determined by experiment, and h height 
due to velocity v, then v 2 = 2 g h, and expression for force becomes 
A C 2 g h = P. 

CENTRIFUGAL PUMPS. (D. K. Clark.) 

Appold 3 ?ump, made with curved receding blades, is the form of 
centrifugal pump most widely known and accepted. M. Morin tested three 
kinds of centrifugal or revolving pumps: 

1st, on model of Appold pump; 2d, one having straight receding blades 
inclined at an angle of 45 0 with the radius, and 3d, one having radial blades. 
They were 12 ins. in diameter and 3.125 ins. in length, and had central open¬ 
ings of 6 ins. Their efficiencies were as follows: 

1. Curved blades.. 48 to 68 per cent. | 2. Inclined blades.. 40 to 43 per cent. 

3. Radial blades.24 per cent. 


Height to which water ascends in a pipe, by action of a centrifugal pump, 
would, if there were no other resistances, be that due to velocity of circum¬ 


ference of revolving wheel, or to 


2 g 


Results of experiments made by the 


author on two pumps, in 1862, yielded following data, showing height to 
which water was raised, without any discharge: 


Gwynne’s Pump 
(blades partly radial, 
curved at ends). 


Diameter of pump-wheel. 4 feet. 

Revolutions per minute. 177 

Velocity of circumference per second... 37.05 feet. 

Head due to the velocity. 21.45 “ 

Actual head. 18.21 “ 

Do. do. in parts of head due to velocity, 85 per cent. 


Appold Pump 
(blades, curved). 


ins. 







58O HYDRODYNAMICS.—IMPACT OR COLLISION. 


Mr. David Thomson made similar experiments with Appold pumps of from 1.25 
to 1.71 feet in diameter, the results of which showed that the actual head was about 
90 per cent, of the head due to the velocity. 

M. Tresca, in 1861, tested two centrifugal pumps, 18 ins. in diameter, with a cen¬ 
tral opening of 9 ins. at each side. The blades were six in number, of which three 
sprung from centre, where they were .5 inch thick; the alternate three only sprung 
at a distance equal to radius of opening from centre. They were radial, except at 
ends, where they were curved backward, to a radius of about 2.25 ins.; and they 
joined the circumference nearly at a tangent. Width of blades was taper, and they 
were 5.75 ins. wide at nave, and only 2.625 ins. at ends: so designed that section of 
outflowing water should be nearly constant. 

M. Tresca deduced from his experiments that, in making from 630 to 700 revolu¬ 
tions per minute, efficiency of the pump, or actual duty in raising water, through a 
height of 31.16 feet, amounted to from 34 to 54 per cent, of work applied to shaft; 
or that, in the conditions of the experiment, the pump could raise upward of 16200 
cube feet of water per hour, through a height of 33 feet, with about 30 IP applied 
to shaft, and an efficiency of 45 per cent. 

According to Mr, Thomson, maximum duty of a centrifugal pump worked by a 
steam-engine varies from 55 per cent, for smaller pumps to 70 per cent, for larger 
pumps. They may be most effectively used for low or for moderately high lifts, of 
from 15 to 20 feet; and, in such conditions, they are as efficient as any pumps that 
can be made. For lifts of 4 or 5 feet they are even more efficient than others. 

At same time, larger the pump higher lift it may work against. Thus, an 18-inch 
pump works well at 20-feet lift, and a 3-feet pump at 30-feet lift. A 21-inch wheel 
at 40-feet lift has not given good results: high lifts demand very high velocities. 

Efficiency is influenced by form of casing of pump. Hon. R. C. Parsons made exper¬ 
iments with two 14-inch wheels on Appold’s and on Rankine’s forms. In Rankine’s 
wheel blades are curved backwards, like those of Appold’s, for half their length; 
and curved forwards, reversely, for outer half of their length. Deducing results of 
performance arrived at, following are the several amounts of work done per lb. of 
water evaporated from boiler, reduced for a speed of 350 revolutions per minute: 

Work done per lb. of 
water evaporated. 

Foot-lbs. Ratio. 


Appold wheel, in concentric circular casing. 6250 1 

“ •• in spiral casing. 9000 1.44 

Rankine wheel, in concentric circular casing. 9700 ■ 1.55 

“ “ in spiral casing. 12500 2 


These data prove:—1st, that spiral casing was better than concentric casing; 2d, 
that Rankine’s ogee-wheel was more efficient by one half than Appold’s wheel. 


IMPACT OR COLLISION. - 

Impact is Direct or Oblique. Bodies are Elastic or Inelastic. The 
division of them into hard and elastic is wholly at variance with these 
properties; as, for instance, glass and steel, which are among hardest 
of bodies, are most elastic of all. 

Product of mass and velocity of a body is the Momentum of the body. 

Principle upon which motions of bodies from percussion or collision are 
determined belongs both to elastic and inelastic bodies; thus there exists in 
bodies the same momentum or quantity of motion, estimated in any one and 
same direction, both before collision and after it. 

Action and reaction are always equal and contrary. If a body impinge 
obliquely upon a plane, force of blow is as the sine of angle of incidence. 

When a body impinges upon a plane surface, it rebounds at an angle equal 
to that at which it impinged the plane, that is, angle of reflection is equal to 
that of incidence. 

Effect of a blow of an elastic body upon a plane is double that of an in¬ 
elastic one, velocity and mass being equal in each; for the force of blow 







IMPACT OR COLLISION. 


581 


from inelastic body is as its mass and velocity, which is only destroyed by 
resistance of the plane; but in an elastic body that force is not only destroyed, 
being sustained by plane, but another, also equal to it, is sustained by plane, 
in consequence of the restoring force, and by which the body is repelled with 
an equal velocity; hence intensity of the blow is doubled. 

If two perfectly elastic bodies impinge on one another, their relative ve¬ 
locities will be same, both before and after impact; that is, they will recede 
from each other with same velocity with which they approached and met. 

If two bodies are imperfectly elastic, sum of their moments will be same, 
both before and after collision, but velocities after will be less than in case 
of perfect elasticity, in ratio of imperfection. 

Effect of collision of two bodies, as B and b, velocities of which are differ¬ 
ent, as v and v', is given in following formulas, in which B is assumed to 
have greatest momentum before impact. 

If bodies move in same direction before and after impact, sum of their 
moments before impact will be equal to their sum after. 

If bodies move in same direction before, and in opposite direction after 
impact, sum of their moments before impact will be equcil to difference of their 
sums after. 

If bodies move in opposite directions before, and in same direction after 
impact, difference of their moments before, impact will be equal to their 
sum after. 

If bodies move in opposite directions before, and in opposite directions 
after impact, difference of their moments before impact xoill be equal to their 
difference after. 


To Compute 'Velocities of Inelastic Bodies after Impact. 

B V + ftli 


When Impelled in Same Direction. 




— r. B and b representing 


weights of the two bodies , V and v their velocities before impact , and r velocity of bodies 
after impact , all in feet. 

y D y — 

Consequently, - Xb= velocity lost by B, and -- X B = velocity gained by b. 
B -J- 0 B -f- b 

Note.— In these formulas it is assumed that V>r. If the result will be 

negative, but may be read as positive if lost and gained are reversed in places. 

Illustration.— An inelastic body, 5 , weighing 30 lbs., having a velocity of 3 feet, 
is struck by another body, B, of 50 lbs., having a velocity of 7 feet; the velocity of 
b after impact will be ' . —— — 

5 ° X 7 "l - 3 ° X 3_ 44 °_ 

—— 80 

B V._ b v 

When Impelled in Opposite Directions. - — r - 

Illustration.— Assume elements of preceding case. 


50 + 30 80 


When One Body is at Rest. 


B V 
B + b 


Illustration.— Assume elements as preceding. 

lr=4.75/^. 

50 -j- 30 80 

When Bodies are inelastic, their velocities after impact will be alike. 

3 C* 












582 


IMPACT OR COLLISION. 


To Compute 'Velocities of Elastic Bodies after Impact. 
TTrz. r 11 1 • n re B — 6Y + 2&U 2 B V— B — bv 

When Impelled m One Direction. - . . -= R, and- .. . : -= r. 


B + 6 

Illustration.— Assume elements as preceding. 


B + 6 


50-30X 7 + 2X30X3 = 3^ = fid and 2 X 50 X 7-50-30 X_3 = 640 = 8 feet. 
50+30 80 50 + 30 


80 


Or, V — 


2 b 


■ V — v = velocity of A, and v + - 


B 


B + 6 ' ’ 1 B + & 

When Impelled in Opposite Directions. 
B — 6 V oj ib v „ , 2 BY — B — b v 


V — v = velocity of r. 


B + & — R, and B + 6 

Illustration.— Assume elements as preceding. 


50 — 30 x 7 'v 2 X 3° X 3 _ 140 + 180 _ 


50 + 30 


80 


— .5 feet, and 


2 X 50 X 7 + 5o — 3° X 3, 
50 + 30 


+ Or,^I±5 = ,efc+!«, S +B. As » X 3°X 7 + 3 = 

80 y B + b 50 + 30 80 

= 7-5 feet. 

When One Body is at Rest 


V B — b 


„ . 2 b y 

E ’ a ” d B+l, = r ' 


B + 6 

Illustration. —Assume elements as preceding. 

7 X 50 — 30 140 , 2X50X7 

- -+-— = -£- = i-75 feet, and ---' 

50+30 80 50 + 30 


= ^ = 8.75 feet. 


To Compute Velocities of Imperfect Elastic Bodies after 

Impact. 

Effect of Collision is increased over that of perfectly inelastic bodies, but 
not doubled, as in case of perfectly elastic bodies; it must be multiplied by 
m-\-n , n 


1 + — or 


•, when — represents degree of elasticity relative to both per¬ 


fect inelasticity and elasticity. 

Moving in same Direction. V — m ~^~ n y 

'Yy) 


B + 6 


(V — r) = R; andr + 


m + n 


X = (V — v) -- r. m and n representing ratio of perfect to imperfect elasticity. 

B -J- b 

Illustration. —Assume elements as preceding. m and n = 2 and 1. 


2 + 1 


30 


30 


2 + 1 


X7-3 = 7~ L 5Xr-X 4 = 7 — 2.25 = 4.75 feet, and 3 + 
5° + 3° »° 


50 


X —V— x 7 
50 + 30 


B + 6 


When One Body is at Rest. 


~3 — 3 + 3-75 — 6-75 feet. 

When Moving in Opposite Directions. 

X (Y + v) — v = r. 


Tr m-t-n b(V 4 ~v) , m-j-n B 

V- - — x ' = R, and - -■ X 


\ m / 
B + & 


B + i!> 


= R, and 


Bv(i+") 

\ ml 


Illustration.— Assume elements of preceding case. 

7 x( 5 o-I X 3 o) ?x — 


15 


B + 6 
50 X 7 X 


50 + 30 
35o X 1.5 


80 


= 3.0625 feet, 


and 


(- + :) 


50 + 30 


= 6.5625 feet. 


80 















































LIGHT, 



LIGHT. 

Light is similar to Heat in many of its qualities, being emitted in 
form of rays, and subject to same laws of reflection. 

It is of two kinds, Natural and Artificial; one proceeding from Sun 
and Stars, the other from heated bodies. 

Solids shine in dark only at a temperature from 6oo° to 700°, and in 
daylight at iooo°. 

Intensity of Light is inversely as square of distance from luminous 
body. 

Velocity of Light of Sun is 185 000 miles per second. 

Standard of Intensity or of comparison of light between different methods 
of Illumination is a Sperm Candle “ short 6,” burning 120 grains per hour. 

Caraclles. 

A Spermaceti candle .85 of a inch in diameter consumes an inch in length 
in 1 hour. 


Decomposition of Light. 


Colors. 

Maximum 

Contrasts. 


Combinations. 


Ray. 

Primary. 

Second’y. 

Tertiary. 

Primary. 

Secondary. 

Tertiary. 

Violet.... 

Chemical. 

_ 

_ _ 

_ 

Blue... 1 

Green.. 

_ 

Indigo.... 

— 

— 

— 

Brown. 

Yellow, j 

Dark. 

Blue. 

Electrical. 

Blue. 

— 

— 

Blue... ) 

Purple. ) 

Green. 

Green.... 

— 

— 

Green. 

Green. 

Red.... j 

Orange. 1 

Gray. 

Yellow... 

Light. 

Yellow. 

— 

— 

— 

Green.. j 

Orange... 

— 

— 

Orange. 

Broken. 

Yellow. 1 

Purple. ) 

Brown. 

Red. 

Heat. 

Red. 

Purple. 

Green. 

Red.... ) 

Orange. J 


All colors of spectrum, when combined, are white. 


Consumption and Comparative Intensity" of Light 

of Candles. 


Candle. 

No. in a 
Lb. 

Diameter. 

Length. 

Consumption 
per Hour. 

Light comp’d 
with Carcel. 

Wax. 

*3 

Inch. 

1 

Ins. 

12 

Grains. 


U 

D 

“3 

•875 

• Q 

15 

15 

13-5 

8.5 

J 135 

.09 

Spermaceti. 

«3 

*3 

| 156 


U 

O 

A 

.8 

.09 

u 

6 

.84 

I 

Tallow. 


12.5 

is 

13-75 

| 204 


U 

O 

3 

•9 

•07 

u 

4 

.8 


Compared with 1000 Cube Feet of Gas. 


Candle. 

Gas=i. 

Con¬ 

sump¬ 

tion. 

Light. 

Con¬ 
sumption 
for equal 
Light. 

Candle. 

Gas=i. 

Con¬ 

sump¬ 

tion. 

Light. 

Con¬ 
sumption 
for equal 
Light. 



Lbs. 

Lbs. 




Lbs. 

Lbs. 


Paraffine. 

.098 

3-5 

35-5 

103 

Adamantine. 

. 108 

5 -i 

47.2 

137 

Sperm... 

•095 

3-9 

41.1 

120 

Tallow. 

.074 

5 -i 

53-8 

155 


In combustion of oil in an ordinary lamp, a straight or horizontally cut wick 
gives great economy over one irregularly cut. 






















































584 


LIGHT. 


Relative Intensity, Consumption, Illumination, and. 
Cost of various IVEodes of Illumination. 


Oil at n cents, Tallow at 14 cents, Wax at 52 cents, and Stearine at 32 cents per 
lb. 100 cube feet coal gas at 14 cents, and 100 cube feet of oil gas at 52 cents. 


Illuminator. 

Illumi¬ 
nation. 
Carcel 
Lamp 
= 100. 

Actual 

Cost 

per 

Hour. 

Cost for 
equal 
Inten¬ 
sity. 

Illuminator. 

Illumi¬ 

nation. 

Carcel 

Lamp 

= IOO. 

Actual 

Cost 

per 

Hour. 

Cost for 
equal 
Inten¬ 
sity. 

Carcel Lamp. 

IOO 

Cents. 

.87 

Per H’r. 
.87 

Stearine Candle 5 to lb. 

66.6 

Cents. 

•59 

Per H’r. 

4- I 3 

Lamp with in-) 
verted reserv’r. j 

5 7 - 8 

.89 

•99 

Tallow “ 6 “ 
Sperm “ 6 “ 

54 

67-5 

•25 

.89 

2-34 

5-7 

Astral Lamp. 

Wax Candle 6 to lb. 

48.7 

61.6 

•56 

.92 

1.78 

6.31 

Coal Gas. 

Oil Gas. 


1.22 

1.25 

.96 

.98 


1000 cube feet of 13-candle coal gas is equal to 7.5 gallons sperm oil, 52.9 lbs. mold, 
and 44.6 lbs. sperm candles. 


Candles, Lamps, Fluids, and Gras. 

Comparison of several Varieties of Candles , Lamps , and Fluids , with Coal * Gas, de¬ 
duced from Reports of Com. of Franklin Institute, and of A. Frye, 31 . D., etc. 


Candle. 

Intensity 

of 

Light.f 

Light 
at Equal 
Costs. 

Cost com¬ 
pared with 
Gas for 
Equal Light. 

Candle. 

Intensity 

Light, t 

Light 

at Equal 

Costs. 

Cost com¬ 

pared with 
Gas for 

Equal Light. 

Diaphane. 


. q 

I 5 -I 

16.2 

Tallow, short 6’s, 1 
double wick .. j 
Wax, short 6’s.... 
Palm oil. 




Spermaceti, short 6 ? s, 
Tallow, short 6’s, ) 
single wick .,. j :' 

.8 

J 

•54 

I 

I 

7- 1 

•53 

•85 

7-5 

.8 

7 

. 6 l 

•77 

^ d 

M H 


* City of Philadelphia. 1 Compared with a fish-tail jet of Edinburgh gas, containing 12 per cent, 
of condensable matter and consuming 1 cube foot per hour. 


Lamp and Fluid. 

Inten¬ 
sity of 
Light. 

Light 

at 

Equal 

Cost. 

Time of 
Burning 
1 Pint 
of Oil. 

Lamp and Fluid. 

Inten¬ 
sity of 
Light. 

Light 

at 

Equal 

Cost. 

Time of 
Burning 
1 Pint 
of Oil. 

Carcel. 

Sperm oil, max^m 

2.15 

1.8 

Hours. 

6.32 

Gas. 

I 

I 

Hours. 

‘ ‘ mean. 

1.22 

i -35 

9.87 

Semi-solar, Sperm oil 

i -15 

•93 

6-75 

“ min’m 

.69 

•77 

1.2 

14.6 

Solar, Sperm oil. 

1.76 

i -55 

8.42 

9 - 3 i 

Lard oil. 

•97 

n -3 

Camphene. 

i -75 

3.08 


Loss of Li fit by Use of Glass Globes. 

Clear Glass, 12 per cent. | Half ground, 35 per cent. | Full ground, 40 per cent. 


Refraction. 

Relative Index of Refraction —Is, Ratio of sine of angle of incidence to sine of 
angle of refraction, when a ray of light passes from one medium into another. 

Absolute Index or Index of Refraction —Is, When a ray passes from a vacuum into 
any medium, the ratio is greater than unity. 

Relative index of refraction from any medium, as A, into another, as B, is always 
equal to absolute index of B, divided by absolute index of A. 

Absolute index of air is so small, that it may be neglected when compared with 
liquids or solids; strictly, however, relative index for a ray passing from air into a 
given substance must be multiplied by absolute index for air, in order to obtain 
like index of refraction for the substance. 


Air at 32 0 . 

Alcohol. 

Canada balsam. t 
Crystalline lens. 


IVIean Indices of Refraction. 


.... 1 

- 1.37 

.... 1.54 
.... 1.34 


Glass, fluid. 

1-58 

1.64 

“ crown. 

i -53 

1.56 


Humors of eye.... 

Salt, rock. 

Water, fresh. 

“ sea. 


i -34 

i -55 

i -34 

1 - 34 — 

































































LIGHT. 


585 


Gras. 

Retort .—A retort produces about 600 cube feet of gas in 5 hours with a 
charge of about 1.5 cwt. of coal, or 2800 cube feet in 24 hours. 

In estimating, number of retorts required, one fourth should be added for 
being under repairs, etc. 

Pressure with which gas is forced through pipes should seldom exceed 2.5 
ins. of water at the Works, or leakage will exceed advantages to be obtained 
from increased pressure. 

The average mean pressure in street mains is equal to that of 1 inch of 
water. 

When pipes are laid at an inclination either above or below horizon, a cor¬ 
rection will have to be made in estimating supply, by adding or deducting 
.01 inch from initial pressure for every foot of rise or fall in the length of pipe. 

It is customary to locate a governor at each change of level of 30 feet. 

Illuminating power of coal-gas varies from 1.6 to 4.4 times that of a tallow 
candle 6 to a lb.; consumption being from 1.5 to 2.3 cube feet per hour, and 
specific gravity from .42 to .58. 

Higher the flame from a burner greater the intensity of the light, the 
most effective height being 5 ins. 

Standard of gas burning is a 15-hole Argand lamp, internal diameter .44 
inch, chimney 7 ins. in height, and consumption 5 cube feet per hour, giving 
a light from ordinary coal-gas of from 10 to 12 candles, with Cannel coal 
from 20 to 24 candles, and with rich coals of Virginia and Pennsylvania of 
from 14 to 16 candles. 

In Philadelphia, with a fish-tail burner, consuming 4.26 cube feet per hour, 
illuminating power was equal to 17.9 candles, and with an Argand burner, 
consuming 5.28 cube feet per hour, illuminating power w T as 20.4 candles. 

Gas, which at level of sea would have a Value of 100, would have but 60 
in city of Mexico. 

Internal lights require 4 cube feet, and external lights about 5 per hour. 
When large or Argand burners are used, from 6 to 10 are required. 

An ordinary single-jet house burner consumes 5 to 6 cube feet per hour. 

Street-lamps in city of New York consume 3 cube feet per hour. In some 
cities 4 and 5 cube feet are consumed. Fish-tail burners for ordinary coal 
gas consume from 4 to 5 cube feet of gas per hour. 

A cube foot of good gas, from a jet .033 inch in diameter and height of 
flame of 4 ins., will burn for 65 minutes. 

Resin Gas .—Jet .033, flame 5 ins., 1.25 cube feet per hour. 

Purifiers .—Wet purifiers require 1 bushel of lime mixed with 48 bushels 
of water for 10 000 cube feet of gas. 

Dry purifiers require 1 bushel of lime to 10000 cube feet of gas, and 1 
superficial foot for every 400 cube feet of gas. 

Intensity of Liglit with. Equal Volumes of Gras from 
different Burners. 


Equal to Spermaceti Candle burning 120 Grains per Hour. 



Exp 

enditure in Cube 


Exp 

enditure in Cube 

Burners. 

E 

'eet per Hour. 

Burners. 

I 

'eet per Hour. 


1 

2 

3 

4 


1 

2 

3 

4 

Single-jet, 1 foot. 

2.6 

_ 

TT^ 

.— 

Argand, 16 holes.... 

•32 

1.9 

3-3 

3-8 

Fish-tail No. 3 . 

3-5 

4 

4.2 

— 

Argand, 24 holes.... 

•33 

2.2 

3-4 

5-3 

Bat’swing. . 

3 

4.1 

4-3 

4-5 

Argand, 28 holes... . 

•34 

2-3 

3-5 

5-8 



















LIGHT. 


586 

"Volume of Gas obtained, from a Ton. of Coal, Resin, etc. 


Material. 

Cube 

Feet. 

Material. 

Cube 

Feet. 

Material. 

Cube 

Feet. 

Boghead Cannel... 
Wigan Cannel. 

Cannel.j 

Cape Breton, ) 
“Cow Bay,” > .. 
etc.) 

13 334 
15 426 
8960 
15 000 

9500 

Cumberland. 

English, mean. 

Newcastle.j 

Oil and Grease. 

Pictou and Sidney.. 
Pine wood. 

9 800 
11 000 
9500 

10 000 

23 000 

8 000 

11 800 

Pittsburgh. 

Resin. 

Scotch.| 

Virginia. 

“ West’n.. 

Walls-end. 

9520 
15 600 
10 300 
15000 
8 960 
9500 
12 000 


1 Chaldron Newcastle coal, 3136 lbs., will furnish 8600 cube feet of gas at 
a specific gravity of .4, 1454 lbs. coke, 14.1 gallons tar, and 15 gallons am- 
moniacal liquor. 


Australian coal is superior to Welsh in producing of gas. 

Wigan Cannel, 1 ton, has produced coke, 1326 lbs.; gas, 338 lbs.; tar, 
250 lbs.; loss, 326 lbs. 

Peat , 1 lb. will produce gas for a light of one hour. 

Fuel, required for a retort 18 lbs. per 100 lbs. of coal. 

In distilling 56 lbs. of coal, volume of gas produced in cube feet when 
distillation was effected in 3 hours was 41.3, in 7, 37.5, in 20, 33.5, and in 
25, 31-7- 

Flow of Gras in Pipes. 

Flow of Gas is determined by same rules as govern that of flow of water. 
Pressure applied is indicated and estimated in inches of water, usually from 
.5 to 1 inch. 

Volumes of gases of like specific gravities discharged in equal times by a 
horizontal pipe, under same pressure and for different lengths, are inversely 
as square roots of lengths. 

Velocity of gases of different specific gravities, under like pressure, are in¬ 
versely as square roots of their gravities. 

By experiment, 30 000 cube feet of gas, specific gravity of .42, were dis¬ 
charged in an hour through a main 6 ins. in diameter and 22.5 feet in length. 

Loss of volume of discharge by friction, in a pipe 6 ins. in diameter and 1 
mile in length, is estimated at 95 per cent. 


Diameter and. Pength of Gras-pipes to transmit given 
Volumes of Gras to Branch-pipes. (Dr. Ure.) 


Volume 
per Hour. 

Diameter. 

Length. 

Volume 
per Hour. 

Diameter. 

Length. 

Volume 
per Hour. 

Diameter. 

Length. 

Cube Feet. 

Ins. 

Feet. 

Cube Feet. 

Ins. 

Feet. 

Cube Feet. 

Ins. 

Feet. 

5 ° 

•4 

IOO 

IOOO 

3.16 

IOOO 

2000 

7 

6000 

250 

I 

200 

1500 

3-87 

IOOO 

6000 

7-75 

IOOO 

500 

1.97 

600 

2000 

5-32 

2000 

6000 

9.21 

2000 

700 

2.65 

IOOO 

2000 

6-33 

4000 

8000 

8-95 

IOOO 


Regulation of Diameter and Extreme Length of Tub¬ 
ing, and IN' umber of Burners permitted. 


Diameter 

of 

Tubing. 

Length. 

Capacity 

of 

Meters. 

Burners. 

Diameter 

of 

Tubing. 

Length. 

Capacity 

of 

Meters. 

Burners 

Ins. 

Feet. 

Light. 

No. 

Ins. 

Feet. 

Light. 

No. 

•25 

6 

3 

9 

•75 

50 

30 

90 

•375 

20 

5 

15 

I 

70 

45 

135 

•5 

30 

IO 

30 

1.25 

IOO 

. 60 

180 

.625 

40 

20 

60 

i -5 

150 

IOO 

300 
























































LIGHT. 


58 ; 


Temperature of Gases. —Combustion of a cube foot of common gas will 
heat 650 lbs. of water i°. 


Services for Lamps. 


Lamps. 

Length 
from Main. 

Diameter 
of Pipe. 

Lamps. 

Length 
from Main. 

Diameter 
of Pipe. 

Lamps. 

Length 
from Main. 

Diameter 
of Pipe. 

No. 

Feet. 

Ins. 

No. 

Feet. 

Ins. 

No. 

Feet. 

Ins. 

2 

40 

•375 

IO 

IOO 

•75 

. 25 

180 

i -5 

4 

40 

•5 

i 5 

130 

I 

30 

200 

!-75 

6 

50 

.625 

20 

150 

1.25 





Volumes of* Gras Li i sc Large cl per Hour under a Pressure 
of LLalf an. IncL of Water. 


Specific Gravity .42. 


Diam. of 
Opening. 

Volume. 

Diam. of 
Opening. 

Volume. 

Diam. of 
Opening. 

Volume. 

Diam. of 
Opening. 

Volume. 

Ins. 

•25 

•5 

Cube Feet. 
80 

321 

Ins. 

•75 

1 

Cube Feet. 
723 
1287 

Ins. 

1-125 

1-25 

Cube Feet. 
1625 
2010 

Ins. 

i -5 

5 

Cube Feet. 
2885 
46 150 


To Compute Volume of Gras DiscLarged through a L*ipe. 

/V 2 G l representing diameter of pipe, and 


h t r ,, . e /V 2 G l . 

J-Ql = and .0635/-*- = * 


h height of water in ins., denoting pressure upon gas, l length of pipe in yards, G 
specific gravity of gas, and V volume in cube feet per hour. 

G may be assumed for ordinary computation at .42, and h .5 to 1 inch. 
Illustration. —Assume diameter of pipe 1 inch, pressure 1.68 ins., and length 
of pipe 1 yard. 

/i X 168 /1.68 

1000 X /-■ = 1000 X / • - ■ = 2000 cube feet, 

V -4 2 X x V • 4 2 

, „ /a 000000 X -42 X 1 -/i 68 000 000 

and .063 X S /* -- — SJ - — Q = 'i.°5 ins. 


1.68 yf 1.68 

Note.—F or tables deduced by above formulas see Molesworth, 1878, page 226. 


Dimensions of HVIains, witL "Weight of One IjeiigtL. 


Diameter in ins. 

4 

6 

8 

9 

IO 

14 

18 

Length in feet. 

9 

9 

9 

9 

9 

9 

9 

Thickness in ins. ... 

•375 

•375 

•5 

•5 

•5 

.625 

•75 

Weight in lbs. 

288 

224 

400 

454 

489 

868 

1316 


20 

9 

1484 


•75 


GAS ENGINES. 


In the Lenoir engine, the best proportions of air and gas are, for common 
gas, 8 volumes of air to 1 of gas, and for cannel gas, 11 of air to 1 of gas. 
The time of explosion is about the 27th part of a second. 

An engine, having a cylinder 4.625 ins. in diameter and 8.75 ins. stroke of 
piston, making 185 revolutions per minute, develops a half horse-power. 

Distribution of Heat Generated in the Cylinder. (M. Tresca.) 

Per cent. Per cent. 


Losses. 


Dissipated by the water and prod¬ 
ucts of combustion. 69 

Converted into work. 4 

Hence efficiency as determined by the brake — 4 per cent. 

Atmospheric Gas Engine. 


27 

100 


A single-acting cylinder 6 ins. in diameter, making 81 strokes per minute, devel¬ 
oped .456 IP, and the gas consumed per minute for cylinder 20 cube feet and for in¬ 
flaming 2 cube feet. (M. Tresca.) 






















































588 LIMES, CEMENTS, MORTARS, AND CONCRETES. 


LIMES, CEMENTS, MORTARS, AND CONCRETES. 

Essentially from a Treatise by Brig.-Gen'l Q. A. Gillmore , U.S.A .* 

Lime. 

Calcination of marble or any pure limestone produces lime ( quick¬ 
lime ). Pure limestones burn white, and give richest limes. 

Finest calcareous minerals are rhombohedral prisms of calcareous 
spar, the transparent double-reflecting Iceland spar, and white or statu¬ 
ary marble. 

Property of hardening under water, or when excluded from air, con¬ 
ferred upon a paste of lime, is effected by presence of foreign sub¬ 
stances—as silicum, alumina, iron, etc.—when their aggregate presence 
amounts to .1 of whole. 

Limes are classed: i. Common or Fat limes, which do not set in water. 

2. Poor or Meagre, mixed with sand, which does not alter its condition. 

3. Hydraulic Lime, containing 8 to 12 per cent, of silica, alumina, iron, 
etc., set slowly in water. 4. Hydraulic, containing 12 to 20 per cent, of 
similar ingredients, sets in water in 6 or 8 days. 5. Eminently Hydraulic, 
containing 20 to 30 per cent, of similar ingredients, sets in water in 2 to 4 
days. 6. Hydraulic Cement, containing 30 to 50 per cent, of argil, sets in a 
few minutes, and attains the hardness of stone in a few months. 7. Natural 
Pozzuolanas, including pozzuolana properly so called, Trass or Terras, Arenes, 
Oclireous earths, Basaltic sands, and a variety of similar substances. 

Indications of Limestones. They dissolve wholly or partly in weak acids 
with brisk effervescence, and are nearly insoluble in water. 

Rich Limes are fully dissolved in water frequently renewed, and they 
remain a long time without hardening; they also increase greatly in vol¬ 
ume, from 2 to 3.5 times their original bulks, and will not harden without 
the action of air. They are rendered Hydraulic by admixture of pozzuolana 
or trass. 

Rich, fat, or common Limes usually contain less than 10 per cent, of im¬ 
purities. 

Hydraulic limestones are those which contain iron and clay, so as to en¬ 
able them to produce cements which become solid when under water. 

Poor Limes have all the defects of rich limes, and increase but slightly in 
bulk, the poorer limes are invariably basis of the most rapidly-setting 
and most durable cements and mortars, and they are also the only limes 
which have the property, when in combination with silica, etc., of indurating 
under water, and are therefore applicable for admixture.of hydraulic cements 
or mortars. Alike to rich limes, they will not harden if in a state of paste 
under water or in wet soil, or if excluded from contact with the atmosphere 
or carbonic acid gas. They should be employed for mortar only when it is 
impracticable to procure common or hydraulic lime qr cement, in which case 
it is recommended to reduce them to powder by grinding. 

Hydraulic Limes are those which readily harden under water. The most 
valuable or eminently hydraulic set from the 2d to the 4th day after immer¬ 
sion; at end of a month they become hard and insoluble, and at end of six 
months they are capable of being worked like the hard, natural limestones. 
They absorb less water than pure limes, and only increase in bulk from 1.75 
to 2.5 times their original volume. 


* See also Ms Treatises on Limes, Hydraulic Cements, and Mortars, in Papers on Practical Engineer¬ 
ing, Engineer Department, U. S. A. 




LIMES, CEMENTS, MORTARS, AND CONCRETES. 589 

Inferior grades, or moderately hydraulic , require a period of from 15 
to 20 days’ immersion, and continue to harden for a period of 6 months. 

Resistance of hydraulic limes increase if sand is mixed in proportion 
of 50 to 180 per cent, of the part in volume; from thence it decreases. 

M. Vicat declares that lime is rendered hydraulic by admixture with it of from 
33 to 40 per cent, of clay and silica, and that a lime is obtained which does not 
slake, and which quickly sets under water. 

Artificial Hydraulic Limes do not attain, even under favorable circum¬ 
stances, the same degree of hardness and power of resistance to compression 
as natural limes of same class. 

Close-grained and densest limestones furnish best limes. 

Hydraulic limes lose or depreciate in value by exposure to the air. 

Pastes of fat limes shrink, in hardening, to such a*degree that they can¬ 
not be used as mortar without a large proportion of sand. 

Arenes is a species of ochreous sand. It is found in France. On account 
of the large proportion of clay it contains, sometimes as great as .7, it can be 
made into a paste with water without any addition of lime; hence it is some¬ 
times used in that state for -walls constructed en pise , as well as for mortar. 
Mixed with rich lime it gives excellent mortar, which attains great hardness 
under water, and possesses great hydraulic energy. 

Pozzuolana is of volcanic origin. It comprises Trass or Terras, the Arenes, 
some of the ochreous earths, and the sand of certain graywackes, granites, 
schists, and basalts; their principal elements are silica and alumina, the 
former preponderating. None contain more than 10 per cent, of lime. 

When finely pulverized, without previous calcination, and combined with paste 
of fat lime in proportions suitable to supply its deficiency in that element, it pos¬ 
sesses hydraulic energy to a valuable degree. It is used in combination with rich 
lime, and may be made by slightly calcining clay and driving off the water of com¬ 
bination at a temperature of 1200 0 . 

Brick or Tile Dust combined with rich lime possesses hydraulic energy. 

Trass or Terras is a blue-black trap, and is also of volcanic origin. It 
requires to be pulverized and combined with rich lime to render it fit for 
use, and to develop any of its hydraulic properties. 

General Gillmore designates the varieties of hydraulic limes as follows: If, after 
being slaked, they harden under water in periods varying from 15 to 20 days after 
immersion, slightly hydraulic; if from 6 to 8 days, hydraulic; and if from 1 to 4 
days, eminently hydraulic. 

Pulverized silica burned with rich lime produces hydraulic lime of ex¬ 
cellent quality. Hydraulic limes are injured by air-slaking in a ratio vary¬ 
ing directly with their hydraulicity, and they deteriorate by age. 

For foundations in a damp soil or exposure, hydraulic limes must be ex¬ 
clusively employed. 

Hydraulic Lime ofTeil is a silicious hydraulic lime; it is slow in setting, 
requiring a period of from 18 to 24 hours. 

Cements. 

Hydraulic Cements contain a larger proportion of silica, alumina, magnesia, 
etc., than any of preceding varieties of lime; they do not slake after calcina¬ 
tion, and are superior to the very best of hydraulic limes, as some of them 
set under water at a moderate temperature (65°) in from 3 to 4 minutes; 
others require as many hours. They do not shrink in hardening, and make 
an excellent mortar without anv admixture of sand. 

3 D 


590 LIMES, CEMENTS, MORTARS, AND CONCRETES. 


When exposed to air, they absorb moisture and carbonic acid gas, and are 
rapidly deteriorated thereby. 

Roman Cement is made from a lime of a peculiar character, found in Eng¬ 
land and France, derived from argillo-calcareous kidney-shaped stones termed 
Sept aria. 

It is about .33 strength of Portland, and is not adapted for use with sand. 

Rosendale Cement is from Rosendale, New York. 

Portland Cement is made in England and France. It requires less water 
(cement 1, water .29) than Roman cement, sets slowly, and can be remixed 
with additional water after an interval of 12 or even 24 hours from its first 
mixture. 

Property of setting slow may be an obstacle to use of some designations of this 
cement, as the Boulogne, when required for localities having to contend against 
immediate causes of destruction, as in sea constructions, having to be executed un¬ 
der water and between tides. On the other hand, a quick-setting cement is always 
difficult of use ; it requires special workmen and an active supervision. A slow- 
setting cement, however, like natural Portland, possesses the advantage of being 
managed by ordinary workmen, and it can also be remixed with additional water 
after an interval of 12 or even 24 hours from its first mixing. 

Conclusions derived from Mr. Grant's Experiments. 

1. Portland cement improves by age, if kept from moisture. 

2. Longer it is in setting, stronger it will be. 

3. At end of a year, 1 of cement to 1 sand is about .75 strength of neat cement; 

1 to 2, .5 strength; 1 to 3, .33; 1 to 4, .25; 1 to 5, .16. 

4. Cleaner and sharper the sand, greater the strength. 

5. Strong cement is heavy; blue gray, slow-setting. Quick-setting has generally 
too much clay in its composition—is brownish and w T eak. 

6. Less water used in mixing cement the better. 

7. Bricks, stones, etc., used with cement should be well wetted before use. 

8. Cement setting under still w r ater will be stronger than if kept dry. 

9. Bricks of neat Portland cement in a few months are equal to Blue bricks, 
Bramley-Fall stone, or Yorkshire landings. 

10. Bricks of 1 cement to 4 or 5 of sand are equal to picked stock bricks. 

11. When concrete is being used, a current of water will wash away the cement. 

Artificial Cement is made by a combination of slaked lime with unburned 
clay in suitable proportions. 

Artificial Pozzuolana is made by subjecting clay to a slight calcination. 

Salt water has a tendency to decompose cements of all kinds, and their 
strength is considerably impaired by their mixture with it. 

WLortaT 1 . 

Lime or Cement paste is the cementing substance in mortar, and its pro¬ 
portion should be determined by the rule that Volume of cementing substance, 
should, be somewhat in excess of volume of voids or spaces in sand or coarse 
material to be united, the excess being added to meet imperfect manipulation 
of the mass. 

Hydraxdic Mortar , if re-pulverized and formed into a paste after having 
once set, immediately loses a great portion of its liydraulicity, and descends 
to the level of moderate hydraulic limes. 

The retarding influence of sea-v T ater upon initial hydraulic induration is 
not very great, if the cement is mixed with fresh water. The strength of 
mortars, however, is considerably impaired by being mixed with sea-water. 

Pointing Mortar is composed of a paste of finely-ground cement and clean 
sharp siliceous sand, in such proportions that the volume of cement paste is 
slightly in excess of the volume of voids or spaces in the sand. The volume 


LIMES, CEMENTS, MORTARS, AND CONCRETES. 59 1 


of sand varies from 2.5 to 2.75 that of the cement paste, or by weight, 1 of 
cement powder to 3 to 3.33 of sand. The mixture should be made under 
shelter, and in small quantities. 

All mortars are much improved by being worked or manipulated; and as rich 
limes gain somewhat by exposure to the air, it is advisable to work mortar in 
large quantities, and then render it fit for use by a second manipulation. 

White lime will take a larger proportion of sand than brown lime. 

Use of salt-water in the composition of mortar injures adhesion of it. 

When a small quantity of water is mixed with slaked lime, a stiff paste 
is made, which, upon becoming dry or hard, has but very little tenacity, but, 
by being mixed with sand or like substance, it acquires the properties of a 
cement or mortar. 

Proportion of sand that can be incorporated with mortar depends partly 
upon the degree of fineness of the sand itself, and partly upon character of 
the lime. For rich limes, the resistance is increased if the sand is in pro¬ 
portions varying from 50 to 240 per cent, of the paste in volume; beyond 
this proportion the resistance decreases. 

Lime, 1, clean sharp sand, 2.5. An excess of water in slaking the lime 
swells the mortar, which remains light and porous, or shrinks in drying; an 
excess of sand destroys the cohesive properties of the mass. 

It is indispensable that the sand should be sharp and clean. 

Stone Mortar. —8 parts cement, 3 parts lime, and 31 parts of sand; or x 
cask cement, 325 lbs., .5 cask of lime, 120 lbs., and 14.7 cube feet of sand= 
18.5 cube feet of mortar. 

Bride Mortar. —8 parts cement, 3 parts lime, and 27 parts of sand; or 1 
cask cement, 325 lbs., .5 cask of lime, 120 lbs., and 12 cube feet of sand= 
16 cube feet of mortar. 

Brown Mortar. —Lime 1 part, sand 2 parts, and a small quantity of hair. 

Lime and sand, and cement and sand, lessen about in volume when mixed 
together. 

Calcareous Mortar , being composed of one or more of the varieties of lime 
or cement, natural or artificial, mixed with sand, will vary in its properties 
with quality of the lime or cement used, the nature and quality of sand, and 
method of manipulation. 

TviiTvislx IPlaster, or LTydravilic Cement. 

100 lbs. fresh lime reduced to powder, 10 quarts linseed-oil, and 1 to 2 
ounces cotton. Manipulate the lime, gradually mixing the oil and cotton, in 
a wooden vessel, until mixture becomes of the consistency of bread-dough. 

Dry, and when required for use, mix with linseed-oil to the consistency of paste, 
and then lay on in coats. Water-pipes of clay or metal, joined or coated with it, 
resist the effect of humidity for very long periods. 

Stu.cco. 

Stucco or Exterior Plaster is term given to a certain mortar designed for 
exterior plastering; it is sometimes manipulated to resemble variegated 
marble, and consists of 1 volume of cement powder to 2 volumes of dry sand. 

In India, to water for mixing the plaster is added 1 lb. of sugar or molas¬ 
ses to 8 Imperial gallons of w r ater, for the first coat; and for second or finish¬ 
ing, 1 lb. sugar to 2 gallons of water. 

Powdered slaked lime and Smith’s forge scales, mixed with blood in suit¬ 
able proportions, make a moderate hydraulic mortar, which adheres well to 
masonry previously coated with boiled oil. 


592 LIMES, CEMENTS, MORTARS, AND CONCRETES. 


Plaster should be applied in two coats laid on in one operation, flrst coat being 
thinner than second. Second coat is applied upon first while latter is yet soft. 

The two coats should form one of about 1.5 inches in thickness, and when fin¬ 
ished it should be kept moist for several days. 

When the cement is of too dark a color for desired shade, it may be mixed with 
white sand in whole or in part, or lime paste may be added until its volume equals 
that of the cement paste. 

Khorassar, or Tmrkisli Mortar, 

Used for the construction of buildings requiring great solidity, .33 pow¬ 
dered brick and tiles, .66 line sifted lime. Mix with water to required con¬ 
sistency, and lay between the courses of brick or stones. 

Mortars. 

Mortars used for inside plastering are termed Coarse, Fine, Gauge or hard 
finish, and Stucco. 

Plastering .—1 bushel, or 1.25 cube feet of cement, mortar, etc., will cover 1.5 
square yards .75 inch thick. 75 volumes are required upon brick work for 70 upon 
laths. 

When full time for hardening cannot be allowed, substitute from 15 to 20 per 
cent, of the lime by an equal proportion of hydraulic cement. 

For the second or brown coat the proportion of hair may be slightly diminished. 

Coarse StafF. — Common lime mortar, as made for brick masonry, 
with a small quantity of hair; or by volumes, lime paste (30 lbs. lime) 1 
part, sand 2 to 2.25 parts, hair .16 part. 

Fine Staff (lime putty).—Lump lime slaked to a paste with a mod¬ 
erate volume of water, and afterwards diluted to consistency of cream, and 
then to harden by evaporation to required consistency for working. 

In this state it is used for a slipped coat , and when mixed with sand or plaster of 
Paris, it is used for finishing coat. 

Grange, or Hard Finish, is composed of from 3 to 4 volumes fine 
stuff and 1 volume plaster of Paris, in proportions regulated by rapidity re¬ 
quired in hardening; for cornices, etc., proportions are equal volumes of 
each, fine stuff and plaster. 

Scratch Coat. —First of three coats when laid upon laths, and is from .25 to 
•375 °f an inch ki thickness. 

One-coat Work. —Plastering in one coat without finish, either on masonry 
or laths—that is, rendered or laid. 

Two-coat Work. —Plastering in two coats is done either in a laid coat 
and set, or in a screed coat and set. 

Screed coat is also termed a Floated coat. Laid first coat in two-coat 
work is resorted to in common work instead of scree ding, when finished sur¬ 
face is not required to be exact to a straight-edge. It is laid in a coat of 
about .5 inch in thickness. 

Laid coat, except for very common work, should be hand-floated. 

Firmness and tenacity of plastering is very much increased by hand-floating. 

Screeds are strips of mortar 6 to 8 inches in width, and of required thick¬ 
ness of first coat, applied to the angles of a room, or edge of a wall and paral- 
lelly, at intervals of 3 to 5 feet over surface to be covered. When these have 
become sufficiently hard to withstand pressure of a straight-edge, the,inter¬ 
spaces between the screeds are filled out flush with them. 

Slipped Coat is the smoothing off of a brown coat with a small quantitv 
of lime putty, mixed with 3 per cent, of white sand, so as to make a compar¬ 
atively even surface. 

This finish answers when the surface is to be finished in distemper, or paper. 


LIMES, CEMENTS, MORTARS, AND CONCRETES. 593 


Concrete or Beton 

Is a mixture of mortar (generally hydraulic) with coarse materials, as 
gravel, pebbles, stones, shells, broken bricks, etc. Two or more of these 
materials, or .all of them, may be used together. As lime or cement paste is 
the cementing substance in mortar, so is mortar the cementing substance in 
concrete or beton. The original distinction between cement and beton was, 
that latter possessed hydraulic energy, while former did not. 

Hydraulic. —1.5 parts unslaked hydraulic lime, 1.5 parts sand, 1 part 
gravel, and 2 parts of a hard broken limestone. 

This mass contracts one fifth in volume. Fat lime may be mixed with concrete, 
without serious prejudice to its hydraulic energy. 

“V^ariovis Compositions of Concrete. 

Hydraulic. —308 lbs. cement = 3.65 to 3.7 cube feet of stiff paste. 12 cube 
feet of loose sand = 9.75 cube feet of dense. 

For Superstructure. —11.75 cube feet of mortar as above, and 16 cube feet 
of stone fragments. 

Sea Wall.—Boston Harbor. — Hydraulic. —308 lbs. cement, 8 cube feet of 
sand, and 30 cube feet of gravel. Whole producing 32.3 cube feet. 

Superstructure. —308 lbs. cement, 80 lbs. lime, and 14.6 cube feet dense 
sands. Whole producing 12.825 cube feet. 

3 ?ise is made of clay or earth rammed in layers of from 3 to 4 ins. in depth. In 
moist climates, it is necessary to protect the external surface of a wall constructed 
in this manner with a coat of mortar. 

-A.sph.alt Composition. 

1. Mineral pitch 1 part, bitumen n, powdered stone, or wood ashes, 7 parts. 

2. Ashes 2 parts, clay 3 parts, and sand 1 part, mixed with a little oil, makes a 
very fine and durable cement, suitable for external use. 

Flooring. —8 lbs. of composition will cover 1 sup. foot, .75 inch thick. 

Asphaltum 55 lbs. and gravel 28.7 lbs. will cover an area of 10.75 sq. feet. 

NEastic. —Pulverized burnt clay 93 parts, litharge, ground very fine, 7 parts, 
mixed with a sufficient quantity of pure linseed oil. 

3. Siliceous sand 14, pulverized calcareous stone 14, litharge 2, and linseed oil 4 
parts by weight. 

The powders to be well dried in an oven, and the surface upon which it is to be 
applied must be saturated with oil. 

4. For Roads. —Bitumen 16.875 parts, asphaltum 225 parts, oil of resin 6.25 parts, 
and sand 135 parts. Thickness, from 1.25 to 1.375 ins. 

Artificial Mastic. —Composition of 1 square yard .9 inch thick: 

Gravel.275 cube ins. 

Slaked lime. 55 “ “ 

1249 cube ins. 

3 VTviral TCfHoresceivce. —White alkaline efflorescence upon the surface 
of brick walls laid in mortar, of which natural hydraulic lime or cement is the basis. 

Mortar mixed with animal fat in the proportion of .025 of its weight will prevent 
its formation. 

Crystallization of these salts within the pores of bricks, into which they have 
been absorbed from the mortar, causes disintegration. 

Distemper is term for all coloring mixed with water and size. 

Grouting. —Mortar composed of lime and fine sand, in a semi-fluid state, 
poured into the upper beds and internal joints of masonry. 

Laitance is the pulpy and gelatinous fluid, of a milky hue, that is washed 
from cement upon its being deposited in water. It is produced more abun¬ 
dantly in sea water than in fresh; it sets very imperfectly, and has a ten¬ 
dency to lessen the strength of the concrete. 

3 D* 


Minei’al tar.205 cube ins. 

Pitch.165 “ “ 

Sand.549 “ “ 









594 LIMES > CEMENTS, MORTARS, AND CONCRETES. 


Slaking. 

Slaked Lime is a hydrate of lime, and it absorbs a mean of 2.5 times its 
volume, and 2.25 times its weight of water. 

Lime ( quicklime ) must be slaked before it can be used as a matrix for 
mortar. 

Ordinary method of slaking is by submitting the lime to its full propor¬ 
tion of water (previously known or attained by trial) in order to reduce it to 
the consistency of a thick pulp. The volume of water required for this pur¬ 
pose will vary with different limes, and will range from 2.5 to 3 volumes 
that of the lime, and it is imperative that it should all be poured upon it so 
nearly at one time as to be in advance of the elevation of the temperature 
consequent upon its reduction. 

This process, when the water used is in an excessive quantity, is termed 
“ drowning,” and when the volume of lime has increased by the absorption 
of water it is termed its “ growth.” 

If too much water is used, the binding qualities of the lime is injured by 
its semi-fluidity; and if too little, it is injurious to add after the reduction of 
the lime has commenced, as it reduces its temperature and renders it granu¬ 
lar and lumpy. 

While lime is in progress of slaking it should be covered with a tarpaulin 
or canvas (a layer of sand will suffice), in order to concentrate its evolved 
heat. 

The essential point in slaking is to attain the complete reduction of the 
lime, and the greater the hydraulic energy of a lime, the more difficult it be¬ 
comes to effect it. 

Whitewash or Grouting. —When lime is required for a whitewash or for 
grouting, it should be thoroughly “ drowned,” and then run off into tight ves¬ 
sels and closed. 

Slaking by Immersion is the method of suspending lime in a suitable ves¬ 
sel in water for a very brief period, and withdrawing it before reduction 
commences. The lime is then transferred to casks or like suitable receptacles, 
and tightly enclosed, until it is reduced to a fine powder, in which condition, 
if secured from absorption of air, it may be preserved for several months 
without essential deterioration. 

Spontaneous or Air Slaking. —When lime is not wholly secured from ex¬ 
posure to the air, it absorbs moisture therefrom, slakes, and falls into a powder. 

Limes and Cements.—A. Cask of Lime = 240 lbs., will make from 7.8 to 
8.15 cube feet of stiff paste. 

A Cask of Cement = 300* lbs., will make from 3.7 to 3.75 cube feet of 
stiff paste. 

A Cask of Portland Cement = 4 bushels or 5 cube feet = 420 lbs. 

A Cask of Roman Cement = 3 bushels or 3.75 cube feet = 364 lbs. 

.5 inch. .75 inch. i inch. 

A Bushel of cement will cover.2.25 yards 1.5 yards 1.14 yards. 


From experiments of General Totten, it appeared that 



One cube foot of dry cemeut, mixed with .33 cube foot of water, will make .63 to 
635 cube foot of stiff paste. 


Lime should be slaked at least one day before it is incorporated with the 
sand, and when they are thoroughly mixed, the mortar should be heaped into 
one volume or mass, for use as required. 


* 300 lbs. net is standard; it usually overruns 8 lbs. 





LIMES, CEMENTS, MORTARS, AND CONCRETES. 595 


Mortar, Cement, «Scc. ( Molesworth .) 

Mortar. —1 of lime to 2 to 3 of sharp river sand. 

Or, 1 of lime to 2 sand and 1 blacksmith’s ashes, or coarsely ground coke. 
Coarse Mortar. —1 of lime to 4 of coarse gravelly sand. 

Concrete 1 of lime to 4 of gravel and 2 of sand. 

Hydraulic Mortar. —1 of blue lias lime to 2.5 of burnt clay, ground to¬ 
gether. 

Or, 1 of blue lias lime to 6 of sharp sand, 1 of pozzuolana and 1 of calcined 
ironstone. 

Beton. —1 of hydraulic mortar to 1.5 of angular stones. 

Cement. — 1 of sand to 1 of cement.—If great tenacity is required, the ce¬ 
ment should be used without sand. 

fortland. Cement 

Is composed of clayey mud and chalk ground together, and afterwards cal¬ 
cined at a high temperature—after calcining it is ground to a fine powder. 


Strength, of jMortars, Cements, and. Concretes. 
Deduced from Experiments of Vicat, Paisley , Treussarf, and Voisin. 

Tensile 


Weight or Power required to Tear asunder One Sq. Inch. 
Cement Nlortar. (42 days old.) 





Proportion of Sand to 1 of Cement. 




0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

Roman. 

284 

284 

199 

166 

142 

128 

Il6 

106 

99 

92 

95 lbs. 

Portland. 

142 

142 

ii 3 

92 

79 

67 

57 

42 

35 

25 

_ <4 


Bricls, Stone, and G-ranite Masonry. (320 days old.) 
Experiments of General Gillmore, TJ. S. A. 

Cement on Bricks. 

Lbs. 

Pure..... 27.5 


Lbs. 

Pure, average. 30.8 


Sand 1 I 
Cement 1 J 
Sand 1 ) 

Cement 2 j 
Sand 1 ) 

Cement 3 j 


I 5-7 


6.8 


Delafield and Baxter. Lbs. 

Pure cement. 68 

Cement 4) . 68 

Sand 1 j 

Cement 81 n 

Siftings 1 j. eo 

Cement 1 1 o„ 

Siftings 1) . 

Cement 1 1 

Siftings 2 J . 

Lawrence Cement Co. 

Pure cement.87 

“ “ 54 


74 


Cement on Granite. 

Lbs. 


Sand 1 j 
Cement 1j 
Sand 1 1 

Cement 2j 
Sand 1 1 

Cement 3J 


20.8 

12.6 

9.2 


James River. Lbs. 

Pure cement. 87 

Cement 4 1 , 

Sand 1 }. 62 

Newark Lime and Cement 
Co. 

Pure cement. 93 

Cement 1 1 

Sand 2 j. 40 

Newark and Rosendale. 

Pure cement. 75 

Cement 1 1 
Sand x J . 


16 


Sand 1 ) 

Cement 4J 
Water 1 ( 
Cement 2 j 
Water .42) 
Cement 1 j 
Water .331 
Cement 1j 


Lbs. 

7-9 

20.5 

37-25 

29.15 


LbB. 


Newark and Rosendale. 
Cement 
Sand 3 

Pure, without 1 
mortar, mean ) 


'} 


Mortar. 


Lime paste 


sand 2.5, 
“ 2 

“ 3 


7 

45 


L 3 ) 

cement paste 5. it 



























































596 LIMES, CEMENTS, MOETAES, AND CONCEETES. 


Pure Cement. 


Lbs. 

Boulogne 100, water 50. 112 

Portland, natural, 1 year. 675 

“ artificial, Eng., 1 year... 462 

“ English, 320 days. 1152 

“ “ 1 month. 393 

Newark and Rosendale. 339 


Lbs. 

Portland, in sea-water, 45 days.266 

“ English, 6 months. 424 

Roman “Septaria,” 1 year. 191 

“ masonry, 5 months. 77 

Rosendale, 9 months... 700 

Lawrence Cement Co. 1210 


Transverse. 

Reduced to a uniform Measure of One Inch Square and One Foot in Length. 

Supported at Both Ends. 

Experiments of Grenera .1 Gfillmore. 

Formed in molds under a pressure of 32 lbs. per sq. inch, applied until mortar 
had set. Exposed to moisture for 24 hours, and then immersed in sea-water. 

Prisms 2 by 2 by 8 ins. between supports, 
a 

: C. C coefficient of rupture, and a weight of 


2 Z W 

Reduced by Formula - 


34 b d 2 2 


portion of prism l. 

Cement. 

Material. 

<0 

Pure. 

Mortar. 

Material. 

CO 

'OC 

<J 

Cement i. 

Sand 1. 

Cement i. 

Sand 2. 


Days. 

Lbs. 


Days. 

Lbs. 

Lbs. 

James River. 



Portland, Eng., stiff paste 

320 

13 

IO 

Thick cream. 

59 

3-9 

Roman, “ “ “ 

20 

2-5 

— 

Thin paste. 

320 

5-8 

u u u u 

IOO 

6 

— 

Stiff paste. 

59 

6.0 

Cumberland, Md. 


12.8 

7. 8 

Rosendale “Hoffman.” 


Akron, N. Y.. 

32° 

8.8 

8.4 

Thin paste. 

•220 

Q 

James River, Ya. 


8.6 

8.8 

Stiff paste. 

320 

8.9 

Pulverized and re-) 




“Delafield and Baxter.” 



mixed after set.... j 

3 

3 -° 


Thin paste. 

320 

8. s 

Fresh. 


Q 


Stiff paste. 

320 

12 

Kingston and Rosendale. 

320 

7.6 

6.6 

English. 



High Falls, Ul-) 




Portland, pure. 

320 

16 

ster Co.,N.Y. j . 

95 


3 - 2 

Stiff paste. 

320 

13 

Fresh w r ater to a stiff) 




Cumberland, Md., pure_ 

320 

13.2 

paste. j 

95 


4.4 

High Falls, Ul- ) 


Q j 

Seaavater to a stiff paste 

95 

— 

2.6 

ster Co., N. Y. ) . 

95 

°-4 

Lawrence Cement Co. 




Complete calcination.... 

95 

4.2 

Fresh . 

320 

10.2 

— 


Crushing. 

Cements, Stones, etc. (Crystal Palace , London .] 
Reduced to a uniform Measure of One Sq. Inch. 


Material. 

Destructive 

Pressure. 

Material. 

Destructive 

Pressure. 

Portl’d cem’t, area 1, height 1. 
“ cement ) 

Lbs. 

1680 

1244 

1144 

Portland cement 1) 

Lbs. 

1244 

692 

342 

“ sand 4 ] . 

“ cement 1) 

“ sand... j 

“ stone... 

“ sand 7 j . 

Roman cement, pure. 



General Deductions. 

1. Particles of unground cement exceeding .0125 of an inch in diameter may be 
allowed in cement paste without sand, to extent of 50 per cent, of whole, without 
detriment to its properties, while a corresponding proportion of sand injures the 
strength of mortar about 40 per cent. 























































LIMES, CEMENTS, MORTARS, ETC.-MASONRY. 597 


2. When these unground particles exist in cement paste to extent of 66 per cent, 
of whole, adhesive strength is diminished about 28 per cent. For a corresponding 
proportion of sand the diminution is 68 per cent. 

3. Addition of siftings exercises a less injurious effect upon the cohesive than upon 
the adhesive property of cement. The converse is true when sand, instead of sift¬ 
ings, is used. 

4. In all mixtures with siftings, even when the latter amounted to 66 per cent, of 
whole, cohesive strength of mortars exceeded their adhesion to bricks. Same re¬ 
sults appear to exist when siftings are replaced by sand, until volume of the latter 
exceeds 20 per cent, of whole, after which adhesion exceeds cohesion. 

5. At age of 320 days (and perhaps considerably within that period) cohesive 
strength of pure cement mortar exceeds that of Croton front bricks. The converse 
is true when the mortar contains 50 per cent, or more of sand. 

6. When cement is to be used without sand, as may be the case when grouting is 
resorted to, or when old walls are to be repaired by injections of thin paste, there is 
no advantage in having it ground to an impalpable powder. 

7. For economy it is customary to add lime to cement mortars, and this may be 
done to a considerable extent when in positions where hydraulic activity and 
strength are not required in an eminent degree. 

8. Ramming of concrete under water is held to be injurious. 

9. Mortars of common lime, when suitably made, set in a very few days, and with 
such rapidity that there is no need of awaiting its hardening in the prosecution of 
work. 

Wire Clay.—The fusibility of clay arises from the presence of impurities, 
such as lime, iron, and manganese. These may be removed by steeping the clay in 
hot muriatic acid, then washing it with water. Crucibles from common clay may 
be made iu this manner. 

Notes by General Gillmore , U. S. A .—Recent experiments have developed that 
most American cements will sustain, without any great loss of strength, a dose of 
lime paste equal to that of the cement paste, while a dose equal to .5 to .75 the vol¬ 
ume of cement paste may be safely added to any Rosendale cemeut without pro¬ 
ducing any essential deterioration of the quality of the mortar. Neither is the 
hydraulic activity of the mortars so far impaired by this limited addition of lime 
paste as to render them unsuited for concrete under water, or other submarine 
masonry. By the use of lime is secured the double advantages of slow setting and 
economy. 

Notes by General Totten , U. S. A .—240 lbs. lime = 1 cask, will make from 7.8 to 
8.15 cube feet of stiff paste. 

1 cube foot of dry cement powder, measured when loose, will measure .78 to .8 
cube foot when packed, as at a manufactory. 

For composition of Concretes, at Toulon, Marseilles, Cherbourg, Dover, Alderney, 
etc., see Treatise of General Gillmore, pp. 253-256. 


MASONRY. 

Brickwork. 

Bond is an arrangement of bricks or stones, laid aside of and above 
each other, so that the vertical joint between any two bricks or stones 
does not coincide with that between any other two. 

This is termed “breaking joints.” 

Header is a brick or stone laid with an end to face of wall. 

Stretcher is a brick or stone laid parallel to face of wall. 

Header Course or Bond is a course or courses of headers alone. 

Stretcher Course or Bond is a course or courses of stretchers alone. 

Closers are pieces of bricks inserted in alternate courses, in order to obtain 
a bond by preventing two headers from being exactly over a stretcher. 

English Bond is laying of headers and stretchers in alternates courses. 



MASONRY. 


598 

Flemish Bond is laying of headers and stretchers alternately in each course. 

Gauged Work. —Bricks cut and rubbed to exact shape required. 

String Course is a horizontal and projecting course around a building. 

Corbelling is projection of some courses of a wall beyond its face, in order 
to support wall-plates or floor-beams, etc. 

Wood Bricks, Pallets, Plugs, or Slips are pieces of wood laid in a wall in 
order the better to secure any woodwork that it may be necessary to fasten 
to it. 

Reveals are portions of sides of an opening in a wall in front of the recesses 
for a door or window frame. 

Brick Ashlar. —Walls with ashlar-facing backed with brick. 

Grouting is pouring liquid mortar over last course for the purpose of filling 
all vacuities. 

Lurrying is filling in of interior of thick walls or piers, after exterior faces 
are laid, with a bed of soft mortar and floating bricks or spawls in it. 

Rendering (Eng.) is application of first coat on masonry, Laying if one 
or two coats on laths, and “ Pricking up ” if three-coat work on laths. 

Bricks should be well wetted before use. Sea sand should not be used in the 
composition of mortar, as it contains salt and its grains are round, being worn by 
attrition, and consequently having less tenacity than sharp-edged grains. 

A common burned brick will absorb 1 pint or about one sixth of its weight of 
water to saturate it. The volume of water a brick will absorb is inversely a test of 
its quality. 

A good brick should not absorb to exceed .067 of its w'eigbt of water. 

The courses of brick w'alls should be of same height in front and rear, whether 
front is laid with stretchers and thin joints or not. 

In ashlar-facing the stones should have a width or depth of bed at least equal to 
height of stone. 

Hard bricks set in cement and 3 months set will sustain a pressure of 40 tons 
per sq. foot. 

The compression to which a stone should be subjected should not exceed .1 of its 
crushing resistance. 

The extreme stress upon any part of the masonry of St. Peter’s at Rome is com¬ 
puted at 15.5 tons per sq. foot; of St. Paul’s, London, 14 tons ; and of piers of New 
York and Brooklyn Bridge, 5.5 tons. 

The absorption of water in 24 hours by granites, sandstones, and limestones of a 
durable description is 1, 8, and 12 per cent, of volume of the stone. 

Color of Bricks depends upon composition of the clay, the molding sand, tem¬ 
perature of burning, and volume of air admitted to kiln. 

Pure clay free of iron will burn white, and mixing of chalk with the clay will 
produce a like effect. 

Presence of iron produces a tint ranging from red and orange to light yellow, 
according to proportion of iron. 

A large proportion of oxide of iron, mixed with a pure clay, will produce a bright 
red, and when there is from 8 to 10 per cent., and the brick is exposed to an intense 
heat, the oxide fuses and produces a dark blue or purple, and with a small volume 
of manganese and an increased proportion of the oxide the color is darkened, even 
to a black. 

Small volume of lime and iron produces a cream, color, an increase of iron pro¬ 
duces red , and an increase of lime brown. 

Magnesia in presence of iron produces yellow. 

Clay containing alkalies and burned at a high temperature produces a bluish green. 

For other notes on materials of masonry, their manipulation, etc., see “Limes, 
Cements, Mortars, and Concretes,” pp. 588-597. 

Pointing.—Before pointing, the joints should be reamed, and in close ma¬ 
sonry they must be open to .2 of an inch, then thoroughly saturated with water, 
and maintained iu a condition that they will neither absorb water from the mortar 
or impart any to it. Masonry should not be allowed to dry rapidly after pointing, 
but it should be well driven in by the aid of a calking iron and hammer. 

In pointing of rubble masonry the same general directions are to be observed. 


MASONRY, 


599 


Sand, is Argillaceous , Siliceous , or Calcareous , according to its composition. 
Its use is to prevent excessive shrinking, and to save cost of lime or cement. Or¬ 
dinarily it is not acted upon by lime, its presence in mortar being mechanical, and 
with hydraulic limes aud cements it weakens the mortal - . Rich lime adheres better 
to the surface of sand than to its own particles; hence the sand strengthens the 
mortar. 

It is imperative that sand should be perfectly clean, freed from all impurities, 
and of a sharp or angular structure. Within moderate limits size of grain does 
not affect the strength of mortar; preference, however, should be given to coarse. 

Calcareous sand is preferable to siliceous. 

Sea and River sand are suitable for plastering, but are deficient in the sharpness 
required for mortar, from the attrition they are exposed to. 

Clean sand will not soil the hands when rubbed upon them, and the presence of 
salt can be detected by its taste. 

ScoriEe, Slag, Clinker, and Cinder, when properly crushed and used, make good 
substitutes for sand. 

Concrete .—In the mixing of concrete, slake lime first, mix with cement, and then 
with the chips, etc., deposit in layers of 6 ins., and hammer down. 

IBriclts. 

Variations in dimensions by various manufacturers, and different degrees 
of intensity of their burning, render a table of exact dimensions of different 
manufactures and classes of bricks altogether impracticable. 


As an exponent, however, of the ranges of their dimensions, following 
averages are given: 


Description. | 

Ins. 

Description. 

Ins. 

Baltimore front 
Philadelphia “ 
Wilmington “ 
Croton “ 

Colabaugh. 

Eng. ordinary... 

“ Lond. stock 
Dutch Clinker... 

i 

8.25 X 4- i2 5 X 2.375 

8.5 X4 X2.25 

8.25 X 3- 62 5 X 2.375 

9 X 4 - 5 X 2.5 

8.75X4.25 X2.5 

6.25 X 3 X i-5 

Maine. 

Milwaukee. 

North River. 

Ordinary. 

Stourbridge I 

fire-brick_) 

Amer. do., N. Y.. 

7-5 X 3-375X2.375 

8.5 X4125X 2.375 

8 X3.5 X 2.25 

( 7-75 X 3-625 X 2.25 
(8 X 4- 12 5 X 2.5 

9.125 X 4-625 X 2.375 
8.875X4 5 X 2.625 


In consequence of the variations in dimensions of bricks, and thickness of 
the layer of mortar or cement in which they may be laid, it is also impracti¬ 
cable to give any rule of general application for volume of laid brick-work. 
It becomes necessary, therefore, when it is required to ascertain the volume 
of bricks in masonry, to proceed as follows : 

To Compeite Volume of Bricks, and. Number in a Cube 

Boot of Masonry. 

Rule. —To face dimensions of particular bricks used, add one half thick¬ 
ness of the mortar or cement in which they are laid, and compute the area; 
divide width of wall by number of bricks of which it is composed ; multiply 
this area by quotient thus obtained, and product will give volume of the 
mass of a brick and its mortar in ins. 

Divide 1728 by this volume, and quotient will give number of bricks in a 
cube foot. 

Example. —Width of a wall is to be 12.75 ins., and front of it laid with Philadel¬ 
phia bricks in courses .25 of an inch in depth; how many bricks will there be in 
face and backing in a cube foot? 

Philadelphia front brick, 8.25 X 2.375 ins. face. 

8.25 _j_ ^25 >( 2 -h2 = 8.25 -f-. 25 = 8.5 = length of brick and joint; 

2.575 -J— • 25 X 2 -f— 2 — 2.375 -h * 25 2.625 width of brick and joint. 

Then 8.5 X 3.625 = 22.3125 ins. — area of face; 12.75 - 4 - 3 ( number of bricks in 
width of wall) ±= 4.25 ins. 

Hence 22.3125 X 4- 25 = 94.83 cube ins. ; and 1728 4 - 94.83 = 18.22 bricks. 















6oo 


MASONRY. 


One rod of brick masonry (Eng.) — u.33 cube yards and weighs 15 tons, or 272 
superficial feet by 13.5 thick, averaging 4300 bricks, requiring 3 cube yards mortar 
and 120 gallons water. 

Bricklayers’ hod will contain 16 bricks or .7 cube feet mortar. 

Fi re- 7 o ricl£ s. 

Fire-clay contains Silica, Alumina, Oxide of Iron, and a small proportion 
of Lime, Magnesia, Potash, and Soda. Its tire-resisting properties depend¬ 
ing upon the relative proportions of these constituents and character of its 
grain. 

A good clay should be of a uniform structure, a coarse open grain, greasy 
to the hand, and free from any alkaline earths. 

The Stourbridge clay is black and is composed as follows: 

Silica.63.3 | Alumina.23.3 | Lime.73 | Protoxide of iron_1.8 

Water and organic matter.10.3 

Newcastle clay is very similar. 


Stone Masonry. 

Masonry is classed as Ashlar or Rubble. 

Ashlar is composed of blocks of stone dressed square and laid with 
close joints. 

Coursed Ashlar consists of blocks of same height throughout each course. 



Fig. 1.— Coursed , with chamfered and Fig. 2 .—Regular Coursed. 

rusticated quoins and plinth. 



Fig. 3 .—Irregular Coursed. 



Fig. 4 .—Random Coursed. 




Fig. 5.—Ranged Random, level, and Fig. 6.—Random, level, and broken, 

broken courses. 















































































































































































MASONRY, 


601 


Rubble -A.sh.lar 

Is ashlar faced stone with rubble backing. 

Rubble Alasonry 

Is composed of small stones irregular in form, and rough. 



Fig. 7. Block Coursed. —Large blocks 
in courses (regular or irregular), Beds 
and Joints roughly dressed. 



Fig. 9. Ranged Random. —Squared 
rubble laid in level and broken 
courses. 



Fig. 8.— Coursed and Ranged Random. 



Fig. 10. Coursed Random. —Stones laid 
in courses at intervals of from 12 to 18 
ins. in height. 




Fig. 11. 


Fig. 11. Uncoursed or Random. — 
Beds and Joints undressed, projections 
knocked off, and laid at random. In¬ 
terstices filled with spalls and mortar. 


Dry Rubble 

Is a wall laid without cement or 
mortar. 

Fig. 12. 


Fig. 12. Dry Rubble. —Without mortar 
or cement. 



Fig. 13. Laced Coursed. —Horizontal Fig. 14. Rustic or Rag. — Stones of 
bands of stone or bricks, interposed to irregular form, and dressed to make 
give stability. close joints. 

Note. —Rustic or Rag work is frequently laid in mortar. 





































































602 


MASONRY. 


Terra, Cotta. 

Terra Cotta in blocks should not exceed 4 cube feet in volume. When 
properly burned, it is unaffected by the atmosphere or by fumes of any acid. 


Arches and. NWalls. 



Springing. —Points, Fig. 15, on each side, 
from which arch springs. 

Croicn. —Highest point of arch. 

Haunches. —Sides of arch, from springing 
half-way up to crown. 

Spandrel. —Space between extrados,a hor¬ 
izontal line drawn through crown and a ver¬ 
tical line through upper end of skewback. 

Skewbaclc is upper surface of an abut¬ 
ment or pier from which an arch springs, 
and its face is on a line radiating from centre of arch. 

Abutment is outer body that supports arch and from which it springs. 

Pier is the intermediate support for two or more arches. 

Jambs are sides of abutments or piers. 

Voussoirs are the blocks forming an arch. 

Key-stone is centre voussoir at crown. 

Span is horizontal distance from springing to springing of arch. 

Rise. —Height from springing line to under side of arch at key-stone. 

Length is that of springing line or span. 

Ring-course of a wall or arch is parallel to face of it, and in direction of 
its span. 

String and Collar courses are projecting ashlar dressed broad stones at 
right angles to face of a wall or arch, and in direction of its length. 

Camber is a slight rise of an arch as .125 to .25 of an inch per foot of 
span. 

Quoin is the external angle or course of a wall. 

Plinth is a projecting base to a wall. 

Footing is projecting course at bottom of a wall, in order to distribute its 
weight over an increased area. Its width should be double that of base of 
wall, diminishing in regular offsets .5 width of their height. 

Blocking Course. —A course placed on top of a cornice. 

Parapet is a low wall, over edge of a roof or terrace. 

Extrados. —Back or upper and outer surface of an arch. 

Intrados or Soffit is underside of lower surface of arch or an opening. 

Groined is when arches intersect one another. 

Invert. —An inverted arch, an arch with its intrados below axis or spring¬ 
ing line. 


Ashlar masonry requires .125 of its volume of mortar. Rubble, 1.2 cube 
yards stone and .25 cube yard mortar for each cube yard. 

Rubble masonry in cement, 160 feet in height, will stand and bear 20000 
lbs. per sq. inch. 

Stones should be laid with their strata horizontal. 

When “through” or “thorough bonds” are not introduced, headers should 
overlap one another from opposite sides, known as dogs' tooth bond. 

Aggregate surface of ends of bond stones should be from .125 to .25 of 
area of each face of wall. 

Weak stones, as sandstone and granular limestone, should not have a 
length over 3 times their depth. Strong or hard stones may have a length 
from 4 to 5 times their depth. 
















MASONRY. 


603 

Gallets are small and sharp pieces of stone stuck into mortar joints, in 
which case the work is termed galleted. 

Snapped work is when stones are split and roughly squared. 

Quarry or Rock-faced. —Quarried stones with their faces undressed. 

Pitch-faced. —Stones on which the arris or angles of their face, with their 
sides and ends, is defined by a chisel, in order to show a right-lined edge. 

Drafted or Drafted Margin is a narrow border chiselled around edges of 
faces of a block of rough stone. 

Diamond-faced is when planes are either sunk or raised from each edge 
and meet in the centre. 

Squared Stones. —Stones roughly squared and dressed. 

Rubble. —Unsquared stones, as taken from a quarry or elsewhere, in their 
natural form, or their extreme projections removed. 

Cut Stones. —Stones squared and with dressed sides and ends. 

Dressed. Stones. 

The following are the modes of dressing the faces of ashlar in engineering: 

Rough Pointed. —Rough dressing with a pick or heavy point. 

Fine Pointed. —Rough dressing, followed by dressing with a fine point. 

Crandalled. —Fine pointing in right lines with a hammer, the face of 
which is close serried with sharp edges. 

Cross Crandalled. —When the operation of crandalling is right angled. 

Hammered. —The surface of stone may be finished or smooth dressed by 
being Axed or Bushed; the former is a finish by a heavy hammer alike to a 
crandall, the latter is a final finish by a heavy hammer with a face serried 
with sharp points at right angles. 


Thickixess of Brick "Walls for Warehouses. ( Molesworth.) 


Length. 

Height. 

Thickness. 

Length. 

Height. 

Thickness. 

Length. 

Height. 

Thickness. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Unlimited. 

25 

13 

Unlimit’d. 

loo 

34 

45 

30 

13 

do. 

30 

W -5 

60 

40 

17-5 

30 

40 

13 

do. 

40 

21.5 

70 

50 

21.5 

40 

50 

i 7-5 

do. 

50 

26 

50 

60 

21.5 

35 

60 

I 7-5 

do. 

60 

26 

45 

70 

21.5 

30 

70 

i 7-5 

do. 

70 

26 

60 

8O 

26 

45 

80 

21.5 

do. 

80 

30 

70 

90 

30 

60 

90 

26 

do. 

90 

34 

70 

IOO 

30 

55 

IOO 

26 


For drawings aDd a description of stone-dressing tools, see a paper by J. R. Cross, 
W. E. Merrill, and E. B. Van Winkle, “A. S. Civil Engineer Transactions,” Nov. 1877. 


Walls not exceeding 30 feet in height, upper story walls may be 8.5 ins. thick. 
From 16 feet below top of wall to base of it, it should not be less than the space 
defined by two right lines drawn from each side of wall at its base to 16 feet from 
top. 

Thickness not to be less in any case than one fourteenth of height of story. 


Baths. 

Laths are 1.25 to 1.5 ins. by 4 feet in length, are usually set .25 of an inch 
apart, and a bundle contains 100. 

















604 


MASONRY. 


IPlastering;. 


Volum es required for Various Thickness. 


Material. 

Sq 

•5 

uare Yar 

•75 

ds. 

1 

Material. 

Sq 

•5 

uare Yar 

•75 

ds. 

1 

Cube Feet. 

Cement 1. 

Ins. 

2.25 

4-5 

6-75 

Ins. 

1. ^ 

Ins. 

I-I 5 

2.25 

3-33 

Cube Feet. 

Lime 1, sand 2 ,) 
hair 3.75.j •* 

Ins. 

75 yai 
dered 
brick 

Ins. 

'ds. sup 
and s 
or 70 or 

Ins. 

T ren¬ 
et on 
lath. 

Cement 1, sand 1... 
Cement 1, sand 2... 

3 

4-5 


Estimate of HVTeiterials and. Labor for lOO Sq. Yards of 

Latli and IPlaster. 


Materials 

Three Coats 

Two Coats 

Materials 

Three Coats 

Two Coats 

and Labor. 

Hard Finish. 

Slipped. 

and Labor. 

Hard Finish. 

Slipped. 

Lime. 

4 casks. 

3.5 casks. 

White sand.... 

2.5 bushels. 


Lump lime. 

'.66 “ 


13 lbs. 

4 days. 

13 lbs. 

3.5 days. 

Plaster of Paris.. 
Laths. 

•5 “ 


Masons. 




Hni r 

4 bushels. 

7 loads. 

3 bushels. 
6 loads. 

Laborer. 

3 

2 £t 

Sand. 

Cartage. 

1 “ 

•75 “ 


Rough Cast is washed gravel mixed with hot hydraulic lime and 
water and applied in a semi-fluid condition. 


Arches and Abutments. 

To Compute Deptli of Key-stone of Circular or Elliptic 

Acrcli. 


VH + S-4-2 

4 


+ - 2 5 —d. 


R representing radius, s span, and d depth, all in feet. 


This is for a rise of about .25 of span; when it is reduced, as to .125, add. 5 instead 
of .25. 

Illustration. —Arch of Washington aqueduct at “Cabin John” has a span of 220 
feet, a rise of 57.25, and a radius of 134.25; what should be depth of its keystone? 


V134. 25 -f- 220 -r- 2 


.25 = 


15'6.3 


■ .25 = 4.16 feet. Depth is 4.16 feet. 


Viaducts of several arches increase results as determined above by add¬ 
ing .125 to .15 to depth. 

For arches of 2d class materials and work, and for spans exceeding 10 
feet, add .125 to depth of keystone, and for good rubble or brick-work 
add .25. 

Note. —It is customary to make the keystones of elliptic arches of greater depth 
than that obtained by above formula. Trautwine, however, who is high authority 
in this case, declares it is unnecessary. 


To Compute Radius of an 



.A_rcli, Circular or Ellipse. 
r representing rise. 


Railway A-rclies. 

For Spans between 25 and 70 feet. Rise .2 of span. Depth of arch .055 of span. 
Thickness of abutments .2 to .25 of span, and of pier .14 to .16 of span. 

Abutments. 


When height does not exceed 1.5 times base. R - 4 - 5 -p. 1 r -j- 2 — thickness at spring 
of arch in feet. ( Trautwine.) 

Batter .—From .5 to 1.5 ins. per foot of height of wall. 










































MASONRY.-MECHANICAL CENTRES.-GRAVITY. 605 


To Compute Depth, of Arch. (Hurst.) 
c \/K = D, c = Stone (block) .3. Brick = .4. Rubble = -45. 

Wh#n there are a series of arches, put .3 — .35, .4 = .45, and .45 --.5. 

NX ini m nm Thickness of Abutments fox- Bridge and 
similar Arches of 120 °. (Hurst.) 

When depth of crown does not exceed 3 feet. Computed from formula. 
/ /3 R\ 2 0 R 

W6R{ I2H) — = ^ representing height of abutment to springing in feet. 


Radius 

Height of Abutment to Spring 

ing. 

Radius 

Height of Abutment to Springing. 

of Arch. 

5 

7-5 

10 

20 

3 ° 

of Arch. 

5 

7-5 

IO 

20 

30 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

4 

3-7 

4.2 

4-3 

4.6 

4-7 

12 

5-6 

6.4 

6.9 

7.6 

7-9 

4-5 

3-9 

4.4 

4.6 

4.9 

5 

15 

6 

7 

7-5 

8.4 

8.8 

5 

4.2 

4.6 

4.8 

5 -i 

5-2 

20 

6.5 

7-7 

8.4 

9.6 

IO 

6 

4-5 

4-7 

5-2 

5-6 

5-7 

25 

6.9 

8.2 

9.1 

10.5 

II. I 

7 

4-7 

5-2 

5-5 

6 

6.1 

30 

7.2 

8.7 

9-7 

11.4 

12 

8 

4.9 

5-5 

5-8 

6.4 

6.5 

35 

7-4 

9 - 1 

10.2 

11.8 

I2.9 

9 

5 -i 

5-8 

6.1 

6.7 

6.9 

40 

7.6 

9.4 

10.6 

12.8 

13.6 

IO 

5-3 

6 

6.4 

7 -i 

7-3 

45 

7.8 

9-7 

II 

ii .4 

G -3 

II 

5-5 

6.2 

6.6 

7-3 

7.6 

50 

7-9 

IO 

11.4 


15 


Note.— Abutments in Table are assumed to be without counterforts or wing- 
walls. A sufficient margin of safety must be allowed beyond dimensions here 
given. 


Culverts for a road having double tracks are not necessarily twice the 
length for a single track. 

For other and full notes, tables, etc., see Trautwine’s Pocket Book, pp. 341-356. 


MECHANICAL CENTRES. 


There are four Mechanical centres of force in bodies, namely, Centre 
of Gravity, Centre of Gyration, Centre of Oscillation, and Centre of 
Percussion. 

Centre of Grravity. 

Centre op Gravity of a body, or any system of bodies rigidly con¬ 
nected together, is point about which, if suspended, all parts will be in 
equilibrium. 

A body or system of bodies, suspended at a point out of centre of gravity, 
will rest with its centre of gravity vertical under point of suspension. 

A body or system of bodies, suspended at a point out of centre of gravity, 
and successively suspended at two or more such points, the vertical lines 
through these points of suspension will intersect each other at centre of 
gravity of body or bodies. 

Centre of gravity of a body is not always within the body itself. 

If centres of gravity of two bodies, as B C, be connected by a line, dis¬ 
tances of B and C from their common centre of gravity, c, is as the weights 
of the bodies. Thus, B : C :: C c : c B. 

To Ascertain Centre of Gravity of any Plane Figure Mechanically. 

Suspend the figure by any point near its edge, and mark on it direction 
of a plumb-line hung from that point; then suspend it from some other 
point, and again mark direction of plumb-line. Then centre of gravity of 
surface will be at point of intersection of the two marks of plumb-line. 

3 E* 





















6o6 


MECHANICAL CENTRES.—GRAVITY. 


Centre of gravity of parallel-sided objects may readily be found in this 
way. For instance, to ascertain centre of gravity of an arch of a bridge, 
draw elevation upon paper to a scale, cut out figure, and proceed with it as 
above directed, in order to find position of centre of gravity in elevation of 
the model. In actual arch, centre of gravity will have same relative position 
as in paper model. 

In regular figures or solids, centre of gravity is same as their geometrical 
centres. 

Line. 

7 * C 

Circular Arc. — = distance from centre, r representing radius, c chord, and l 
i 

length of arc. 

Surfaces. 

Square, Rectangle, Rhombus , Rhomboid, Gnomon , Cube, Regular Polygon, 
Circle, Sphere, Spheroid or Ellipsoid , Spheroidal Zone , Cylinder , Circidar 
Ring, Cylindrical Ring , Link , Helix, Plain Spiral, Spindle , a/!/ Regular Fig¬ 
ures, and Middle Frusta of all Spheroids, Spindles, etc. 

The centre of gravity of the surfaces of these figures is their geometri¬ 
cal centre. 


Triangle .—Cto a line drawn from any angle to the middle of opposite side, 
at two thirds of the distance from angle. 

Trapezium .—Draw two diagonals, and ascertain centres of gravity of each 
of four triangles thus formed; join each opposite pair of these centres, and it 
is at intersection of the lines. 


Trapezoid. X — — distance from B on a line joining middle of two 

\ B + fi / 3 


parallel sides B b, m representing middle line. 


C V 

Circular Arc. — — distance from centre of circle. 

i 

Sector of a Circle. .4244 r = distance from centre of circle. 

Semicircle. .4244 r = distance from centre. 

Semi-semicircle. .4244 r = distance from both base and height and at their inter¬ 
section. 


Segment of a Circle. 


c3 


= distance from centre, a representing area of segment. 


a gjU c? / 7*3 — 7*'3 

Sector of a Circular Ring. — X — X - y- = distance from centre of 

2f arc / t —■ 7* 

arcs, r and r' representing the radii. 

Illustration. —Radii of surfaces of a dome are 5 and 3.5 feet, and angle «^) at 
centre 130°. 

4 X Sin ' 65 ° X I2 ~* 4 2 - 8 75 _ 4 -9063 82.125 f 

3 arc 130 0 25 —12.25 3 2.2689 * 6.8067 X 12.75 3 - 43 / 

Hemisphere, Spherical Segment, and Spherical Zone, At centre of their 
heights. 


Circular Zone .—Ascertain centres of gravity of trapezoid and segments 
comprising zone; draw a line (equally dividing zone) perpendicular to 
chords; connect centres of segments by a line cutting perpendicular to 
chords. 


Then centre of gravity of figure will be on perpendicular, toward lesser 
chord, at such proportionate distance of difference between centres of gravity 
of trapezoid and line connecting centres of segments, as area of segments 
bears to area of trapezoid. 









MECHANICAL CENTRES.-GRAVITY. 


607 

Prism and Wedge .—When end is a Parallelogram, in their geometrical 
centres; when the end is a Triangle, Trapezium, etc., it is in middle of its 
length, at same distance from base, as that of triangle or trapezoid of which 
it is a section. 

Parabola in its axis — .6 distance from vertex. 

Prismoid.—At same distance from its base as that of the trapezoid or 
trapezium, which is a section of it. 

Lune.—On a line connecting centres of gravity of arcs at a proportionate 
r' v V°iid to respective areas of arcs. 

( . / r i 

V ^ r + r’j 
/ 2 V '±f\ * =a! 

r ') 3 



Co-ordinates. 


and 


r + r 

Solids. 

Cube , Parallelopipedon, Hexahedron, Octahedron, Dodecahedron , Icosahe¬ 
dron, Cylinder , Sphere , Right Spherical Zone, Spheroid or Ellipsoid, Cylin¬ 
drical Ring , im/c, Spindle, all Regular Bodies, and Middle Frusta of all 
Spheroids and Spindles, etc. Centre of gravity of these figures in their 
geometrical centre. 

Tetrahedron.—In common centre of centres of gravity of the triangles made by a 
section through centre of each side of the figures. 

Cone and Pyramid. .25 of line joining vertex and centre of gravity of base = dis¬ 
tance from base. 

(?* —|— 2 —1— 2 9* 2 I 

Frustum of a Cone or Pyramid. - ; ——- , X - h = distance from centre 

J u ( } ._|.yj2_ rr ' A 4 ^ 

0/ Zesser end , r and ?•', in a cone representing radii, and in a pyramid sides, and & 
height. 

Cone, Frustum of a Cone, Pyramid, Frustum of a Pyramid, and Ungula .— 
A t same distance from base as in that of triangle, parallelogram, or semicir¬ 
cle, xohich is a right section of them. 

Hemisphere. .375 r — distance from centre. 

( vs\ 2 

r -j -r-v — distance from centre, vs repre¬ 

senting versed sine, and v volume of segment. ( —- — ;> - ) X h = distance from 

\i2 r — 4 h) 

2 r T" 3 h 


vertex. 

Spherical Sector. .75 (r — .5 h) — distance from centre. — distance 

from vertex. 

Spirals. —Plane, in its geometrical centre. Conical, at a distance from the 
base, .25 of line joining vertex and centre of gravity of base. 

9 * 2 — y * 2 

Frustum of a Circular Spindle. ——-—— = distance from centre of spindle, 

2 yib I_). Zj 

h representing distance between two bases, D distance of centre of spindle from centre 
of circle, and z generating arc, expressed in units of radius. 

r 2 

Segment of a Circular Spindle. ——- = distance f rom centre of spindle. 

Semi-spheroids. — Prolate. .375 a.—Oblate. .375 a — distance from centre. 
Semi-spheroid or Ellipsoid and its Segment. — See HasweWs Mensuration, pages 
281 and 282. 

Frusta of Spheroids or Ellipsoids. Prolate. .75 


h (2 a 2 — h 2 ) 


: distance from 


3 a 2 — h 2 

centre of spheroid, a representing semi-transverse diameter in a prolate frustum, and 
semi-conjugate in an oblate frustum. 













6o8 


MECHANICAL CENTRES.-GRAVITY. 


(a + d') 2 ,. . 

Segments of Spheroids.—Prolate. .75 -.— Oblate. .75 - : —— = distance 


(afdf 
2 a d 


2 a-fid' 

from centre of spheroid, d and d' representing distances of base of segments from 
centre of spheroid. 


Any Frustum. .75 


(tZ -)- d') X (2 a- — rZ' 2 4 ~ d 2 ) 


distance from centre of sphe- 


3 a 2 — d' 2 -\-d' d-\-d 2 
roid, d and d' representing distances of base and end of segments from centre of the 
spheroid. 

Segment of an Elliptic Spindle at two thirds of height from vertex. 

Paraboloid of Revolution, at two thirds of height from vertex. 

Segment of a Hyperbolic Spindle, at 75 of height from vertex. 

2 7* 2 J- y' Ji 

Frustum of Paraboloid of Revolution. — 2 -ft' ^ ~ — distance from base, r and 

r' representing radii of base and vertex. 

Segment of Paraboloid of Revolution , at two thirds of height from vertex. 

Segments of a Circular and a Parabolic Spindle. —See Haswell's Mensuration, 
pages 192 and 199. 

Parabola. .4 of height = distance from base. 

Hyperboloid of Revolution. \ f i ~~ -~r Xh = distance from vertex, b representing . 

6 b -j- 4 h 

diameter of base. 

_i_ d') (2 a 2 _ d ' 2 -f- d 2 ) 

Frustum of Hyperboloid of Revolution. .75 -—— d' 2 -\- d 'd -|- cl 2 ~ = ^ stance 

from centre of base, a representing semi-transverse axis, or distance from centre of 
curve to vertex of figure; cl and d’ distances from centre of curve to centre of lesser 
and greater diameter of frustum. 


Segment of Hyperboloid of Revolution, 
d v 


4 ft 4 ~ 3 ft 

6 b -j- 4 h 


X h — distance from vertex. 


Of Two Bodies. 


distance from V or volume or area of larger body, d rep- 


V + a' 

resenting distance between centres of gravity of bodies, and v volume or area of less 
body. 


Cycloid. — .833 of radius of generating circle = distance from centre of 
chord of curve. 

A ny Plane Figure .—Divide it into triangles, and ascertain centre of grav¬ 
ity of each ; connect two centres together, and ascertain their common cen¬ 
tre ; then connect this common centre and centre of a third, and ascertain 
■the common centre, and so on, connecting the last-ascertained common centre 
to another centre till whole are included, and last common centre will give 
centre required. 


Of an Irregular Body of Rotation. 

Divide figure into four or six equidistant divisions; ascertain volume of 
each, their moments with reference to first horizontal plane or base, and 
then connect them thus : 


(A-J-4 A1 + 2 A2 + 4 A3 + A4) ~ =r V, A A X) etc., representing volume of divis¬ 
ions, and h height of body from base; 

(o A -(-1 X 4 -A 1 -(-2X2 A2 -f- 3X4 A3 -(- 4 A4) h 

and --- t — 7 - t — ;- -— : - r—:— - X - = distance of centre of 

A + 4 Aj -f-2 A 2 + 4 A 3 -f A 4 4 J J 

gravity from base. 










MECHANICAL CENTRES.-GYRATION. 


609 


Centre of Gryration. 

Centre of Gyration is that point in any revolving body or system 
of bodies in which, if the whole quantity of matter were collected, the 
Angular velocity would be the same ; that is, the Momentum of the body 
or system of bodies is centred at this point, and the position of it is a 
mean proportional between the centres of Oscillation and Gravity. 

If a straight bar of uniform dimensions was struck at this point, the 
stroke would communicate the same angular velocity to the bar as if the 
whole bar was collected at that point. 

The A ngular velocity of a body or system of bodies is the motion of a line 
connecting any point and the centre or axis of motion: it is the same in all 
parts of the same revolving body. 

In different unconnected bodies, each oscillating about a common centre, 
their angular velocity is as the velocity directly, and as the distance from 
the centre inversely. Hence, if their velocities are as their radii, or distances 
from the axis of motion, their angular velocities will be equal. 

When a body revolves on an axis, and a force is impressed upon it suffi¬ 
cient to cause it to revolve on another, it will revolve on neither, but on a 
line in the plane of the axes, dividing the angle which they contain; so that 
the sine of each part will be in the inverse ratio of the angular velocities 
with which the bodies would have revolved about these axes separately. 

Weight of revolving body, multiplied into height due to the velocity with 
which centre of gyration moves in its circle, is energy of body, or mechani¬ 
cal power, which must be communicated to it to give it that motion. 

Distance of centre of gyration from axis of motion is termed the Radius 
of gyration; and the moment of inertia is equal to product of square of 
radius of gyration and mass or weight of body. 

The moment of inertia of a revolving body is ascertained exactly by as¬ 
certaining the moments of inertia of every particle separately, and adding 
them together; or, approximately, by adding together the moments of the 
small parts arrived at by a subdivision of the body. 

To Compute Nloment of Inertia of a Revolving Body-. 

Rule. —Divide body into small parts of regular figure. Multiply mass 
or weight of each part by square of distance of its centre of gravity from 
axis of revolution. The sum of products is moment of inertia of body. 

Note. —The value of moment of inertia obtained by this process will be more 
exact, the smaller and more numerous the parts into which body is divided. 

To Compute Radius of Gfyration of a Revolving Body 
about its Axis of Revolution. 

Rule. —Divide moment of inertia of body by its mass, or its weight, and 
square root of quotient is length of radius of gyration. 

Note. —When the parts into which body is divided are equal, radius of gyration 
may be determined by taking mean of all squares of distances of parts from axis 
of revolution, and taking square root of their sum. 

Or, VR 2 -f- r 2 -r- 2 = G. R and r representing radii. 

Example.— A straight rod of uniform diameter and 4 feet in length, weighs 4 lbs.; 
what is its inertia, and where is its radius or centre of gyration? 

Each foot of length weighs 1 lb., and if divided into 4 parts, centre of gyration of 
each is respectively .5, 1.5, 2.5, and 3.5 feet. Hence, 

1 X -S 2 = -25 1 

iX i. 5 2 = 2.25 i 21 — inertia, which4 = 5.25, and f 25 = 2.291 
1 X 2. s 2 =: 6.25 f feet radius. 

1 X 3-5 2 = 12.25 J 



6 io 


MECHANICAL CENTRES.—GYRATION. 




Following are distances of centres of gyration from centre of motion in 
various revolving bodies: 

Straight, uniform Rod or Cylinder or thin Rectangular Plate revolving about one 
end; length x -5773, and revolving about their centi’e; length x .2886. 

The general expression is, when revolving at any point of its length, 

r l 3 1' 3 \ 

- !-) . I and V representing length of the two points. 

<3 (l + )) 

Circular Plane , revolving on its centre; radius of circle X -7071 ; Circle Plane, as 
a Wheel or Disc of uniform Thickness , revolving about one of its diameters as an 
axis; radius X .5. 

Solid Cylinder , revolving about its axis; radius X -7071. 

Solid Sphere , revolving about its diameter as an axis; radius X -6325. 

Thin , hollow Sphere , revolving about one of its diameters as an axis; radius 
X .8164. . Surface of sphere .8615 r. 

Sphere and Solid Cylinder (vertical), at a distance from axis of revolution = 
V£ 2 + .4 r 2 for sphere, and yjl- +. 5 r- for cylinder, l representing length of connec¬ 
tion to centre of sphere and cylinder. 

Cone , revolving about its axis; radius of base X -5447; revolving about it§ ver¬ 
tex = V12 /t 2 + 3 r 2 -4-20, h representing height, and r radius of base; revolving 
about its base = V 2 h 2 -f- 3 r 2 — 20. 

Circular Ring , as Rim of a Fly-wheel or Hollow Cylinder, revolving about its 

diameter = \/R 2 + r 2 4 - 2, R representing radius of periphery, and r of inner circle 
of ring. 

/6 W (R 2 4 - ?' 2 ) 4 - w (4 r 2 4 - Z 2 ) 

Fly-wheel = ^ --, W and w representing weights of 

rim and of arms and hub, and l length of arms from axis of wheel. 

U d 2 4 -c 

Section of Rim. -—— 

of rim. 


-j- ?* 2 -f- r d. d representing depth and c periphery 


4 l 2 + b 2 


, b rep- 


Parallelopiped, revolving about one end, distance from end = 
resenting breadth. 

Illustration. —In a solid sphere revolving about its diameter, diameter being 
2 feet, distance of centre of gyration is 12 X .6325 = 7.59 ins. 


To Compute Elements of Gyration. 


Gffs 

rtg 

Trig 


P; 


T rtg 
W v '' 


G; 


GWv 

"rtg 


= r: 


P r t g 
G» : 


:W; 


GWv 

Prg' 


t; 


G W 


v. G representing distance of centre of gyration from axis of rotation, 

W weigh t of body, t time power acts in seconds, v velocity in feet per second acquired 
by revolving body in that time, and r distance of point of application of power from 
axis of body, as length of crank, etc. 

Illustration i. —What is distance of centre of gyration in a fly-wheel, power 
224 lbs., length of crank 7 feet, time of rotation 10 seconds, weight of wheel 5600 
lbs., and velocity of it 8 feet per second? 


224 X 7 X 10 X 32.166 504 373 


5600 X 8 


42 800 


= 11.78 feet. 


2.—What should be weight of a fly-wheel making 12 revolutions per minute, its 
diameter 8 feet, power applied at 2 feet from its axis 84 lbs., time of rotation 6 sec¬ 
onds, and distance of centre of gyration of wheel 3.5 feet? 

-= -= 5.0265 feet = velocity. Then— -- -=1843.2 lbs. 

60 3-5X5-0265 





















MECHANICAL CENTRES.-GYRATION. 


61 I 


When the Body is a Compound one. Rule.— Multiply weight of several 
particles or bodies by squares of their distances in feet from centre of mo¬ 
tion or rotation, and divide sum of their products by weight of entire mass; 
the square root of quotient will give distance of centre of gyration from 
centre of motion or rotation. 


Example. —If two weights, of 3 and 4 lbs. respectively, be laid upon a lever (which 
is here assumed to be without weight) at the respective distances of 1 and 2 feet, 
what is distance of centre of gyration from centre of motion (the fulcrum) ? 


3 X x 2 = 3; 


4 X 2 2 = i6; 


3jfi6__i9 

3 + 4 7 


2.71, and 3/2.71 = 1.64 feet. 


That is, a single weight of 7 lbs., placed at 1.64 feet from centre of motion, and re¬ 
volving in same time, would have same momentum as the two weights in their 
respective places. 


When Centre of Gravity is given. Rule.— Multiply distance of centre of 
oscillation from centre or point of suspension, by distance of centre of grav¬ 
ity from same point, and square root of product will give distance of centre 
of gyration. 


Example. —Centre of oscillation of a body is 9 feet, and that of its gravity 4 feet 
from centre of rotation or point of suspension; at what distance from this point is 
centre of gyration ? 

9 X 4 = 36, and 3/36 = 6 feet. 


To Compuite Centre of Gryration of a Water-wheel. 

Rule.— Multiply severally twice weight of rim, as composed of buckets, 
shrouding, etc., and twice that of arms and that of water in the buckets 
(when wheel is in operation) by square of radius of wheel in feet; divide 
sum by twice sum of these several weights, and square root of quotient will 
give distance in feet. 

Example. — In a wheel 20 feet in diameter, weight of rim is 3 tons, weight of 
arms 2 tons, and weight of water in buckets 1 ton; what is distance of centre of 
gyration from centre of wheel ? 


Rim = 3 tons x io 2 X 2 = 600 
Buckets == 2 tons X io 2 X 2 = 400 
Water = 1 ton X io 2 = 100 

1100 


3 + 2 + 1X2 = 12 sum of weights. 
Hence = 3/91.67 = 9,57 feet. 


General Formulas.—P representing power , H horses ’ power , F force applied to 
rotate body in lbs., M mass of revolving body in lbs., r radius upon which F acts in 
feet, d distance from axis of motion to centre of gyration in feet, t time force is ap¬ 
plied in seconds, n number of revolutions in time t, x angular velocity, or number of 
’ 32.166 Fr 2 

revolutions per minute at end of time t, and G = ' 




4 pm 


t 


G 

153-5 Hr 
• M d 2 


2 p r 2 x 
60 G 


t 


M x d 2 


153-5 tr 


244 t P 
x 2 d 2 


M 


M d 2 
M n d 2 
2.56 t 2 F 
x 2 M d 2 „ 


= F; 


2.56 t 2 F r 


M d 2 
■ M d 2 


= n 


244 


134100 t 


— H. 


Illustration. —Rim of a fly-wheel weighing 7000 lbs. has radii of 6.5 and 5.75 
feet; what is its centre of gyration, and what force must be applied to it 2 feet, 
from axis of motion to give it an angular velocity of 130 revolutions per minute in 
40 seconds? how many revolutions will it make in 40 seconds? and what is its 
power ? 


130 2 X 7000 X 6.14 2 _ 4459 862680 

134100 X 40 5 364 000 


: 829.7 horses. 


Centre of gyration = -~^ 5 — = 6- hM- Then F = / ~ 


34 306 636 
12 280 


= 2793.7 lbs., and 


2.56 x 40 2 X 2793.7 X 2 
7000 X 6.14 2 


: 86.67 revolutions. 














6 12 MECHANICAL CENTRES.—OSCILLATION, ETC. 


Centres of Oscillation and. Percussion. 

Centre of Oscillation of a body, or a system of bodies, is that point 
in axis of vibration of a vibrating body in which, if, as an equivalent 
condition, the whole matter of vibrating body was concentrated, it would 
continue to vibrate in same time. It is resultant point of whole vibrat¬ 
ing energy, or of action of gravity in producing oscillation. 

As particles of a body further from centre of its suspension have greater 
velocity of vibration than those nearer to it, it is apparent that centre of 
oscillation is further from its centre than centre of gravity is from axis of 
suspension, but it is situated in centre of a line drawn from axis of a body 
through its centre of gravity. It further differs from centre of gyration 
in this, that while motion of oscillation is produced by gravity of a body, 
that of gyration is caused by some other force acting at one place only. 

Radius of oscillation, or distance of centre of oscillation from axis of sus¬ 
pension, is a third proportional, to distance of centre of gravity from axis 
of suspension and radius of gyration. 

Centre of Percussion of a body, or a system of bodies, revolving 
about a point or axis, is that point at which, if resisted by an immov¬ 
able obstacle, all the motion of the body, or system of bodies, wmuld be 
destroyed, and without impulse on the point of suspension. It is also 
that point which would strike any obstacle with greatest effect, and 
from this property it has been termed percussion. 

Centres of Oscillation and Percussion are in same point .—If a blow is 
struck ‘by a body oscillating or revolving about a fixed centre, percussive 
action is same as if its entire mass was concentrated at centre of oscillation. 
That is, centre of percussion is identical with centre of oscillation, and its 
position is ascertained by same rules as for centre of oscillation. If an ex¬ 
ternal body is struck so that the mean line of its resistance passes through 
centre of percussion, then entire force of percussion is transmitted directly 
to the external body ; on the contrary, if a revolving body is struck at its 
centre of percussion, its motion will be absolutely destroyed, so that the body 
will not incline either way. 

As in bodies at rest, the entire weight may be considered as collected in 
centre of gravity; so in bodies in vibration, the entire force may be consid¬ 
ered as concentrated in centre of oscillation; and in bodies in motion, the 
whole force may be considered as concentrated in centre of percussion. 

If centre of oscillation is made point of suspension, point of suspension 
will become centre of oscillation. 

Angle of Oscillation or Percussion is determined by angle delineated by 
vertical plane of body in vibration, in plane of motion of body. 

Velocity of a Body in Oscillation or Percussion through its vertical plane 
is equal to that acquired by a body freely falling through a vertical line 
equal in height to versed sine of the arc. 

To Compute Centre of Oscillation or Percussion of a 
Body of Uniform Density and Ifignre. 

Rule.—M ultiply weight of body by distance of its centre of gravity from 
point of suspension; multiply also weight of body by square of its length, 
and divide product by 3. 

Divide this last quotient by product of weight of body and distance of 
its centre of gravity, and quotient is distance of centres from point of sus¬ 
pension. 


MECHANICAL CENTRES.-OSCILLATION, ETC. 6 1 3 


Or,- t- W X g = distance from axis. Or, square radius of gyration of body 

3 

and divide by distance of centre of gravity from axis of suspension. 

Example. —Where is centre of oscillation in a rod 9 feet in length from its point 
of suspension, and weighing 9 lbs. ? 

Q Q Q 2 

9 X - == 40.5 = product of weight and its centre of gravity; -—— = 243 = quo- 
2 3 

tient of product of weight of body and square of its length -4- 3; -A- — 6 feet. 

4 o -5 

When Point of Suspension is not at End of Rod. Rule. — To cube of 
distance of point of suspension from top of rod or bar, add cube of its dis¬ 
tance from lower end, and multiply sum by 2. 

Divide product by three times difference of squares of these distances, and 
quotient is distance of point of oscillation from point of suspension. 


Example. —A homogeneous rod of uniform dimensions, 6 feet in length, is sus¬ 
pended 1.5 feet from its upper end; what is distance of point of oscillation from 
that of suspension ? 



2 (4-5 3 + i-5 3 ) 189 

3<4.5‘->V) = 


Centres of Oscillation and. Percussion in Bodies of 

Various Figures. 

When Axis of Motion is in Vertex of Figure , and when Oscillation or Motion 

is Facewise. 

Right Line , or any figure of uniform, shape and density — .66 l. 

Isosceles Triangle — .75 h. Circle = 1.25 r. 

Parabola = .714 h. Cone = .&h. 


When Axis of Motion is in Centre of Body. Wheel = .75 radius. 


When Oscillation or Motion is Sideuise. Right Line , or any figure of uni¬ 
form shape and density ■= 66 l. Rectangle , suspended at one angle = . 66 of di¬ 
agonal. 

Parabola , if suspended by its vercex = .714 of axis-f.33 parameter; if suspended 
by middle of its base =. 57 of axis -f. 5 parameter, 
o £irc v 

Sector of a Circle =1 —- } c representing chord of arc , and r radius of base. 

4 c 

4 T ^ 

Circle =. 75 d. Cone = - axis 4 - --—. 

5 5 axis 


Sphere = 


5 (c + r) 


r-f-c, c representing length of cord by which it is suspended. 


To Ascertain Centres of Oscillation and Percussion 

experimentally. 

Suspend body very freely from a fixed point, and make it vibrate in small arcs, 
noting number of vibrations it makes in a minute, and let number made in a min¬ 
ute be represented by n; then will distance of centre of oscillation from point of 
140830 

suspension be = —= rns. 


For length of a pendulum vibrating seconds, or 60 times in a minute, being 
39.125 ins., and lengths of pendulums being reciprocally as the squares of number 

6o 2 X39-i25 140850 

of vibrations made in same time, therefore n 2 :6o 2 :: 39.125: --- T 


being length of pendulum ivhich vibrates n times in a minute , or distance of centre 
of oscillation below axis of motion. 

3 F 










6 14 MECHANICAL CENTRES.-MECHANICS. 


To Compute Centres of Oscillation or Percussion of a 
System of Barticles or Bodies. 

Rule.—M ultiply weight of each particle or body by square of its distance 
from point of suspension, and divide sum of their products by sum of weights, 
multiplied by distance of centre of gravity from point of suspension, and 
quotient will give centre required, measured from point of suspension. 


Or, 


W d 2 + W' d ' 2 


— distance of centre. 


WgfW'g' 

Example i.— Length of a suspended rod being 20 feet, and weight of a foot in length 
of it equal 100 oz., has a ball attached at under end weighing 100 oz.; at what point 
of rod from point of suspension is centre of percussion? 

20 

100 X 20 = 2000 = iccight of rod ; 2000 X — = 20 000 — momentum of rod , or prod- 

2000 X 20 2 


uct of its weight, and distance of its centre of gravity ; 
force of rod ; 1000 X 20 2 400000 —force of ball. 

266 666.66 -f- 400 000 


266 666.60 = 


Then 


= 16.66 feet. 


20 000 -f- 20 000 

2.—Assume a rod 12 feet in length, and weighing 2 lbs. for each foot of its length, 
with 2 balls of 3 lbs. each—one fixed 6 feet from the point of suspension, and the 
other at the end of the rod; what is the distance between the points of suspension 
and percussion? 

12 X 2 X — 144 — momentum of rod. 24 X 12 2 _ 3456 

3X6 =18= “ of 1st ball. 3 3 

3X12 = 36= “ of 2d ball. 3X 6 2 r=3X36=io8= li ofistball. 

198 sum of moments. 3Xi2 2 =3 X 144=432 = “ of 2d ball. 

Then 1692 -1- 198 = 8.545 feet. 1692 sum of forces. 


:n52 =force of rod. 


MECHANICS. 

Mechanics is the science which treats of and investigates effects of 
forces, motion and resistance of material bodies, and of equilibrium: 
it is divided into two parts—S tatics and Dynamics. 

Statics treats of equilibrium of forces or bodies at rest. Dynamics 
of forces that produce motion, or bodies in motion. 

These bodies are further divided into Mechanics of Solid, Fluid , and A eri- 
form bodies; hence the following combinations: 

x. Statics of Solid Bodies , or Geostatics. 

2. Dynamics of Solid Bodies , or Geodynamics. 

3. Statics of Fluids , or Hydrostatics. 

4. Dynamics of Fluids , or Hydrodynamics. 

5. Statics of Aeriform Bodies , or Aerostatics. 

6. Dynamics of Aeriform Bodies , Pneumatics or Aerodynamics. 

Forces are various, and are divided into moving forces or resistances; as 

Gravity , Heat or Caloric , Tnertia, 

Muscular, Magnetism , Cohesion , 

Elasticity and Contractility , Percussion , Adhesion , 

Central , Expansion , and Explosion. 

Couple .—Two forces of equal magnitude applied to or operating upon 
same body in parallel and opposite directions, but not in same line of action, 
constitute a couple , and its force is sum or magnitude of the two equal forces. 

Moment .—Quantity of motion in a moving body, which is always equal 
to product of quantity of matter and its velocity. 

When velocities of two moving bodies are inversely as their quantities of 
matter, their momenta are equal. 







MECHANICS.-STATICS. 


615 


STATICS. 

Composition and. Resolution of Forces. 

When two forces act upon a body in same or in an opposite direc¬ 
tion, effect is same as if only one force acted upon it, being sum or 
difference of the forces. Hence, when a body is drawn or projected in 
directions immediately opposite, by two or more unequal forces, it is affected 
as if it were drawn or projected by a single force equal to difference between 
the two or more forces, and acting in direction of greater force. 

This single force, derived from the combined action of two or more forces, 
is their Resultant. 

The process by which the resultant of two or more forces, or a single 
force equidistant in its effect to two or more forces, is determined, is termed 
the Composition of Forces , and the inverse operation ; or, when combined 
effects of two or more forces are equivalent to that of a single given force, 
the process by which they are determined is termed the Decomposition or 
Resolutio?i of Forces. Two or more forces which are equivalent to a single 
force are termed Components. 

When two forces act on same point their intensities are represented by sides 
of a parallelogram , and their combined effect will be equivalent to that of a 
single force acting on point in direction of diagonal of parallelogram , the 
intensity of which is proportional to diagonal. 

Illustration. —Attach three cords to a fixed point, c. Fig. 1; let c a and c b pass 
over fixed rollers, and suspend weights A and B therefrom. 

Point c will be drawn by the forces A and B in directions ac 
and b c. Now, in order to ascertain which single force, P, would 
produce the same effect upon it, set off the distances c m and 
c n on the cords in the same proportion of length as weights 
of A and B; that is, so that cm : cn\ \ A : B; then draw par¬ 
allelogram cm 0 n and diagonal 0 c, and it will represent a sin¬ 
gle force, P, acting in its direction, and having same ratio to 
weights A or B as it has to sides cm or cn of parallelogram. 
Consequently, it will produce same effect on point c as com¬ 
bined actions of A and B. 

A parallelogram, constructed from lateral forces, and diagonal of which is 
mean force, is termed a Parallelogram of Forces. 

Illustration. —Assume a weight. W, Fig. 2, to be 
suspended from a; then, if any distance, a 0, is set 
off in numerical value upon the vertical line, a\V, 
and the parallelogram, 0 r a s, is completed, a s and 
a r , measured upon the scale, a o , will represent 
strain upon ac and ae in same proportion that ao 
bears to weight W. 

If several forces act upon same point , and their intensities taken in order 
are represented by sides of a polygon , except one, a single force proportioned 
to and acting in direction of that one side icill be their resultant. 

To Resolve a Single Force into a Pair of Forces— Figs. 3 and 4. 

The ends of a cord, Fig. 3, are led over two points, a and b , and in centre of 
cord at c a weight of 4 lbs. is suspended. If distances a c, be, are each r foot, dis- 
Fig. 3. tance a b should be 18 ins. 

When cord is in this posi¬ 
tion, weight at c draws upon 
c a and c b with a force of 
3 lbs.; hence c of 4 lbs. is 
equal to two forces of 3 lbs. 
each in direction of a c and b c. 

Apply ends of cord to ef Fig. 4, distance being 22 ins., then the strain on ce , cf, 
are each 5 lbs.; hence one force of 4 lbs. is equal to two of 5 lbs. each. 










6i6 


MECHANICS.-STATICS.-DYNAMICS. 


Eci-uilitoriiaxn. of UTorces. 

Two bodies which act directly against each other in same line are in equi¬ 
librium when their quantities of motion are equal; that is, when product of 
mass of one, into velocity with which it moves or tends to move, is equal to 
product of mass of other, into its actual or virtual * velocity. 

When the velocities with which bodies are moved are same, their forces 
are proportional to their masses or quantities of matter. Hence, when equal 
masses are in motion, their forces are proportional to their velocities. 

Relative magnitudes and directions of any two forces may be represented 
by two right lines, which shall bear to each other the relations of the forces, 
and which shall be inclined to each other in an angle 
equal to that made by direction of the forces. 

Illustration.— Assume a body, W, to weigh 150 lbs., and 
resting upon a smooth surface, to be drawn by two forces, a 
and b, Fig. 5, — 24 and 30 lbs., which make with each other 
an angle; a W 6 = 105°, > n which direction and with what 
acceleration will motion occur? 

Cos. a W b = 105°, and cos. i3o° — 105° = cos. 75°, mean 
force. 

P = V 30 2 -)- 24 2 — 2 X 30 X 24 cos. 75° = V 900 -f- 576 — 1440 cos. 75° 

— V 1476 — (1440 X 258 82) — V 1103.3 ” 33-2i lbs. 

m , ,. . P q 33.21 X 32.166 . , 

The acceleration is ~ = —-—-= 7.1215 feet. 

W 150 ' 

Angle of Repose is greatest inclination of a plane to horizon at which a 
body will remain in equilibrium upon it. 

Hence greatest angle of obliquity of pressure between two planes, consist¬ 
ent with stability, is the angle tangent of which is equal to coefficient of 
friction of the two planes. 

Inertia is resistance which a body at rest offers to an external power to 
be put in motion or to change its velocity or direction when in motion. 

To Compute Inertia of a Revolving Body. 

Divide it into small parts of a regular figure, multiply weight of each part 
by square of its distance of its centre of gravity from axis of revolution, 
and sum of products will give moment of inertia of body. 

DYNAMICS. 

Dynamics is the investigation of the laws of Motion of Solid Bodies, 
or of Matter, Force, Velocity, Space, and Time. 

Mass of a body is the quantity of matter of wdiich it is composed. 

Force is divided into Motive, Accelerative, or Retardative. 

Motive^ Force , or Momentum, of a body, is the product of its mass and 
its velocity, and is its quantity of motion. This force can, therefore, be 
ascertained and compared in any number of bodies when these two 
quantities are known, f 

Accelerative or Retardative Force is that which respects velocity of 
motion only, accelerating or retarding it; and it is denoted by quotient 
of motive force, divided by mass or weight of body. Thus, if a body 

* Virtual velocity is the velocity which a body in equilibrium -would acquire were the equilibrium 
to be disturbed. 

t It is compared, because it is not referable to anv standard, as a ton, pound, etc. Thus, suppose 
a cannon-ball weighing 15 lbs., projected with a velocity of 1500 feet per second, strike a resisting 
body, its momentum, according to the above rule, would be 15 X 1500 = 22 500: not pounds, for weight 
is a pressure with which it cannot be compared. 









MECHANICS.-DYNAMICS. 


617 


of 5 lbs. is impelled by a force of 40 lbs., accelerating force is 8 lbs.; 
but if a force of 40 lbs. act upon a body of 10 lbs., accelerating force 
is only 4 lbs., or half former, and will produce only half velocity. 

With equal masses, velocities are proportional to their forces. 

With equal forces, velocities are inversely as the masses. 

With equal velocities, forces are proportional to the masses. 

Work is product of force, velocity, and time. 

Motion ,—The succession <Jf positions which a body in its motion pro¬ 
gressively occupies forms a line which is termed the trajectory, or path 
of the moving body. 

A motion is Uniform when equal spaces are described by it in equal 
times, and Variable when this equality does not occur. When spaces 
described in equal times increase continuously with the time, a variable 
motion is termed accelerated , when spaces decrease, retarded , and when 
equal spaces are described within certain intervals only, the motion is 
termed periodic , and intervals periods. Uniform motion is illustrated 
in progressive motion of hands of a watch; variable in progressive ve¬ 
locity of falling and upwardly projected bodies; and periodic by oscil¬ 
lation of a pendulum or strokes of a piston of a steam-engine. 


f s W 

Formulas, fv, , H 550, and — 


V_ 

/: 

and 

W 


H 550 W 

~r' and r =v ' 


Uniform HVTotion. 

P W 
v ’ s 
P t 




- - W H 550 1 

'*? T' 7’ and 7 


fs 

H 55° 
= H. 


= <; 


f s, H 550 l, Ft, and/u< = W; 


and 

H 55 o _ 

= /; 

s 


V 


t ’ 


sf 

s 

W 

- 5 ? 

p ’ 

V ’ 

fv' 

P 

f V 

/* 

, and 

550 

550 

550 1 



_ P representing power in effect, body, or momentum, f force in lbs., v and 

55° ^ 

s velocity and space in feet per second, t time in seconds, H horse-power, and W work 
in foot-lbs. 

Tf two or more bodies , etc., are compared, two or more corresponding letters , 
as P ,p,p’, V, v, v', etc., are employed. 

Illustration 1.— Two bodies, one of 20, the other of 10 lbs., are impelled by same 
momentum, say 60. They move uniformly, first for 8 seconds, second for 6; what 
are the spaces described by both? 

60 - 4 - 20 = 3 = V, and 60 - 4 - 10 = 6 = v. 

Then T V r= 3 X 8 = 24 = S, and tvz=6x6 = i6 = s, spaces respectively. 

2. — If a power of 12 800 effects has a velocity of 10 feet per second, what is its 
force ? 12 800 -rio= 1280 lbs. 


Uniform "V'aviaDle HVTotion. 

Space described by a body having uniform variable motion is represented 
by sum or difference of velocity, and product of acceleration and time, ac¬ 
cording as the motion is accelerated or retarded. 

Illustration i.— A sphere rolling down an inclined plane with an initial velocity 
of 25 feet, acquires in its course an additional velocity at each second of time of 5 
feet; what will be its velocity after 3 seconds? 

25 + 5 X 3 = 4 ofeet. 


2 _a locomotive having an initial velocity of 30 feet per second is so retarded 

that in each second it loses 4 feet; what is its velocity after 6 seconds? 

30 — 4X6 = 6 feet. 

3 F* 





6i8 


MECHANICS.—DYNAMICS, 


XJniforiTL HVIotion Accelerated. 


In this motion, velocity acquired at end of any time whatever is equal to prod¬ 
uct of accelerating force into time, and space described is equal to product of half 
accelerating force into square of time, or half product of velocity and time of ac¬ 
quiring the velocity. 

Spaces described in succtessive seconds of time are as the odd numbers, i, 3, 5, 7, 
9, etc. 

Gravity is a constant force, and its effect upon a body falling freely in a vertical 
line is represented by g, and the motion of such l^ody is uniformly accelerated. 


The following theorems are applicable to all cases of motion uniformly acceler¬ 
ated by any constant force, F: 


.5(»=.5SFi= = -iL =s . 


2 S 


~J — 9 F f= f 2 gFs — v. 


2 S 

V 

/ S 

V 

gv 

V -5 9 F 

V 

2 5 

v 2 

9 t 

~ gt z ~ 

2 gs 


When gravity acts alone , as when a body falls in a vertical line, F is omit¬ 
ted. Thus, 


v 2 

• 5 gt- = — =s. 
2 g 


g t=f 2 g s — v. 


v 

g v 9 t 

t representing time in seconds, and s velocity in feel per second. 




2 s 


V2 
2 S 


= 9 ■ 


If, instead of a heavy body falling freely, it be projected vertically upward 
or downward with a given velocity, v , then s = t v.ff .5 g t 2 ; an expression 
in which — must be taken when the projection is upward, and -j- when it is 
downward. 


Illustration i. — If a body in 10 seconds has acquired a velocity by uniformly 
accelerated motion of 26 feet, what is accelerating force, and what space described, 
in that time? 


26 10 = 2.6 — accelerating force ; 


X 10 2 •= 130 feet — space described. 


2.—A body moving with an acceleration of 15.625 feet describes in 1.5 seconds a 
15.625 X (1.5I 2 

space = --- = 17.578 feet. 

3 - A body propelled with an initial velocity of 3 feet, and with an acceleration 

2 

of 5 feet, describes in 7 seconds a space = 3X7 +5X- — 143.5 feet. 

4. —A body which in 180 seconds changes its velocity from 2.5 to 7.5 feet, trav¬ 
erses in that time a distance of 2, 5 d~ 7-5 ^ i g Q __ g QO 

5. —A body which rolls up an inclined plane with an initial velocity of 40 feet per 
second, bj which it suffers a retardation of 8 feet, ascends only — = 5 seconds , and 

4 o 2 ^- 2 X 8 = 100 feet in height, then rolls back, and returns, after 10 seconds, with 
a velocity o f 40 feet, to its initial point; and after 12 seconds arrives at a distance 

of 40 X 12 — 4 X i2 2 = 96 feet below point, assuming plane to be extended backward. 


Circular NEotion. 

2 prn = 2 prn' _^' gSgog _ W frn_ f 2 prn _ 

6° t ’ rn 2 prn'~ J ' 5500 “"Iso X 60 

. ,ftzprn 

j 2 p 1 n — f • r representing radius in feet, n number of revolutions 

of ci) cle pei minute , n total revolutions, f force in lbs., t time in seconds, and IP 
norse-power. 












MECHANICS.—DYNAMICS. 


619 


Nlotioix on an Inclined. Plane. 



To Ascertain Conditions of Motion by Gravity. 

Fig. 6. 1, Assume A B, Fig. 6, an inclined plane, B C its base, 

A C its height, and b a body descending the plane; from 
dot, centre of gravity of body, draw b a perpendicular 
to B C, representing pressure of b by gravity; draw b 0 
parallel and br perpendicular to "A B, and complete 
a "'' c parallelogram; then force & a is equal to both b o,b r, 
of which b r is sustained by reaction of plane, and 
force b 0 is wholly effective in accelerating motion of body. 

Let this force be represented by f and b a, by g or force of gravity, then by similar 
triangle,/: g::b 0 : b a: AC: AB. Hence, A G * g —f 

A 13 

Put A B — l, AC = h and /.ABC = a, then force which produces motion on the 
plane on/becomes g - , and g sin. a. 

L 

Therefore, accelerating force on an inclined plane is constant, and equations of 
motion will be obtained by substituting its value of / for g in equations 1, 2, and 
3, page 618. 

ght 2 l v 2 . v 2 

-——, -p, .5 tv, .5 of 2 sin. a, and - T -=s. 

2I ’ 2 gh' 3 ’ 3 y ’ 2 g sin. a 

2 s ght Izghs , -r./mrTT— . ■ > ' • 

—, ifi \/~^~L —’ andV2fssm.a = u 


2 $ 

IT’ 


l V 

gh’ 


J2I s 

/ 


V 9 


g sin. a 


, and 


V; 


2 s 


g sin. a 


= t. a representing /.ABC. 


When a Body is projected down or up an Inclined Plane, with a given Ve¬ 
locity. — The distance which it will be from point of projection in a given 
time will be a hi 2 t 

t v ±—t j and —- (2 lv±ght)=s. 

2 L 2 i 

Illustration l— Length of an inclined plane is 100 feet, and its angle of inclina¬ 
tion 6o°; what is time of a body rplling down it, and velocity acquired ? 

sin.. 60 0 = .866. 

= -1/7.18 = 2.68 seconds, and 32.16 X 2.68 X .866 = 74.64 feet. 
32.16 X -866 v 

2.—If a body is projected up an inclined plane, which rises 1 in 6, with a velocity 
of 50 feet per second, what will be its place and velocity at end of 6 seconds ? 

Q2. x6 X1 X 6^ / 1 \ 

6 X 50-'- - -= 203.52 feet from bottom , and 50 — I 32.16 X 6 X -7 ) = 

2X6 \ 6/ 

50 — 32.16 = 17.84 feet. t 

To effect an ascent up an inclined plane in least time, its length, to its height, 
must be as twice weight to power. 


V: 


Work Accumulated in Nloving Bodies. 

Quantity of work stored in a body in motion is same as that which would 
be accumulated in it by gravity if it fell from the height due to the velocity. 
Accumulated work expressed in foot-lbs. is equal to product of height so 
found in feet, and weight of body in lbs. Height due to velocity is equal 
to square of velocity divided by 64.4, and work and velocity may be de¬ 
duced directly from each other by following rules: 

To Compute Accumulated Work. 

Rule. —Multiply weight in lbs. by square of velocity in feet per second, 
and divide by 64.4, and quotient is accumulated work in foot-lbs. 

*1)2 x w 

Or, W =—-, or, =wXh. W representing work, w weight in lbs., and 

64.4 

h height due to velocity in feet per second. 














620 


MECHANICS.-DYNAMICS. 


Work by Percussive Force. 

If a wedge is driven by strokes of a hammer or other heavy mass, effect 
of percussive force is measured by quantity of work accumulated in stricken 
body. This work is computed by preceding rules, from weight of body 
and velocity with which a stroke is delivered, or directly from height of 
fall, if gravity be percussive power. 

Useful work done through a wedge is equal to work expended upon it, 
assuming that there is no elastic or vibrating reaction from the stroke, as if 
the work had been exerted by a constant pressure equal to weight of strik¬ 
ing bodjq exerted through a space equal to height of fall, or height due to 
its final velocity. 

If elastic action intervenes, a portion of work exerted is absorbed in an 
elastic stress to resisting body; and the elastic action may be, in some cases, 
so great as to absorb the work expended. 

The principle of action of a blow on a wedge is alike applicable to action 
of the stroke of a monkey of a pile-driver upon a pile. 

If there be no elastic action, the work expended being product of weight 
of monkey by height of its fall, is equal to work performed in driving the 
pile: that is, to product of resistance to its descent by depth through which 
it is driven by each blow of monkey. 

Illustration. —If a horse draws 200 lbs. out of a mine, at a speed of 2 miles per 
hour, how many units of work does he perforin in a minute, coefficient of friction .05 ? 

2 X S^So - 

— -rj: 176 feet per minute. Hence, 176 X 200-f- .05 X 200 = 35 210 units. 

Decomposition of Force. 

By parallelogram of force it is il¬ 
lustrated how a vessel is enabled to 
be sailed with a free wind and against 
one. 

Assume wind to be free or in direction 
of arrows, Fig. 7, and perpendicular to 
line A "B, the course of vessel. 

Let line mo represent direction and 
2 force of wind, and rs plane of sail; from 
0 draw 0 u perpendicular to r s, and 
from m perpendicular, m v on r s , and 
m u on o u. 

By principle of parallelogram of forces, 
force m o may be decomposed into 0 v 
and on, since they are the sides of parallelogram of which m 0, representing force 
of wind, is diagonal. Force of wind, therefore, is measured by 0 u, both in magni¬ 
tude and direction, and represents actual pressure on sail. 

Draw un and u x parallel to oA and om, thus forming parallelogram unox. 

Hence force 0 u is equal to the two, 0 n 
and 0 x. Force 0 n acts in a direction 
perpendicular to vessel’s course and that 
of o x is to drive vessel onward. 

It can thus be shown that when di¬ 
rection of sail bisects angle m o B, the 
effect of 0 x is greater than when sail is 
in any other position. 

Assume wind to be ahead as in direc¬ 
tion of arrows, Fig. 8. Let 0 m repre¬ 
sent direction and force of wind, and r s 
direction of sail; from 0 draw ou, and 
proceed as before, and 0 u represents the 
effective force that acts upon the sail, 
on that which drives her to leeward, and 
0 x that which drives her on her course. 

For full treatises on this subject, see John C. Trautwine’s Engineer’s Pocket-booh, 1872 ; Bull’s Ex¬ 
perimental Mechanics, London, 1871; and Dynamics, Construction of Machinery, etc., by (J. Finden 
Warr, London, 1851. 



Fig. 8. 
















MECHANICS.-MOMENTS OF STRESS ON GIRDERS, ETC. 621 


MOMENTS OF STRESS. 

To Describe and. Compute Moments of* Stress in. Girders 

or Beams. 


Beam Supported at Both Ends. 


Fig. i 



Loaded in Middle , Fig. i. — Assume 
A B beam. At middle erect W c — 
W l 

-. Connect A c and c B, and ver- 

4 

tical distances between them and A B 
2 will give moment required. 

Thus, -= M at any point. W rep¬ 


VT 


resenting weight or load , l length of 
span , x horizontal distance from nearest support at which M is required , and M mo¬ 
ment of stress. 

Illustration. —Assume l— io feet, W = io lbs., and x = 3 feet. 

10 10 25 lbs. at centre of span ; 10 3 = 15 lbs. at x. 

Loaded at Any Point , Fig 2.— 
Proceed as for previous figure. 

W a b 



l 

W xb 


i 

W x a 


or W c — maximum load. 
M between A and W. 

M between W and B. 


a representing least distance of W to support , 
and b greatest distance. I 

Illustration. —Take elements as before with a — 3 feet, and x = 1.5 and 3.5 feet. 

Then, W c — 10 *** 3 7 = 21 lbs. at point of stress; 10 * 1 5 X 7 _ IO ^ ifts. at x 

10 10 

jq \/ o r ^ o 

between A and W, and -—-- = 10.5 lbs. at x between W and B. 

IO 

Note. — x must be taken from the pier which is on the same side of W as x is. 

Loaded with Two Equal Weights at Equal Distances from Ends , alike to a Trans¬ 
verse Girder as for a Single Line of Railway.— Fig. 3. 


Fig. 3- 


A i 




1 


I w 

^A—a —->• 7 

>- 


W 


At point of stress of weights 
erect W c and W d, each = W a. 
Connect A c d and B, and vertical 
»—I distances between A B, as defined 
—by c d, will give moments. 

= Wa = W6=;M at 


iff 

W W (l — o) 


any point between weights. 


Loaded with Four Equal Weights , symmetrically bearing from Centre , alike to a 
Transverse Girder as for a Double Line of Railway.— Fig. 4. 

At W and w" erect W c, and 
w” i = 2 W a, and at w and w' 
erect wd, w' e, each = W (2 a fa'). 

Connect A c d ei and B, and or¬ 
dinates to A B will give mo¬ 
ments. 

W (2 a + a') = M at w and w’\ 
2 W a — M at W and w". 

Illustration. —Assume W each 
10 lbs. 2 feet apart, and 1 10 feet, 

Then, 10 (2 X 2 -f- 2) = 60 at w or w', and 2 X 10 X 2 — 40 at W or w". 











































622 MECHANICS.-MOMENTS OF STRESS ON GIRDERS, ETC. 

Fig. 5. ?n Loaded at Different Points. —Fig. 5. 

Locate three weights, W, w, and 
w’, as at a b, a x bi, a 2 b 2 - 
Draw A c B, ArfB, and A e B, for 
three separate cases, as by formula, 
W a b 

—’ tlg ' 2 - 

Produce W c until Wo — W r, W s, 
and W c ; W d until ivu--wn , w v 
and w d, and w'e to w'm in like 
manner. 

Connect A oum and B, and an or- 



1 


0 © 

c W w 

... 7, . 



1 


_ _y. 

- b 


1 


r- 


Ul J, 

- x-0 2 - 

— -ri 

1 


moment or stress at the point taken. 

Illustration.— Take a —2 feet, a r = 4, a 2 = 6, 6 = 8, &i = 6, b 2 —. 4, x = 2, W, 
w, and w' each 10 lbs., and l = 10 feet. 


Then — (W a x -j- w a z x -}- w’ £>2 x) = M at x. 

i 


280 


Take x = 2. Then — (10X2X2 + 10X4X2 + 10X8X2) = — = 28 lbs. 

10 10 

I - - 44O 

x = 4. Then — (10 X 2 X 2 -f- 10 X 4 X 2 + 10 X 8 X 4) = — = 44 lbs. 

10 10 


500 


Take x = 5. Then — (10 x 2 X 5 + 10 X 4 X 5 + 10 X 4 X 5) = — 5 ° lbs. 

Loaded with a Rolling Weight .— 
Fig. 6. 

Define parabola A c B as deter- 
W i 

mined by — = the ordinate at c, 
4 



W x (l — x) 
. - 


M at any point. 


Loaded Uniformly its Entire Length .—Define parabola as at Fig. 6, ordinate of 
w l 2 

which at c— ■— . L representing stationary or dead load per unit of length. 

Lx w l 2 

- (I —2;) = M at any point, and - =; M at centre. 

2 8 

Loaded with Two Connected Weights, moving in either Direction , alike to a Locomo¬ 
tive or Car on a Railway. —Fig. 7. 

Fig- 7 - c Define parabola A c B as deter- 

11 (W-fw)l 
mined by 1- -—— — c. 

4 

At A and B erect A e, B i = w d, 
connect A i and B e, and vertical 
distances between A 0 B and A c B 



at any point. 

Position of W at greatest moment , when x 


will give moments. 

7. [(W-fw) (l — x) — w d\ 


M 


equal, when x = - ± — 
2 4 


1 , w d 

~ ± -. Or if W and w are 

2 2 (W-j-wq 


Illustration. —Assume x = 3, d = 4, and W w each 10 lbs., and l 10 feet. 

any point , as at W r, w r. 


Then — (10-fioX 10 — 3 — 10X4) = Mai 

IO 

































MECHANICS.-MOMENTS OF STRESS ON GIRDERS, ETC. 623 


Shearing Stress. 

To Determine Shearing Stress at any Bart of a Grirder 
or Beam and under any DistriDntion of Load. 

Fig. 8. 

A J~ 






Required to determine stress of a 
beam at any point as c, Fig. 8. 

Assume W = load between A and 
c, and w that between B and c. 

Then S x at c = P — W, or P' — w. 

The greater of the two values to be taken. 

S x representing shearing stress at any point x, P and P' the reaction on supports 
due to total load on beam between supports , W and w loads or stress concentrated at 
any point. 

Stress 


To IDescrihe 


and Ascertain Shearing 
Grii’der or Beam. 

Supported or Fixed at Both Ends. 


in a 


Fig. 9. 


V. 



1 v ,4 

8 9 8 69 


A 

t I 


-z~ 


t 




Loaded Uniformly. Fig. 9. 

At A and B, erect Ac, Be, each 
W l 

equal to -. Connect c and e at 

2 

middle of span as at n , and vertical 
distances between A B and cne will 
give shearing stresses as determined 
by the ordinates to cne. 


be disregarded. L representing distributed load per unit of length. 
Illustration. —Assume W = 10 lbs. per foot, l — io, and x = 2. 5 feet. 


— x'j — S. Sign of result to 

unit 
io, a 

Then 10 — 2.5^ = 25 lbs. 

Note.— The moment of rupture at any point, produced by several loads acting 
simultaneously on a beam, is equal to the sum of the moments produced by the 
several loads acting separately. 

For other Formulas and Diagrams see Strains in Girders, by William Humber, 
A.L.C.E, London, 1872. 

Operation deduced by Graphic Delineation of Greatest Stress , with a 
Uniformly Distributed Load 0/4000 Lbs. —Fig. 10. 


Fig. 10. 
A 


ft ft ft ® ft # 


Determine moment of weights by 


ID 


, Wmn wr s 
formulas — ; —, - 







w 

% 





r 


■ 

6\ 






w" 


VUT 



l 


l 


and 


w 0 v 
l ' 


Assume W = 7000 lbs., w == 4000, 
and w' = 3000, to — 7 feet, n = 13, 
r= 13. s=7, 0= 3, u = i7, and l=z 20. 

Then W — 7 °°° X X -- — 31 850, 
- a" 20 J 0 ’ 

w = 4000 >< — 3 -X -7 — 18 200, and w' = 3 °°° X 3 X 17 _ and let f a n perpendic- 
20 20 

ulars thereto, as 3 d, 2 c, and 1 b. 

Connect d , c, and b with A B, and sum of distances of intersections of these lines 
upon perpendiculars, from 3, 2, and 1 respectively, will give stress upon A B at 
these points. 

To determine Greatest Stress at Greatest Load. 

Stress at 3 d — 31 850 I Stress at 1 b = 17 : 7650 : 3 = 1350 

“ “ 2 0 = 13 : 18 200 : 7 = 9 800 I . 43000 

000 4 - 7X 13 X 400° X-5 _ iQO lb$ concentrated load at W, and proportion 
20 

of uniformly distributed load of 4000 lbs. 































624 


MECHANICAL POWERS.-LEVER. 


MECHANICAL POWERS. 

Mechanical Power is a compound of Weight , or Force and Velocity: 
it cannot be increased by mechanical means. 

The Powers are three in number —viz., Lever, Inclined Plane, and 
Pulley. 

Note. —A Wheel and Axle is a continuous or revolving lever , a Wedge a double in¬ 
clined plane, and a Screw a revolving inclined plane. 

LEVER. 

Levers are straight, bent, curved, single, or compound. 

To Compute Length of a Lever. 

When Weight and Power are given. Rule.— Divide weight by power, 
and quotient is leverage, or distance from fulcrum at which power supports 
weight. 

w 

Or, y~P w presenting weight , P power , and p distance of power from fulcrum. 

Example.—A weight of 1600 lbs. is to be raised by a power or force of 80; re¬ 
quired length of longest arm of lever, shortest being 1 foot. 

1600 -T- 80 = 20 feet. 

To Compute Weiglit fhat can be raised by a Lever. 

When its Length , Power , and Position of its Fulcrum are given. Rule.— 
Multiply power by its distance from fulcrum, and divide product by dis¬ 
tance of weight from fulcrum. 

P V 

Or, ——= W. w representing distance of weight from fulcrum. 

Example.—W T hat weight can be raised by 375 lbs. suspended from end of a lever 
8 feet from fulcrum, distance of weight from fulcrum being 2 feet? 

375 X 8 - 4 - 2 = 1500 lbs. 


To Compute Position of Fulorum. 

When Weight and Poiver and Length of Lever are given , and when Ful¬ 
crum is between Weight and Power. Rule.—D ivide weight by power, add 
1 to quotient, and divide length by sum thus obtained. 

/W \ 

Or, L -r- I — + 1 ) — w . L representing entire length of lever . 

Example.— A weight of 2460 lbs. is to be raised with a lever 7 feet long and a 
power of 300; at what part of lever must fulcrum be placed ? 

2460-7- 300 =7 8.2, and 8.2 -j- 1 = 9.2. Then (7 X 121.84 - 4 - 9.2 = 9.13 ins . 

When Weight is between Fulcrum and Power. Rule.—D ivide length 
by quotient of weight, divided by power. 

„ T W 

Or, L + p- = w - 

4 

To Compute Length of Arm of Lever to 'wliicli 
Weight is attached. 

When Weight , Power , and Length of Arm of Lever to which Power is ap¬ 
plied are given. Rule. — Multiply power by length of arm to which it is 
applied, and divide product by weight. 


MECHANICAL POWERS.-LEYER. 


625 


Example. —A weight of 1600 lbs., suspended from a lever, is supported by a power 
of 80, applied at other end of arm, 20 feet in length; what is length of arm ? 

80 X 20 -r-1600 = 1 foot. 

Note. —These rules apply equally When fulcrum, (or support ) of lever is between 
weight and power ;* when fulcrum is at one extremity of lever, and power, or weight, 
at the other yf and when arms of lever are equally or unequally bent or curved. 

To Compute Tower Required to Raise a given "Weight. 

When Length of Lever and Position of Fulcrum are given. Rule. —Mul¬ 
tiply weight to be raised by its distance from fulcrum, and divide product 
by distance of power from fulcrum. 

W w „ 

Or, -= P. 

P 

Example. —Length of a lever is 10 feet, weight to be raised is 3000 lbs., and its 
distance from fulcrum is 2 feet; what is power required? 

3000 X 2 6000 

2 - = -5- = 750 lbs. 


To Compute Length op Arm of Lever to which Rower 

is applied. 

When Weight , Power , and Distance of Fulcrum are given. Rule. —Mul¬ 
tiply weight by its distance from fulcrum, and divide product by power. 

^ W w 
Or, -|- *= 1 >. 

Example.— A weight of 400 lbs., suspended 15 ins. from fulcrum, is supported by 
a power of 50, applied at other; what is length of the arm ? 

400 X 15 - 4 - 50 = 120 ins. 


Fig. 1. 



When A rms of a Lever are bent or curved , 
Distances taken from perpendiculars, drawn 
from lines of direction of weight and power, 
must be measured on a line running horizon¬ 
tally through fulcrum, as a b c, Figs. 1 and 2. 

When Arms of a Lever are at Right Angles, 
and Power and Weight are applied at a Right 


Angle to each other , 
Fig. 3, The moments 
are computed directly as a b to b c. 

Thrust, or press¬ 
ure on fulcrum, 
is in this case less 
than sum of pow¬ 
er and weight; 
and it may be 
determined by 
drawing a paral¬ 
lelogram upon 
the two arms of 
lever, arms repre- 
sentinsr inverse- 



1 


Fig. 3. 


O w 

five forces. That is, a b represents magnitude and direction of weight W, 
and be of power P. Diagonal 0 b of parallelogram represents magnitude 
and direction of third force, or thrust upon fulcrum. 


ly their respec- 


* Pressure upon fulcrum is equal to sum of weight and power. 

+ Pressure upon fulcrum is equal to difference of weight and power. 

3 G 






















626 MECHANICAL POWERS.-LEVER.-WHEEL. 



When same Lever is borne into an Oblique 
Position , Power continuing to act Horizontally , 
Fig. 4, Draw vertical a v through end 0 of 
lever, and produce the power line pc to meet 
it at a. Complete parallelogram avbr; then 
sides r b and b v are perpendiculars to direc¬ 
tions to power and weight, on which moments 
are computed. 

Consequently, moment Pxri = moment 
Wxa v, and a diagonal, b a, is resultant thrust 
at fulcrum. 


Fig- 5 - 


When Power does not act Horizon¬ 
tally , Fig. 5, but in some other direc¬ 
tion, a p, produce the power-line p a 
and draw b c perpendicular to it; draw 
(o\p b o, then moments are computed on 
perpendiculars b c. b 0, and P x c b = 
W X b 0. 

If several weights or powers act 
upon one or both ends of a lever, con¬ 
dition of equilibrium is 

K -p P p -(- P' p' -(- P "p etc., = W w -)- 
W' w', etc. 

In a system of levers, either of similar, compound, or mixed 
kinds, condition is P p p' p" 

,, — W. 

IV w w 

Illustration. —Let P = i lb., p and p' each 10 feet, p" 1 foot; and if iv and w’ 
be each 1 foot, and w" 1 inch, then 



= 1200; that is, 1 lb. will support 1200, with levers 


1 X 120 X 120 x 12 172 800 

12 X 12 X 1 144 

of the lengths above given. 

Note.—W eights of levers in above formulas are not considered, centre of gravity 
being assumed to be over fulcrums. 


General Rule, therefore, for ascertaining relation of Power to 
Weight in a lever, whether straight or curved, is, Power multiplied by its 
distance from fulcrum is equal to weight multiplied by its distance from 

fulci um. Or, P : W :; w : p, or P p = W w ; and 

W to P p _ W w _ Pp_ 


W 


"WHEEL AND AXLE. 

-A- "Wlieel and Axle is a revolving lever. 

Power, multiplied by radius of wheel, is equal to weight, multiplied by 
radius of axle. 


As radius of wheel is to radius of axle, so is effect to power. 

Or, P It = W r. Or, P V = W v. Or, R : r W : P. Or, P-^-W; ^ = 

W v 

—- = R. R and r represent ing radii, and V and v velocities of wheel and axle. 












MECHANICAL POWERS. 


-WHEEL AND AXLE. 627 


When a series of wheels and axles act upon each other, either by belts or 
teeth, weight or velocity will be to power or unity as product of radii, or 
circumferences of wheels, to product of radii, or circumferences of axles. 

Illustration.— If radii of a series of wheels are 9, 6, 9, 10, and 12, and their pin¬ 
ions have each a radius of 6 ins., and power applied is 10 lbs., what weight will 
they raise? 

10 X 9 X 6 X 9 X 10 X 12 _ 583 200 _ 

6X6X6X6X6 7776 — 75 

Or, if 1st wheel make 10 revolutions, last will make 75 in same time. 


To Compute Power of a Combination of Wheels and. an 
.Axle or Axles, as in Cranes, etc. 

Rule. —Divide product of driven teeth by product of drivers, and quo¬ 
tient is their relative velocity; which, multiplied by length of lever or arm 
and power applied to it in pounds, ancl divided by radius of barrel, will give 
weight that can be raised. 

v l W j* 

Or, —— — = W; Or, W r = v l P; Or, —— = P. I representing length of lever or 
arm, r radius of barrel, P power, v velocity, and W weight. 

Example i. —A power of 18 lbs. is applied to lever or winch of a crane, length of 
it being 8 ins., pinion having 6 teeth, driving-wheel 72, and barrel 6 ins. diameter. 

— = 12, and 12 X 8 X iB = 1728, which, -f- 3, radius of barrel, = 576 lbs. 

6 


2.—A weight of 94 tons is to be raised 360 feet in 15 minutes, by a power, velocity 
of which is 220 feet per minute; what is power required? 

360 -=-15 = 24 feet per minute. Hence = 10.2545 tons. 

220 

Compound Axle, or* Chinese Windlass. 

Axle or drum of windlass consists of two parts, diameter of one 
being less than that of the other. 

The operation is thus: At a revolution of axle or drum, a portion of sus¬ 
taining rope or chain equal to circumference of larger axle is wound up, and 
at same time a portion equal to circumference of lesser axle is unwound. 
Effect, therefore, is to wind up or shorten rope or chain, by which a weight 
or stress is borne, by a length equal to difference between circumferences of 
the two axles. Consequently, half that portion of the rope or chain will be 
shortened by half difference between circumferences. 


To Compute Elements of a, W'lieel ancl Compound 
Axle, or Chinese Windlass.— Fig. 6 - 

Rule. —Multiply power by radius of wheel, arm, or Fig. 6. 
bar to which it is applied, and divide product by half 
difference of radii of axle, and quotient is weight that a \ 
can be sustained. F 

P R 

Or -— W. R representing radius of wheel, etc., and r and r' 

’ -5 {r — r') 

radii of axle at its greatest and least diameters. 

Example.— What weight can be raised by a capstan, radius of its bar, a, 

5 feet, power applied 50 lbs., and radii, r r', of axle or drum 6 and 5 ins. ? 





50 X 5 x 12 
• 5 (6 — 5) 


= — = 6000 lbs. 
•5 


W 












628 MECHANICAL POWERS.-INCLINED PLANE. 


NVlieel and. IPinion Combinations, or Complex 
Wbeel-work. 

Power, multiplied by product of radii or circumferences, or number of 
teeth of wheels, is equal to weight, multiplied by product of radii or circum¬ 
ferences, or number of teeth or leaves of pinions. 

Or, PRR' R", etc., = W r r' r", etc. 

Note. —Cogs on face of wheel are termed teeth, and those on surface of axle are 
termed leaves ; the axle itself in this case is termed a pinion. 

Rack and Pinion. 


To Compute Tower of a, Raok and. Pinion. 

Rule. —Multiply weight, to be sustained by quotient of radius of pinion, 
divided by radius of crank, and product is power required. 

Or,w£ = P. 

When Pinion on Cranlc Axle communicates with a Wheel and Pinion. 
Rule. —Multiply weight to be sustained by quotient of product of radii of 
pinions, divided by radii of crank and wheel, and product is power required. 

r r' 

° r ’ W RR 7 ~ P- 

Example. —If radii of pinions of a jack-screw are each one inch; of crank and 
wheel io and 5 ins.; what power will sustain a weight of 750 lbs. ? 


750 X 


1 X x 
10 X 5 


75° . 

:- =15 lOS. 

50 


INCLINED PLANE. 

To Compute Length of Base, Height, or Length. 
When any Two of them are given , and when Line o f Direction of Power 
or Traction is Parallel to Face of Plane. —Proceed as in Mensuration or 
Trigonometry to determine side of a right-angled triangle, any two of three 
being given. 

To Compute Power necessary- to Support a "Weight 011 
an Inclined. Plane. 

When Height and Length are given. Rule.- 
of plane, and divide product by length. 


-Multiply weight by height 


o,Ai = P . 


h and l representing height and length of plane. 


Example.— What is power necessary to support 1000 lbs. on an inclined plane 
4 feet in height and 6 feet in length? 

1000 X 4 -r- 6 = 666.67 lbs. 

To Compute Weight that may L>e Sustained by a given 
I 3 oAver on an Inclined IPlane. 

When Height and Length of Plane are given. Rule.—M ultiply power 
by length of plane, and divide product by height. 

p 7 

Or, f- =z W. 
h 

Example. —What is weight that can be sustained on an inclined plane 5 feet in 
height and 7 feet in length by a power of 700 lbs. ? 

700 X 7 -f- 5 = 980 lbs. 

Note. —In estimating power required to overcome resistance of a body being 
drawn up or supported upon an inclined plane, and contrariwise, if body is de¬ 
scending; weight of body, in proportion of power of plane (i. e., as its length to its 
height), must be added to resistance , if being drawn up or supported, or to the mo¬ 
ment if descending. 



MECHANICS.—INCLINED PLANE. 


629 


To Compute TIeiglit or Length, of a-n Inclined. Plane. 


When Weight and Power and one of required Elements are given , and 
when Height is required. Rule. — Multiply power by length, and divide 
product by weight. 

When Length is required. Rule. — Multiply weight by height, and divide 
product by power. 


Or, y = h, and 



To Compute Pressure on an Inclined Tlane. 

Rule.— Multiply weight by length of base of plane, and divide product 
by length of face. 

W b 

Or, —j- = pressure, b representing length oj base of plane. 

Example.—W eight 011 an inclined plane is 100 lbs., base of plane is 4 feet, and 
length of it 5; required pressure on plane. 

100 X 4 -T- 5 = 80 lbs. 


When Two Bodies on Two Inclined Planes sustain each other , as by Connection 
of a Cord over a Pulley , their Weights are directly as Lengths of Planes. 
Illustration. —If a weight of 50 lbs. upon an inclined plane, of 10 feet rise in 100 
of an inclination, is sustained by a weight on another plane of 10 feet rise in 90, 
what is the weight of the latter? 

100 : 90 :: 50 : 45 — weight that on shortest plane would sustain that on largest. 


When a Body is Supported by Two Planes , as Fig. 7, pressure upon them 
j ;itr will be reciprocally as sines of inclinations of planes. 

Thus, weight is as sin. A B D. 

Pressure on A B as sin. D B i. 

Pressure on B D as sin. A B h. 

Assume angle A B D to be 90 0 , and D B i , 6o°; then angle 
A B h will be 30 0 ; and as sines of 90 0 , 6o°, and 30 0 are respec¬ 
tively .1, .866, and .5, if weight =r 100 lbs., then pressures on 
A B and B D will be 86.6 and 50 lbs., centre of gravity of weight assumed to be in its 
centre. 

When Line of Direction of Power is parallel to Base of Plane , power is 
to weight as height of plane to length of its base. 

Or, P : W :: h : b. 



Hence, P: 


W h 


W: 


P b 

IP 


h = 


P b 
W 


6 = 


W h 


When Line of Direction of Power is neither parallel to Face of Plane nor 
to its Base , but in some other Direction , as P', Fig. 8, power is to weight as 
sine of angle of plane’s elevation to cosine of angle which line of power or 
traction describes with face of plane. 

Fig. 8. s\ Thus, P' : W :: sin. A : cos. P' e c. 

P' Sin. A : cos. P' e c :: P' : W. 

Cos. P ' e c : sin. A :: W : P'. 
Illustration. —A weight of 500 lbs. is required to be 
sustained on a plane, angle of elevation of which, 
c A B, is io°; line of direction of power or traction, 
P' e c, is 5 0 ; what is sustaining power required? 

Cos. P' e c (5 0 ) = .996 19 : sin. A (io°) — .173 65 ;; 500 : 87.16 lbs. 

Or, draw a line, B s, perpendicular to direction of power’s action from end 
of base line (at back of plane), and intersection of this line on length, A c, 
will determine length and height (n r) of the plane. 

3 G* 










MECHANICS.—WEDGE.—SCEEW. 


63O 


Illustration.— By Trigonometry (page 385), A B, assumed to be 1, A r and n r are 
= .985 and .171. 

^00 X 1 71 

Hence -- —l— = 86.8 lbs. =product of weight X height of plane -r- length of it. 

• 9 8 5 

Note. —When line of direction of power is parallel to plane, power is least. 

'W'eclge. 

A Wedge is a double inclined plane. 


To Compute Power. 

1. When One Body is to be Forced or Sustained. Rule. —Multiply weight 
or resistance to be sustained by depth of back of wedge, and divide product 
by length of its base. 

Example. —What power, applied to the back of a wedge 6 ins. deep, will raise a 
weight of 15000 lbs., the wedge being 100 ins. long on its base? 

15000X6 00 000 ,, 

-- = --— 900 lbs. 

100 100 


2. When Two Bodies or Two Parts of a Body are Forced or Sustained in a 
Direction Parallel to Back of Wedge. Rule. —Multiply weight or resist¬ 
ance to be sustained by half depth of back of wedge, and divide product by 
length of wedge. 


Or, 



d representing depth of back , and l length. 


Note.—T he length of a single wedge is measured on its base, and of a double 
wedge, from centre of its head to its point. 

Example. —The depth of the back of a double-faced wedge is 6 ins., and the 
length of it through the middle 10; what power applied to it is necessary to sus¬ 
tain or overcome a resistance of 150 lbs. ? 

150 X 6-f- 2 450 

-= = 45 lbs. 

10 10 

To Compute Elements of a Wedge. 


W d 


=zl. 


F l 


: W. 


W d 2 


= 1 . 


P l TTT 

— = W. 
d 

P l 


w 


W d 

~~r 


= d-v- 2. 


= P. 
W d 


l 


-t-2 = P. 


Note.—A s power of wedge in practice depends upon split or rift in wood to be 
cleft, or in rise of body to be raised, the above rules as regards length of wedge are 
only theoretical when a rift or rise exists. 


SCREW. 

A Screw is a revolving inclined plane. 

To Compute Length and. Height of Flane of a Screw. 

As a screw is an inclined plane wound around a cylinder, length of plane 
is ascertained by adding square of circumference of screw to square of dis¬ 
tance between threads, and taking square root of sum. 

The Pitch or height of a screw is distance between its consecutive threads. 

To Compute Tower. 

Rule. —Multiply weight or resistance, to be sustained by pitch of threads, 
and divide product by circumference described by power. * 


Or, 


W_P_ p 

c 


p representing pitch , and c circumference. 


Example. —What is power requisite to raise a 
ins. circumference and 1 inch pitch? 


weight of 8000 lbs. by a screw of 12 
8oco X 1 -T-12 = 666.66 lbs. 









MECHANICS.—SCREW. 


631 


To Compute "Weiglit. 

Rule. —Multiply power by circumference described by it, and divide 
product by pitch of threads. 

Or,— = W. 

P 


To Compute TPiteli. 

Rule. —Multiply power by circumference described by it, and divide 
product by weight. 

„ p c 

0 r ’y? =p - 

To Compute Circumference. 

Rule. —Multiply weight by pitch, and divide product by power. 

W p 


„ Wp ^ 

Or,- F = c. Or, 6 . 28p 


: r. r representing radius. 


When Power is applied by a Lever or Wheel , substitute radius of power 
for circumference. 

Illustration.— If a lever 30 ins. in length was added to circumference of screw 
in preceding example, 

Then, 12 -f- 3.416 = 3.819, and —— -|~ 3 ° — 31-9095 — radius of power. 


Fig. 9. 


TT 8000 X 1 „ 

Hence --—- = 39-92 lbs. 

r X 6.28 y 

Compound Screw. 

When a Lever and Endless Screw or a Series of 
Wheels are applied to a Screw, as Fig. 9. Rule. 
—Ascertain result of each application, and take 
their continued product. 

Note. — If there is more titan one thread to a screw , 
0 pitch must be increased as many times as there are 
threads. 

Illustration. — What weight can be raised with a power of 
10 lbs., applied to a crank, c, Fig 9, 32 ins. long, turning an end¬ 
less screw, 6, of 3.5 ins. diameter and 1 inch pitch, applied to a 
wheel, d, of 20 ins. diameter, upon an axle, a, of 5 ins. ? 

10 X 32 X 6.28 __ 20Q ^ g _ quotient of product of power and 

1 

circumference described by it, and pitch, and 2 °° 9 - x _Jgg _ 8038.4 lbs. = quotient of 
power applied to wheel, divided by its axle. 

When a Series of Wheels and Axles are in Connection with each other , 
Weight is to power, as continued product of radii of wheels is to continued 
product of radii of axles. 

r W : P : *. R n : r n . 



Or r n : R n :: P : W. n representing continued product of number of wheels or 
axles. 

Illustration.— If a power of 150 lbs. is applied to a crank of 20 ins. radius, turn¬ 
ing an endless screw with a pitch of half an inch, geared to a wheel, pinion of 
which is geared to another wheel, and pinion of second wheel is geared to a third 
wheel to axle or barrel of which is suspended a weight; it is required to know 
what weight can be sustained in that position, diameter of wheels being 18, and 
pinions and axle 2 ins. 

150 X 20 X 2 X 3 - I 4 I j _ 27699.2 lbs. — power applied to face of first wheel. 

•5 

Diameters of wheels and pinions being 18 and 2, their radii are 9 and 1. 

Hence, 1 X 1 X 1 : 9 X 9 X 9 •• 37 6 99' 2 : 27482716.8 lbs. 










632 


MECHANICS.—SCREW.-PULLEY. 


Differential Screw. 

When a hollow screw revolves upon one of less diameter and pitch (as 
designed by Mr. Hunter), effect is same as that of a single screw, in which 
the distance between threads is equal to difference of distances between 
threads of the two screws. 

Therefore power, to effect or weight sustained, is as difference between 
distances of threads of the two screws to circumference described by power. 

Illustration. — If external screw has 20 threads, and internal one 21 threads in 
pitch of 1 inch, and power applied describes a circumference of 35 ins., the result or 

power is as co — = —, or .002 38. Hence —= 14 706. 

21 20 420 .00238 


PULLEY. 

Pulleys are designated as Fixed and Movable , according as cord is passed 
over a lixed or a movable pulley. A movable pulley is when cord passes 
through a second pulley or block in suspension; a single movable pulley is 
termed a runner; and a combination of pulleys is termed a system of pulleys. 

A Whip is a single cord over a fixed pulley. 


To Compute Dower Required to Raise a given Weiglit. 

When Number of Parts of Cord supporting Lower Bloch are given , and 
when only one Cord or Rope is used. Rule. —Divide weight to be raised by 
number of parts of cord supporting lower or movable block. 

Or, W-t- n — P. Or, n P = W. n representing number of parts of cord sustain¬ 
ing lower block. 


Example.— What power is required to raise 600 lbs. when lower block contains 
six sheaves? 

When Cord is attached to Upper or Fixed Bloch. 

= 50 lbs. — weight - 3 - number of parts of rope sustaining lower block. 

When Cord is attached to Lower or Movable Bloch. 

= 46.15 lbs. = weight = number of parts of rope sustaining lower block. 


600 

6X2 

600 


6X2 + 1 


To Compute "Weiglit a given Dower -will Raise. 

When Number of Parts of Cord supporting Lower Bloch are given. Rule. 
—Multiply power by number of parts of cord supporting hover block. 

Or, P n = W. 


To Compute Number of Cords necessary to Sustain 
Lower Block. 

When Weight and Power are given. Rule. —Divide weight by power. 

Or, W - 4 - P = n. 


Fig. 10. 



When more than one Cord is used. 

In a Spanish Burton , Fig. 10, where ends of 
one cord, a P, are fastened to support and power, 
and ends of the other, c 0, to lower and upper 
blocks, weight is to power as 4 to 1. 

In another, Fig. 11, where there are two cords, 
a and 0, two movable pulleys, and one fixed 
pulley, with ends of one rope fastened to sup¬ 
port and upper movable pulley, and ends of 
other fastened to lower block and power, weight 
is to power as 5 to 1. 


Fig. 11. 










MECHANICS-PULLEY. 


633 


Fig. 12. 


Compound or Fast and. Loose 3?nlleys. 

When Cord is attached to Fixed Bloch, Fig. 12. Rule.— 
Multiply power by the power of 2, of which the index is 
number of movable pulleys. 

Or, P2 n = W. 

Or, Multiply power successively by 2 for each pulley. 

Example i. —What weight will one pound support in a system 
of three movable pulleys, the cords being connected to a Axed 
block on Fig. 12. 1 X 23 ~ 8 lbs. 

Example 2. —What would a like power support, fixed block be¬ 
ing made movable and cord attached thereto? 

1X2 4 —1 = 15 lbs . 

If fixed pulleys were substituted for hooks a be, Fig. 12, power 
would be increased threefold; hence 1 X3 3 — 2y. 

In a System of Pulleys, Figs. 13 and 14, with any Number of Cords, 00, e e, 
Ends being fastened to Support. 

W 

W-r2 ,l -P; 2 n X P = W; y = 2 n . n rep¬ 
resenting number of distinct cords. 

Illustration. —What weight will a power 
of 1 lb. sustain in a system of two,movable pul¬ 
leys and two cords ? 

1 X 2 X 2 = 4 lbs. 




W 


When fixed Pulleys , e e, are used in Place 
of Hooks, to Attach Ends oj Rope to Sup¬ 
port. —Fig. 14. 

W-F 3 n = P; 3»XP = W; W-fP = 3». 

Illustration. —What weight will a power of 5 lbs. sustain with two movable and 
three fixed pulleys, and two cords ? 2X3X3 = 45 lbs. 



When Ends of Cord or Fixed Pulleys are fastened to Weight, as by an Inver¬ 
sion of the last Figures, putting Supports for Weights, and contrariwise .— 
Figs. 13 and 14. 

w w 

Fig. 13. _ —=z'P\ ( 2 *-i)P = W; _ —( 2 «_i). 

Fig. 14 - — P 5 ( 3 m — i)P = W; y=(3 w — 1 ). 

Illustration. —What weight will a power of 1 lb. sustain in a system of two mov¬ 
able pulleys and two cords, and one of two movable and tw 7 o fixed pulleys and two 
cords ? 1X2X2 — 1 = 3 lbs. 1 X 3 X 3 — 1 = 8 lbs. 


When Cords sustaining Pulleys are not in a Vertical Direction. —Fig. 15. 

Fig. 15. eo, Fig. 15, is vertical line through which weight bears, and 

from o draw 0 r,o s parallel to D e and A e. 
n Forces acting at e are represented by lines e s, e r, and eo; 
u and as tension of every part of cord is same, and equal to 
power P, sides 0 s and or of parallelogram must be equal, and 
therefore diagonal e 0 divides the angle r o s into two equal 
portions. Hence the weight will always fall into the position 
in which the two parts of cord A e and e D will be equally 
inclined to vertical line, and it will bear to power same ratio 
as e 0 to e s. 

Therefore W : P :: 2 cos. .56:1. e representing angle A e D. 
Or, 2 P X cos. .5 e — W. That is, twice power, multiplied by cosine of half angle 
of cord, at point of suspension of weight, is equal to weight. 


























634 METALS.-ALLOYS AND COMPOSITIONS. 


Illustration. —What weight will he sustained by a power of 5 lbs., with an ob¬ 
lique movable pulley, Fig. 15, having an angle, A e D, of 30 0 ? 

5 X 2 X .965 93 — 9.6593 lbs. =. twice power X cos. 15 0 . 


When Direction of Cord is Irregular , Weight not resting in Centre of it. 


P sin. a _ P sin. (a-f-b) _ 
W ~ sip. (a -j- b) ’ sin. a — 
greater and lesser angles of cord at e. 


W sin. a 
sin. (a-j- b) 


a and b representing 


METALS. 

ALLOYS AND COMPOSITIONS. 

Alloy is the proportion of a baser metal mixed with a finer or purer, 
as copper is mixed with gold, etc. 

Amalgam is a compound of Mercury and a metal—a soft alloy. 

Compositions of copper contract in admixture, and all Amalgams ex¬ 
pand. 

In manufacture of Alloys and Compositions, the less fusible metals 
should be melted first. 

In Compositions of Brass, as proportion of Zinc is increased, so is 
malleability decreased. 

Tenacity of Brass is impaired by addition of Lead or Tin. 

Steel alloyed with one five-hundredth part of Platinum, or Silver, is 
rendered harder, more malleable, and better adapted for cutting instru¬ 
ments. 

Specific gravity of alloys* does not follow the ratios of those of their 
components ; it is sometimes greater and sometimes less than the mean. 


Composition for 'Welding Cast Steel. 

Borax, 91 parts; Sal-ammoniac, 9 parts. Grind or pound them roughly together; 
fuse them in a metal-pot over a clear fire, continuing heat until all spume has dis¬ 
appeared from surface. When liquid is clear, pour composition out to cool and con¬ 
crete, and grind to a fine powder; then it is ready .for use. 

To use this composition, the steel to be welded should be raised to a bright yellow 
heat; then dip it in the welding powder, and again raise it to a like heat as before; 
it is then ready to be submitted to the hammer. 


FnsiDle Compounds. 


Compounds. 

Zinc. 

Tin. 

Lead. 

Bismuth, 

Cadmium. 

Rose’s fusing at 200 0 . 


25 

25 



Fusing at less than 200 0 . 

33-3 

33-4 

50 

50 


Newton’s, fusing at less than 212 0 . 
Fusing at 150 0 to 160 0 . 

12 

JJ* J 

3i 

25 

*3 


Solders. 

Solder is an alloy used to make joints between metals, and it must be 
more fusible than the metals it is designed to unite, and it is distinguished 
as hard and soft , according to the temperature of its fusing. 

The addition of a small portion of Bismuth increases its fusibility. 


* For a table of Alloys, having densities different from a mean of their components, see D. K. Clark’s 
Manual, London, 1877, page 201. 



















METALS.-ALLOYS AND COMPOSITIONS. 635 


.A.lloys and. Compositions. 



Copper. 

Zinc. 

Argentan . 

55 

2 4 

Aluminum, brown . 

95 

— 

Babbitt’s metal * . 

3-7 

— 

Brass, common . 

84-3 

5 - 2 

u u 

75 

25 

“ “ hard . 

79-3 

6.4 

“ instruments . 

02. 2 

— 

“ locomot. bearings. 

9 ° 

I 

“ Pinchbeck . 

80 

20 

“ red Tombac . 

88.8 

II .2 

“ rolled . 

74-3 

22.3 

“ Tutenag . 

50 

31 

“ very tenacious... 

88.9 

2.8 

“ wheels, valves. ... 

9 ° 

— 

“ white . 

IO 

80 

U U 

3 

90 

it u 

7 

— 

“ wire . 

67 

33 

“ yellow, fine . 

66 

34 

Britannia metal . 

— 

— 

When fused add . 

— 

— 

Bronze, red . 

87 

13 

U U 

86 

II. I 

“ yellow . 

67.2 

31.2 

“ Gun metal, large 

90 

— 

“ “ small 

93 

— 

“ “ soft. 

95 

— 

“ Cymbals . 

80 

— 

“ Medals . 

93 

— 

“ Statuary . 

9 1 * 4 

5-5 

Chinese silver . 

58.1 

17.2 

“ white copper. .. 

40.4 

25-4 

Church bells . 

80 

5-6 

U U 

69 


Clocks, Musical bells. .. . 

87-5 

— 

Clock bells. 

7 2 

— 

German silver. 

33-3 

33-4 

U U 

40.4 

25-4 

“ “ fine. 

49-5 

24 

Gongs. 

81.6 

— 

House bells . 

77 

— 

Lathe bushes . 

80 

— 

Machinery bearings . 

8 7-5 

— 

“ “ hard. 

77-4 

7 

Metal that expands in 
cooling . ) 


— 

Muntz metal, 10 oz. lead. 

60 

40 

Pewter, best . 

— 


U 

— 

— 

Sheathing metal . 

56 

45 

Speculum “ . 

66 

— 

U U 

50 

21 

Telescopic mirrors . 

66.6 

;— 

Temper f. . 

33-4 

— 

Type metal and stereo-) 

— 

— 

type plates . j 

— 

— 

White metal. 

7-4 

7-4 

“ “ hard. 

69.8 

25.8 

Oreiae . 

73 

12.3 


Tin. 

Nickel. 

Lead. 

Anti¬ 

mony. 

Bis¬ 

muth. 

Alu¬ 

minum. 

— 

21 

— 

— 

— 

• — 

— 

— 

— 

— 

— 

5 

89 

— 

— 

7-3 



10.5 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

i 4-3 

— 

— 

— 

— 

— 

7.8 

— 

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— 

' - 

— 

9 

— 

— 

— 

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— 


— 

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— 

— 

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3-4 

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19 

— 

— 

— 

— 

0° 

CO 


— 

— 

— 

— 

10 

— 

— 

— 

— 

— 

10 

— 

— 

— 

— 

— 

— 

— 

— 

7 

— 

— 

— 

— 

46 

47 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

25 

— 

— 

25 

— 

— 

— 

— 

— 

25 

25 

— 

— 

— 

— 

— 

— 

— 

2.9 

— 

—■ 

— 

— 

— 

1.6 

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— 

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c 

10 

— 

— 

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0 

►-« 

7 

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O 

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20 

— 

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— 

£ 

0 3 

4=1 

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7 




> 

0 

1.4 

— 

i -7 

— 

GQ 


— 

11.6 

— 

— 

2 

II.I 

2.6 

31.6 

— 

— 

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— 

10.1 

— 

4-3 

— 

— 

C 

O 

31 

— 

— 

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12.5 

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26.5 

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33-3 

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00 

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s 

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75 

16.7 

8-3 

0 

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— 


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— 

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03 

80 

— 

20 

— 

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22 

—. 

— 

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12 

29 

— 

— 

— 

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33-4 

— 

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— 

— 

— 

66.6 

— 

— 

— 

— 

— 

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75 

25 

— 

— 

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— 

87-5 

12.5 

— 

— 

28.4 

— 


56.8 

— 

— 

4.4 

— 

— 

— 

— 

— 


Magnesia.4.4 Cream of tartar .6,5 

Sal-ammoniac . 2.5 Quicklime.r. 3 


See page 636 for directions. 


t For adding small quantities of copper. 

































































636 METALS.-ALLOYS AND COMPOSITIONS. 


Solders. 




Copper. 

| Tin. 

Lead. 

Zinc. 

Silver. 

Bis¬ 

muth. 

Gold. 

Cad¬ 

mium. 

Anti¬ 

mony. 

Tin. 


_ 

25 

75 

_ 

_ 

— 

— 

_ 

_ 

t t 


— 

58 

l6 

— 

— 

l6 

— 

— 

10 

“ coarse, melts 
at 500 0 ... 


— 

33 

67 

— 

— 

— 

— 

— 

— 

“ ordi’y, melts 
at 360°... 


— 

67 

33 

— 

— 

— 

— 

— 

— 

Spelter, soft. 


50 

— 

— 

50 

— 

— 

— 

— 

— 

“ hard.... 


63 

— 

— 

35 

— 

— 

— 

— 

— 

Lead. 



33 

67 


— 

— 

— 

— 

— 

Steel. 


13 



5 

82 

— 

— 

— 

— 

Brass or Copper. 


50 

— 

— 

50 

— 

— 

— 

— 

—- 

Fine brass. 


47 

— 

_ 

47 

6 

— 

— 

— 

-— 

Pewterers’ or Soft 


33 

45 


— 

22 

— 

— 

— 

U U 


— 

5 o 

25 

— 

— 

25 

— 

— 

— 

Plumbers’ pot-' 
metal.. 


— 

33 

67 

— 

— 

— 

— 

— 

— 

M coarse.. 


— 

25 

75 

— 

— 

— 

— 

— 

— 

“ fine. 


— 

67 

33 

— 

— 

— 

— 

— 

— 

“ fusible.. 


— 

50 

50 

— 

— 

— 

— 

— 

— 

“ very “ .. 


— 

25 

25 

— 

— 

50 

— 

— 

—■ 

Gold. 


4 

— 

— 

— 

7 

— 

89 

— 

— 

“ hard . 


66 

— 

— 

34 



— 

— 

“ soft. 


_ 

66 

34 


— 

— 

— 

— 

— 

Silver, hard. 


20 

— 

— 

80 

— 

— 


— 

“ soft. 


12 

— 

— 

— 

67 

— 

— 

21 

— 

Pewter. 


— 

40 

20 

— 

40 

— 

— 

— 

Iron. 


66 


— 

33 

— 

— 

— 

1 

Copper. 


53 

47 

— 


— 

— 

— 

— 

— 


A Plastic Metallic Alloy. —See Journal of Franklin Institute, vol. xxxix., page 55, 
for its composition and manufacture. 


Soldering Fluid for use with Soft Solder. 

To 2 fluid oz. of Muriatic acid add small pieces of Zinc until bubbles cease to rise. 
Add .5 a teaspoonful of Sal-ammoniac and two fluid oz. of Water. 

By the application of this to Iron or Steel, they may be soldered without their sur¬ 
faces being previously tinned. 


Iron. 

Tinned iron. 

Copper and Brass 


Fluxes for Soldering or Welding. 


Borax. 

Resin. 

Sal-ammoniac. 


Zinc. 

Lead. 

Lead and tin 


Chloride of zinc. 
Tallow or resin. 
Resin and sweet oil. 


BaD'bitt’s Anti-attrition IVIetal. 

Melt 4 lbs. Copper; add by degrees 12 lbs. best Banca tin, 8 lbs. Regulus of anti¬ 
mony, and 12 lbs. more of Tin. After 4 or 5 lbs. Tin have been added, reduce heat 
to a dull red, then add remainder of metal as above. 

This composition is termed hardening; for lining , take 1 lb. of this hardening , 
melt with it 2 lbs. Banca tin, which produces the lining metal for use. Hence, the 
proportions for lining metal are 4 lbs. of copper, 8 of regulus of antimony, and 06 
of tin. 

Brass. 

Brass is an alloy of copper and zinc, in proportions varying with purpose 
of metal required, its color depending upon the proportions. 

It is rendered brittle by continued impacts, more malleable than copper 
when cold, but is impracticable of being forged, as its zinc melts at a low 
temperature. 

Its fusibility is governed by its proportion of zinc; a small quantity of 
phosphorus gives it fluidity. 






















































METALS.—ALLOYS AND COMPOSITIONS.—IKON. 637 


Bronze. 

Bronze is an alloy of copper and tin; it is harder, more fusible, and 
stronger than copper. It is usually known as Gun-metal. 

Aluminum Bronze contains 90 to 95 per (jent. of copper, and 5 to 10 per 
cent, aluminum. 

Phosphor Bronze contains copper and tin and a small proportion of phos¬ 
phorus. It wears better than bronze. 

IRON. 

Foreign substances which iron contains modify its essential proper¬ 
ties. Carbon adds to its hardness, but destroys some of its qualities, 
and produces Cast Iron or Steel, according to proportion it contains. 
Thus, .25 per cent, renders it malleable, .5 steel, 1.75 is limit of weld¬ 
ing steel, and 2 is. lowest limit of cast iron. Sulphur renders it fusible, 
difficult to weld, and brittle when heated, or “ hot short.” Phosphorus 
renders it “ cold short” but may be present in proportion of .002 to 
.003, without affecting injuriously its tenacity. Antimony , Arsenic , and 
Copper have same effect as sulphur, the last in a greater degree. Sili¬ 
con renders it hard and brittle. Manganese , in proportion of .02, ren¬ 
ders it “ cold short” and Vanadium adds to its ductility. 

Cast Iron. 

Process of making Cast Iron depends much upon description of fuel used; 
whether charcoal, coke, bituminous, or anthracite coals. A larger yield from 
same furnace, and a great economy in fuel, are effected by use of a hot blast. 
The greater heat thus produced causes the iron to combine with a larger 
percentage of foreign substances. 

Cast Iron for purposes requiring great strength should be smelted with 
a cold blast. Pig-iron , according to proportion of carbon which it contains, 
is divided into Foundry Iron and Forge Iron , latter adapted only to conver¬ 
sion into malleable iron; while former, containing largest proportion of car¬ 
bon, can be used either for castings or bars. 

High temperature in melting injures gun-metal. 

There are many varieties of Cast Iron, differing by almost insensible 
shades; the two principal divisions are gray and white, so termed from 
color of their fracture. Their properties are very different. 

Gray Iron is softer and less brittle than white; it is in a slight degree 
malleable and flexible, and is insonorous; it can easily be drilled or turned, 
and does not resist the file. It has a brilliant fracture, of a gray, or some¬ 
times a bluish-gray, color; color is lighter as grain becomes closer, and its 
hardness increases. It melts at a lower heat than white, and preserves its 
fluidity longer. Color of the fluid metal is red, and deeper in proportion as 
the heat is lower; it does not adhere to the ladle; it fills molds well, con¬ 
tracts less, and contains fewer cavities than white; edges of its castings 
are sharp, and surfaces smooth and convex. It is used for machinery and 
ordnance where the pieces are to be bored or fitted. Its tenacity and specific 
gravity are diminished by annealing. 

White Iron is very brittle and sonorous; it resists file and chisel, and is 
susceptible of high polish ; surface of its castings is concave; fracture pre¬ 
sents a silvery appearance, generally fine grained and compact, sometimes 
radiating or lamellar. When melted it is white, throws off a great number 
of sparks, and its qualities are the reverse of those of gray iron ; it is there¬ 
fore unsuitable for machinery purposes. Its tenacity is increased , and its 
specific gravity diminished , by annealing. 


6 3 8 


METALS.-IRON. 


Mottled Iron is a mixture of white and gray; it has a spotted appear¬ 
ance ; flows well, and with few sparks; its castings have a plane surface, 
with edges slightly rounded. It is suitable for shot, shells, etc. A fine mot¬ 
tled is only kind suitable for castings which require great strength. The 
kind of mottle will depend much upon volume of the casting. A medium¬ 
sized grain, bright gray color, fracture sharp to touch, and a close, compact 
texture, indicate a good quality of iron. A grain either very large or very 
small, a dull, earthy aspect, loose texture, dissimilar crystals mixed together, 
indicate an inferior quality. 

Besides these general divisions, the different varieties of pig-iron are more 
particularly distinguished by numbers, according to their relative hardness. 

No. i.—Fracture dark gray, crystals large and highly lustrous, alike to 
new surface of lead. It is the softest iron, possessing in highest degree the 
qualities belonging to gray iron; it has not much strength, but on account 
of its fluidity when melted, and of its mixing advantageously with scrap 
iron and with the harder kinds of cast iron, it is of great use to a foundry. 

No. 2 is harder, closer grained, and stronger than No. i; it has a gray 
color and considerable lustre. It is most suitable for shot and shells. 

No. 3 is harder than No. 2. Fracture white, crystals larger and brighter 
at centre than at the sides; color gray, but inclining to white; has consid¬ 
erable strength, but is principally used for mixing with other kinds of iron 
and for large castings. 

No. 4 or Bright .—Fracture light gray, with small crystals and little lustre, 
and not being sufficiently fusible for castings it is used for conversion to 
wrought iron. 

No. 5. Mottled. — Fracture dull white, with gray specks, and a line of 
white around edge or sides of fracture. 

No. 6. White .—Fracture white, with little lustre, granulated with radiat¬ 
ing crystalline surface. It is hardest and most brittle of all descriptions, 
and is unfit for use unless mixed with other grades, or for being converted 
to an inferior wrought iron. 

Qualities of these descriptions depend upon proportion of carbon, and upon 
state in which it exists in the metal; in darker kinds of iron, wdiere propor¬ 
tion is sometimes 7 per cent., it exists partly in state of graphite or plumbago, 
which makes the iron soft. In white iron the carbon is thoroughly com¬ 
bined with the metal, as in steel. 

Cast iron frequently retains a portion of foreign ingredients from the ore, 
such as earths or oxides of other metals, and sometimes sulphur and phos¬ 
phorus, which are all injurious to its quality. 

Foreign substances, and also a portion of the carbon, are separated by 
melting iron in contact with air, and soft iron is thus rendered harder and 
stronger. Effect of remelting varies with nature of the iron and character 
of ore from which it has been extracted; that from hard ores, such as mag¬ 
netic oxides, undergoes less alteration than that from hematites, the latter 
being sometimes changed from No. 1 to white by a single remelting in an 
air furnace. 

Color and texture of cast iron depend greatly upon volume of casting and 
rapidity of its cooling; a small casting, which cools quickly, is almost ahvavs 
white , and surface of large castings partakes more of the qualities of wdiite 
metal than the interior. 

All cast iron expands at moment of becoming liquid, and contracts in cool¬ 
ing; gray iron expands more and contracts less than other iron. 

Remelting iron improves its tenacity; thus, a mean of 14 cases for two 
fusions gave, for 1st fusion, a tenacity of 29284 lbs.; for 2d fusion, 33 790 
lbs. For tw T o cases—for first fusion, 15 129 lbs.; for 2d fusion, 35 786 lbs." 


METALS.-IRON. 


639 


ALallea'ble Castings. 

Malleable cast iron is made by subjecting a casting to a process of anneal¬ 
ing, by enclosing it in a box with hematite iron ore or black oxide of iron, 
and maintaining it in an equable heat for a period depending upon form and 
volume of casting. 

Wrought Iron. 

Wrought iron is made from pig-iron in a Bloomei'y Fire or in a Puddling 
Furnace —generally in latter. Process consists in melting and keeping it 
exposed to a great heat, constantly stirring the mass, bringing every part of 
it under action of the flame until it loses its remaining carbon, when it be¬ 
comes malleable iron. When, however, it is desired to obtain iron of best 
quality, pig-iron should be refined. 

Refining. —This operation deprives iron of a considerable portion of its 
carbon; it is effected in a Blast Furnace , where iron is melted by means of 
charcoal or coke, and exposed for some time to action of a great heat; the 
metal is then run into a cast-iron mold, by which it is formed into a large 
broad plate. As soon as surface of plate is chilled, cold water is poured on 
to render it brittle. 

A Bloomery resembles a large forge fire, where charcoal and a strong blast 
are used ; and the refined metal or pig-iron, after being broken into pieces of 
proper size, is placed before the blast, directly in contact with charcoal; as 
the metal fuses, it falls into a cavity left for that purpose below the blast, 
where the “ bloomer ” works it into the shape of a ball, which he places again 
before the blast, with fresh charcoal; this operation is generally again re¬ 
peated, when ball is ready for the “ shingler.” 

Shingling is performed in a strong squeezer or under a trip-hammer. Its 
object is to press out as perfectly as practicable the liquid cinder which a 
ball contains; it also forms a ball into shape for the puddle rolls. A heavy 
hammer, weighing from 6 to 7 tons, effects this object most thoroughly, but 
not so cheaply as the squeezer. A ball receives from 15 to 20 blows of a 
hammer, being turned from time to time as required: it is now termed a 
Bloom. , and is ready to be rolled or hammered; or a ball is passed once 
through the squeezer, and is still hot enough to be passed through the puddle 
rolls. 

A Puddling Furnace is a reverberatory furnace, where flame of bituminous 
coal is brought to act directly upon the melted metal. The “ puddler ” then 
stirs it, exposing each portion in turn to action of flame, and continues this 
as long as he is able to work it. When it lias lost its fluidity, he forms it into 
balls, weighing from 80 to 100 lbs., which are then passed to the “shingler.” 

Puddle Rolls. — By passing through different grooves in these rolls, a 
bloom is reduced to a rough bar from 3 to 4 feet in length, its term convey¬ 
ing an idea of its condition, which is rough and imperfect. 

Piling. —To prepare rough bars for this operation, they are cut, by a pair 
of shears , into such lengths as are best adapted to the volume of finished bar 
required; the sheared bars are then piled one over the other, according to 
volume required, when pile is ready for balling. 

Balling. —This operation is performed in balling furnace, which is similar 
to puddling furnace, except that its bottom or hearth is made up, from time 
to time, with sand; it is used to give a welding heat to piles to prepare 
them for rolling. 

Finishing Rolls. —The balls are passed successively between rollers of va¬ 
rious form’s and dimensions, according to shape of finished bar required. 

Quality of iron depends upon description of pig-iron used, skill of the 
“ puddler,” and absence of deleterious substances in the furnace. 


METALS.—IRON.-LEAD.—STEEL. 


64O 

Strongest cast irons do not produce strongest malleable iron. 

For many purposes, such as sheets for tinning, best boiler-plates, and bars 
for converting into steel, charcoal iron is used exclusively; and, generally, 
this kind of iron is to be relied upon, for strength and toughness, with greater 
confidence than any other, though iron of a superior quality is made from 
pigs made with other fuel, and with a hot blast. Iron for gun-barrels has 
been lately made from anthracite hot-blast pigs. 

Iron is improved in quality by judicious working, reheating, hammering, 
or rolling: other things being equal, best iron is that which has been wrought 
the most. 

Best quality of iron has greatest elasticity. 

Tests. —It will not blacken if exposed to nitric acid. Long silky fibres in 
a fracture denote a soft and strong metal; short black fibres denote a badly 
refined metal, and a fine grain denotes hardness and condition known as 
“ cold short.” Coarse grain with bright and crystallized fracture, with dis¬ 
colored spots, also denotes “ cold short ” and brittle metal, working easily and 
welding well. Cracks upon edges of a bar, etc., indicate “ hot short.” Good 
iron heats readily, is worked easily, and throws off but few sparks. 

A high breaking strain may not be conclusive as to quality, as it may be 
due to a hard, elastic metal, or a low one may be due to great softness. 

When iron is fractured suddenly, a crystalline surface is produced, and 
when gradually, a fibrous one. Breaking strain of iron is increased by heat¬ 
ing it and suddenly cooling it in water. Iron exposed to a welding or white 
heat and not reduced by hammering or rolling is weakened. 

Specific gravity of iron is a good indication of its quality, as it indicates 
very correctly its relative degree of strength. 

LEAD. 

Sheet Lead is either Cast or Milled , the former in sheets 16 to 18 feet in 
length and 6 feet in width; the latter is rolled, is thinner than the former, 
is more uniform in its thickness, and is made into sheets 25 to 35 feet in 
length, and from 6 to 7.5 feet in width. 

Soft or Rain Water, when aerated, Silt of rivers, Vegetable matter, Acids, 
Mortar, and Vitiated Air will oxidize lead. The waters which act with 
greatest effect on it are the purest and most highly oxygenated, also nitrites, 
nitrates, and chlorides, and those which act with least effect are such as con¬ 
tain carbonate and phosphate of lime. 

Coating of Pipes , except with substances insoluble in water, as Bitumen 
and Sulphide of lead, is objectionable. 

Lead-encased Pipes. —An inner pipe of tin is encased in one of lead. 

STEEL. 

Steel is a compound of Iron and Carbon, in which proportion of latter 
is from 1 to 5 per cent., and even less in some descriptions. It is dis¬ 
tinguished from iron by its fine grain, and by action of diluted nitric 
acid, which leaves a black spot upon it. 

There are many varieties of steel, principal of which are: 

Natural Steel , obtained by reducing rich and pure descriptions of iron 
ore with charcoal, and refining cast iron, so as to deprive it of a sufficient 
portion of carbon to bring it to a malleable state. It is used for files and 
other tools. 

Indian Steel, termed Wootz, is said to be a natural steel, containing a small 
portion of other metals. 


METALS. —STEEL. 


64I 

Blistered Steel , or Steel of Cementation , is prepared by direct combination of 
iron and carbon. For this purpose, iron in bars is put in layers, alternating 
with powdered charcoal, in a close furnace, and exposed for 7 or 8 days to 
a high temperature, and then put to cool for a like period. The bars, on 
being taken out, are covered with blisters, have acquired a brittle quality, 
and exhibit in fracture a uniform crystalline appearance. The degree of 
carbonization is varied according to purposes for which the steel is intended, 
and the very best qualities of iron are used for the finest kinds of steel. 

Tilted Steel is made from blistered steel moderately heated, and subjected 
to action of a tilt hammer, by which means its tenacity and density are in¬ 
creased. 

Shear Steel is made from blistered or natural steel, refined by piling thin 
bars into fagots, which are brought to a welding heat in a reverberatory 
furnace, and hammered or rolled again into bars ; this operation is repeated 
several times to produce finest kinds of shear steel, which are distinguished 
by the terms of Half shear, Single shear, and Double shear , or steel of 1, 2, or 
3 marks , etc., according to number of times it has been piled. 

Spring Steel is blister steel heated to an orange red color and rolled or 
hammered. 

Cast or Crucible Steel is made by breaking blistered steel into small pieces 
and melting it in close crucibles, from which it is poured into iron molds; 
ingot is then reduced to a bar by hammering or rolling. Cast steel is best 
kind of steel, and best adapted for most purposes; it is known by a very 
fine, even, and close grain, and a silvery, homogeneous fracture; it is very 
brittle, and acquires extreme hardness, but is difficult to weld without use 
of a flux. Other kinds of steel have a similar appearance to cast steel, but 
grain is coarser and less homogeneous; they are softer and less brittle, and 
weld more readily. A fibrous or lamellar appearance in fracture indicates 
an imperfect steel. A material of great toughness and elasticity, as well as 
hardness, is made by forging together steel and iron, forming the celebrated 
Damasked Steel , which is used for sword-blades, springs, etc.: damask ap¬ 
pearance of which is produced by a diluted acid, which gives a black tint to 
the steel, while the iron remains white. 

With cast steel, breaking strength is greater across fibres of rolling than 
with them. 

HeaWs Process is an improvement on this method, and consists in adding to 
molten metal a small quantity of carburet of manganese. 

Heaton's Process consists in adding nitrate of soda to molten pig-iron, in order to 
remove carbon and silica. 

Mushet's Process.— -Malleable iron is melted in crucibles with oxide of manganese 
and charcoal. 

Puddled Steel is produced by arresting the puddling in the manufacture 
of the wrought iron before all the carbon has been removed, the small 
amount of carbon remaining, .3 to 1 per cent., being sufficient to make an 
inferior steel. 

Mild Steel contains from .2 to .5 per cent, of carbon; when more is pres¬ 
ent it is termed Hard Steel. 

Bessemer Steel is made direct from pig-iron. The carbon is first removed, 
in order to obtain pure wrought iron, and to this is added the exact quantity 
of carbon required for the steel. The pig should be free from sulphur and 
phosphorus. It is melted in a blast or cupola, and run into a converter (a 
pear-shaped iron vessel suspended on hollow trunnions and lined with fire¬ 
brick or clay), where it is subjected to an air blast for a period of 20 min¬ 
utes, in order to dispel the carbon, after which from 5 to 10 per cent, of spie- 
geleisen is added. 


METALS.-STEEL. 


642 

The blast is then resumed for a short period, to incorporate the two metals, 
when the steel is run off into molds. The moment at which all the carbon 
has been removed is indicated by color of the flame at mouth of converter. 
The ingots, when thus produced, contain air holes, and it becomes necessary 
to heat them and render them solid under a hammer. 

Siemen's Process. —Pig-iron is fused upon open hearth of a regenerative 
furnace, and when raised to a steel-melting temperature, rich and pure ore 
and limestone are added gradually, whereby a reaction is established between 
the oxygen of the ferrous oxide and the carbon and silicon in the metal. The 
silicon is thus converted into silicic acid, which with the lime forms a fusible 
slag, and the carbon, combining with oxygen, escapes as carbonic acid, and 
induces a powerful ebullition. 

Modification of this process .—The ore is treated in a separate rotatory furnace 
with carbonaceous material, and converted into balls of malleable iron, which are 
transferred from the rotatory to the bath of the steel-melting furnace. 

This process is adapted to the production of steel of a very high quality, because 
the sulphur and phosphorus of the ore are separated from the metal in the rotatory 
furnace. 

Siemen's-Martin Process. —Scrap-iron or steel is gradually added in a 
highly heated condition to a bath of about .25 its weight, of highly heated 
pig, and melted. Samples are occasionally taken from the bath, in order to 
ascertain the percentage of carbon remaining in the metal, and ore is added 
in small quantities, in order to reduce the carbon to about .1 per cent. 

At this stage of the process, siliceous iron, spiegeleisen, or ferro-manganese 
is added in such proportions as are necessary to produce steel of the required 
degree of hardness. The metal is then tapped into a ladle. 

Landore-Siernen's Steel is a variety of steel made by the Modification of 
Siemen's Process. Its great value is due to its extreme ductility, and its 
having nearly like strength in both directions of its plates. 

Whitworth's Compressed Steel is molten steel subjected to a pressure of 
about 6 tons per square inch, by which all its cavities are dispelled, and it is 
compressed to about .875 of its original volume, its density and strength be¬ 
ing proportionately increased. 

Chrome and Tungsten Steel are made by adding a small percentage of 
Chromium or Tungsten to crucible steel, the result producing a steel of 
great hardness and tenacity, suitable for tools, such as drills, etc. 

Homogeneous Steel is a variety of cast steel containing .25 per cent, of 
carbon. 

Remarks on Manufacture of Steel, and Mode of Working it. 

ID. Chernoff , 1868). 

Steel, when cast and allowed to cool quietly, assumes a crystalline structure. 
Higher temperature to which it is heated, softer it becomes, and greater is liberty 
its particles possess to group themselves into crystals. 

Steel, however hard it may be, will not harden if heated to a temperature low r er 
than what may be distinguished as dark cherry-red, a , however quickly it is cooled- 
on contrary, it will become sensibly softer, and more easily worked with a file. ’ 

Steel, heated to a temperature lower than red, but not sparkling, b, does not 
change its structure whether cooled quickly or slowly. When temperature has 
reached b , substance of steel quickly passes from granular or crystalline condition 
to amorphous, or wax-like structure, which it retains up to its melting-point, c. 

Points a, b, and c have no permanent place in scale of temperature, but their posi¬ 
tions vary with quality of steel; in pure steel, they depend directly on quantity of 
constituent carbon. Harder the steel, lower the temperatures. Tints above speci¬ 
fied have reference only to hard and medium qualities of steel; in very soft kinds 
of steel, nearly approaching to wrought iron, points a and b range very high, and in 
wrought iron point b rises to a white heat. 


METALS.-STEEL. 


643 

Assumption of the crystalline structure takes place entirely in cooling, between 
temperatures c and b ; when temperature sinks below b there is no change of struc¬ 
ture. For successful forging, therefore, heated ingot, after it is taken out of furnace, 
must be forged as quickly as practicable, so as not to leave any spot untouched by 
hammer, where the steel might crystallize quietly, as formation of crystals should 
be hindered, and the steel should be kept in an amorphous condition until tem¬ 
perature sinks below point b. 

Below this temperature, if piece is cooled in quiet, mass will no longer be disposed 
to crystallize, but will possess great tenacity and homogeneousness of structure. 

When steel is forged at temperatures lower than 6, its crystals or grains, being 
driven against each other, change their shapes, becoming elongated in one direction, 
and contracted in another; while density and tensile strength are considerably in¬ 
creased. But available hammer-power is only sufficient for treatment of small steel 
forgings; and object of preventing coarse crystalline structure in large forgings 
is more easily and more certainly effected, if, after having given forging desired 
shape, its structure be altered to an homogeneous amorphous condition by heating 
it to a temperature somewhat higher than 6, and the condition be fixed by rapid 
cooling to a temperature lower than b , the piece should then be allowed to finish 
cooling gradually, so as to prevent, as far as practicable, internal strains due to 
sudden and unequal contraction. 

Alloys of steel with Silver , Platinum , Rhodium , and Aluminum have been 
made with a view to imitating Damascus steel, Wootz, etc., and improving 
fabrication of some finer kinds of surgical and other instruments. 

Properties of Steel. —After being tempered it is not easily broken; it welds 
readily; does not crack or split; bears a very high heat, and preserves the 
capability of hardening after repeated working. 

Hardening and Tempering. —Upon these operations the quality of manu¬ 
factured steel in a great measure depends. 

Hardening is effected by heating steel to a cherry-red, or until scales of 
oxide are loosened on surface, and plunging it into a cooling liquid; degree 
of hardness depends upon heat and rapidity of cooling. Steel is thus ren¬ 
dered so hard as to resist files, and it becomes at same time extremely 
brittle. Degree of heat, and temperature and nature of cooling medium, 
must be chosen with reference to quality of steel and purpose for which it 
is intended. Cold water gives a greater hardness than oils or like sub¬ 
stances, sand, wet-iron scales, or cinders, but an inferior degree of hardness 
to that given by acids. Oil, tallow, etc., prevent cracks caused by too rapid 
cooling.^ Lower the heat at which steel becomes hard, the better. 

Tempering. —Steel in its hardest state being too brittle for most purposes, 
the requisite strength and elasticity are obtained by tempering—or “ letting 
down the temper ’—which is performed by heating hardened steel to a certain 
degree and cooling it quickly. Requisite heat is usually ascertained by color 
which surface of the steel assumes from film of oxide thus formed. Degrees 
of heat to which these several colors correspond are as follows : 

At 430 0 , very faint yellow.. (Suitable for hard instruments; as hammer - faces, 
At 450 0 ’ pale straw color_ ( drills, lancets, razors, etc. 

At 470 0 , full yellow.(For instruments requiring hard edges without elastici- 

At 490°,’ brown color. { ty; as shears, scissors, turning tools, penknives, etc. 

At 510 0 , brown, with purple ^p or ^ 00 j g f or cu tting w r ood and soft metals; such as 

s P ots .,•. ) plane-irons, saws, knives, etc. 

At 538°, purple.( y ’ 

At 550 0 , dark blue. (For tools requiring strong edges without extreme 

At 560°, full blue. ( hardness; as cold-chisels, axes, cutlery, etc. 

At 600 0 ’ grayish blue, verg- ( For spring-temper, which will bend before breaking; 

ing on black. ( as saws, sword-blades, etc. 

If steel is heated to a higher temperature than this, effect of the hardening 
process is destroyed. 

A high breaking strain may not be conclusive as to quality, as it may be 
due to a hard, elastic metal, or a low one may be due to great softness. 









644 


METALS.-TIN.-ZINC.-MODELS. 


Case-hardening. 

This operation consists in converting surface of wrought iron into steel, 
by cementation, for purpose of adapting it to receive a polish or to bear fric¬ 
tion, etc.; it is effected by heating iron to a cherry-red, in a close vessel, in 
contact with carbonaceous materials, and then plunging it into cold water. 
Bones, leather, hoofs, and horns of animals are generally used for this pur¬ 
pose, after having been burned or roasted so that they can be pulverized. 
Soot is also frequently used. 

The operation reduces strength of the iron. 

TIN. 

Tin is more readily fused than any other metal, and oxidizes very slowly. 

Its purity is tested by its extreme brittleness at high temperature. 

Tinplate is iron plate coated with tin. 

Block Tin is tin plate with an additional coating of tin. 

ZINC. 

Zinc, if pure, is malleable at 220° ; at higher temperatures, such as 400°, 
it becomes brittle. It is readily acted upon by moist air, and when a film 
of oxide is formed, it protects the surface from further action. When, how¬ 
ever, the air is acid, as from the sea or large towns, it is readily oxidized to 
destruction. 

Iron, Copper, Lead, and Soot are very destructive of it, in consequence of 
the voltaic action generated, and it should not be in contact with calcareous 
water or acid woods. 

The best quality, as that known as “ Vielle Montagne,” is composed of zinc 
.995, iron .004, and lead .001. Its expansion and contraction by differences 
of temperature is in excess of that of any other metal. 


STRENGTH OF MODELS. 

The forces to which Models are subjected are, 

1. To draw them asunder by tensile stress. 2. To break them by trans¬ 
verse stress. 3. To crush them by compression. 

The stress upon side of a model is to corresponding side of a structure as 
cube of its corresponding magnitude. Thus, if a structure is six times greater 
than its model, the stress upon it is as 6 3 to 1 =216 to 1: but resistance of 
rupture increases only as squares of the corresponding magnitudes, or as 
6 2 to 1 =36 to 1. A structure, therefore, will bear as much less resistance 
than its model as its side is greater. 

To Compute Dimensions of a Beam, etc., 'vvh.icli a 
Structure can Dear. 

Rule. —Divide greatest weight which the beam, etc. (including its weight), 
in the model can bear, by the greatest weight which the structure is required 
to bear (including its weight), and quotient, multiplied by length of beam, 
etc., in model, will give length of beam, etc., in structure. 

Example.— A beam in a model 7 inches in length is capable of bearing a weight 
of 26 lbs., but it is required to sustain only a weight or stress of 4 lbs.; what is the 
greatest length that a corresponding beam can be made in the structure? 

26 - 4 - 4 = 6.5, and 6.5 X 7 = 45- 5 bis. 



MODELS.-MOTION OF BODIES IN FLUIDS. 


645 


Resistance in a model to crushing increases directly as its dimensions; 
but as stress increases as cubes of dimensions, a model is stronger than the 
structure, inversely as the squares of their comparative magnitudes. 

Hence, greatest magnitude of a structure is ascertained by taking square 
root of quotient, as obtained by preceding rule, instead of quotient itself. 

Example.—I f greatest weight which a column in a model can sustain is 26 lbs., 
and it is required to bear only 4 lbs.; height of column being 18 ins., what should 
be height of it in structure? 



= V 6 -5 = 


55, and 2.55 X 18 = 45.9 ins., height of column in structure. 


If, when length or height and breadth are retained, and it is required to 
give to the beam, etc., such a thickness or depth that it will not break in con¬ 
sequence of its increased dimensions, 


Th “V(f) 

ness required. 


= 3/6.5 = 2.55, which, X square of relative size of model =t= thick- 


To Compute Resistance of a Bridge from a NIodel. 
n 2 W — J^— (n — 1 ) iwj == load bridge will bear in its centre. 

Example.— If length of the platform of a model between centres of its repose 
upon the piers is 12 feet, its weight 30 lbs., and the weight it will just sustain at its 
centre 350 lbs., the comparative magnitudes of model and bridge as 20, and actual 
length of bridge 240 feet; what weight will bridge sustain ? 

= 140 000 — 3800 X 30 — 26 000 lbs. 


20 2 X 35 ° ■ 


4 °° 


X (20 — 1) X 3° 


MOTION OF BODIES IN FLUIDS. 

If a body move through a fluid at rest, or fluid move against body at 
rest, resistance of fluid against body is as square of velocity and density 
of fluid ; that is, R = cl v 2 . For resistance is as quantity of matter or 
particles struck, and velocity with which they are struck. But quan¬ 
tity or number of particles struck in any time are as velocity and density 
of fluid; therefore, resistance of a fluid is as density and square of 
velocity. 

oj2 ft -y 2 

— = h and -= R. h representing height due to velocity , d density of fluid, 

2 g 2 g 

and R resistance or motive force. 

Resistance to a plane is as plane is greater or less, and therefore resistance 
to a plane is as its area, density of medium, and square of velocity; that is, 
R = a d v 2 . 

Motion' is not perpendicular, but oblique, to plane or to face of body in any 
angle, sine of which is s to radius 1; then resistance to plane, or force of 
fluid against plane, in direction of motion, will be diminished in triplicate 
ratio of radius to sine of angle of inclination, or in ratio of 1 to s 3 . 

Hence, a ^- V — — = R, and a ^ V — S — = F. w representing weight of body , and F 
’ 2 g 2 gw 

retarding force. 

Progression of a solid floating body, as a boat in a channel of still water, 
gives rise to a displacement of water surface, which advances with an un¬ 
dulation in direction of body, and this undulation is termed Wave of Dis¬ 
placement. 






MOTION OF BODIES IN FLUIDS. 


646 

Resistance of a fluid to progression of a floating body increases as velocity 
of body attains velocity of wave of displacement, and it is greatest when the 
two velocities are equal. 

In the motion of elastic fluids, it appears from experiments that oblique 
action produces nearly same effect as in motion of water, in the passage of 
curvatures, apertures, etc. 


Resistance to an Area of One Sq. Foot moving tlirongli 
Water, or Contrariwise. 


Angle of 
Surface 
with 
Plane of 
Current. 

Presswi 

lo 

120 

e per Sq. 
'ities per F 

240 

hoi for folio 
oot per Min 

480 

winq Ve¬ 
nt e. 

900 

Angle of 
Surface 
with 
Plane of 
Current. 

Pressur 

loc 

120 

per Sq. F 
ities per Ft 

240 

oot for folk 
<ot per Mir 

480 

ming Ve¬ 
nts. 

goo 

O 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

O 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

6 

.09 

•359 

1-435 

5.046 

45 

2.66 

10.639 

42-557 

149.614 

8 

•133 

•53 

2.122 

7-459 

5 o 

2-995 

11.981 

47-923 

168.48 

9 

.156 

.624 

2.496 

8 -775 

55 

3- 2 49 

12.995 

51-979 

182.739 

10 

.179 

.718 

2.87 

IO. O9I 

60 

3-455 

13.822 

55.286 

194.366 

x 5 

•355 

1.42 

5.678 

19.963 

65 

3.607 

14-43 

57-72 

202.922 

20 

.608 

2-434 

9-734 

34.222 

70 

3.728 

14.914 

59- 6 54 

209. 722 

25 

•94 

3 - 7 6 

15.038 

52.869 

75 

3.81 

15.241 

60.965 

214.329 

30 

i -353 

5 - 4 I 3 

21.653 

76.123 

80 

3- 8 57 

15.428 

61.714 

216.926 

35 

i- 79 8 

7.192 

28.766 

101.132 

85 

3.892 

15-569 

62.275 

218.936 

40 

2.258 

9.032 

36.13 

127.018 

90 

3-9 

15.6 

62.4 

219-375 


Resistance to a plane, from a fluid acting in a direction perpendicular to 
its face, is equal to weight of a column of fluid, base of which is plane and 
altitude equal to that which is due to velocity of the motion, or through 
which a heavy body must fall to acquire that velocity. 

Resistance to a plane running through a fluid is same as force of fluid in 
motion with same velocity on plane at rest. But force of fluid in motion is 
equal to weight or pressure which generates that motion, and this is equal to 
weight or pressure of a column of fluid, base of which is area of the plane, 
and its altitude that which is due to velocity. 

Illustration.— If a plane 1 foot square be moved through water at rate of 32.166 
n 2 . l66^ 

feet per second, then -— X6.083, space a body would require to fall to acquire 

64-333 

a velocity of 32.166 feet per second; therefore 1 x 62.5 (weight of a cube foot of 

32. l66 ^ 

water) x v— -• = 100=; lbs. = resistance of plane. 

64-333 


Resistance of different Riguires at different Velocities in 

Air. 


Veloci¬ 
ty per 
Second. 

Co 

Vertex. 

ne. 

Base. 

Sphere. 

Cylin¬ 

der. 

Hemi- 
sph ere. 
Round. 

Veloci¬ 
ty per 
Second. 

Co 

Vertex. 

ne. 

Base. 

Sphere. 

Cylin¬ 

der. 

Hemi¬ 

sphere. 

Round. 

Feet. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Feet. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

3 

.028 

.064 

.027 

•05 

.02 

12 

•376 

•85 

•37 

.826 

•347 

4 

.048 

. 109 

.047 

.09 

•039 

14 

.512 

1.166 

•505 

i-i 45 

.478 

5 

.071 

. 162 

.068 

•M 3 

.063 

15 

•589 

1.346 

.581 

x - 3 2 7 

•552 

8 

.168 

.382 

. 162 

•36 

.16 

16 

•673 

1.546 

.663 

1.526 

•634 

9 

.211 

•478 

.205 

•456 

.199 

18 

.858 

2.002 

.848 

1.986 

.818 

IO 

.26 

•587 

•255 

•56S 

.242 

20 

1.069 

2-54 

1-057 

2.528 

1-033 


Diameter of all the figures was 6.375 ins., and altitude of the cone 6.625 ins. 
Angle of side of cone and its axis is, consequently, 25 0 42' nearly. 


From the above, several practical inferences may be drawn. 

1. That resistance is nearly as surface, increasing but a very little above 
that proportion in greater surfaces. 












































MOTION OF BODIES IN FLUIDS. 


647 


2. Resistance to same surface is nearly as square of velocity, but gradu¬ 
ally increasing more and more above that proportion as velocity increases. 

3. When after parts of bodies are of different forms, resistances are differ¬ 
ent, though fore parts be alike. 

4. The resistance on base* of a cone is to that on vertex nearly as 2.3 to 
1. And in same ratio is radius to sine of angle of inclination of side of cone 
to its path or axis. So that, in this instance, resistance is directly as sine 
of angle of incidence, transverse section being same, instead of square of sine. 

Resistance on base of a hemisphere is to that on convex side nearly as 
2.4 to 1, instead of 2 to 1, as theory assigns the proportion. 


Spliere.— Resistance to a sphere moving through a fluid is but half re¬ 
sistance to its great circle, or to end of a cylinder of same diameter, moving 
with an equal velocity, being half of that of a cylinder of same diameter. 


V 


2 g X - d X — —- = V. d representing diameter of sphere, and N and n spe- 

3 n 


cific gravities of sphere and resisting fluid. 


JN 4 

X ~ d — S. S representing space through which a sphere passes while acquir- 
n 3 

ing its maximum velocity , in falling through a resisting fluid. 

Illustration.—I f a ball of lead 1 inch in diameter, specific gravity 11.33, be set 
free in water, specific gravity 1, what is greatest velocity it will attain in descend¬ 
ing, and what space will it describe in attaining this velocity? 


£ = 32.166, 


d — — foot, 
12 


N = 11.33, and n = i. 


Then . / 2 X 32.166 X — of — X 

V 3 12 ] 


: 3/7. 148 X 10.33 = 8. S93 feet per sec. 


Hence, 


X — of — = 1.259 feet. 
3 12 


3 nv = f= retardive force = ——. 
8 g N d 2 g s 


n a v 2 , n a v 2 * „ , 

Cylinder. —-= R. and — ———f. a representing area or p r 2 , and 


2 9 


2 g w 


w weight of body. 

Illustration. — Assume a = 32 sq. feet, v = 10 feet per second, and n = .0012. 
.0012 X 32 X io 5 


Then 


6 4-33 


: .06 of a cube foot of water = .06 of 62.5 = 3.75/65. 
: R, also 


nav 2 s* n , npd 2 v 2 s 2 _ . n p d 2 v 2 s r ‘ 

Conical Surface. —— — = R, a ls° - g - ^ —R> an< l g - ^ w 


—f s representing sine of inclination , and a con vex surface of cone. 
Curved End as a Spliere or Idemisplierical End. 


p n v 2 d 2 
16 g 


— R, and Circle .5 of spherical end. 

In general, when n is to water as a standard, result is in cube feet of water, if 
a is in sq. feet; and in cube ins. of water, if a is in sq. ins., v in ins., and g in ins. 

If n i s given in lbs. in a cube foot, a is in sq. feet, v and g are in feet, result is in lbs. 


To Compute Altitn.de of a Column of Air, Pressure of 
wliicli slaall be equal to Resistance of a Body moving 
tlirongL. it, "vvitli any "V" elocity. 

— X — = x = altitude in feet, ax — volume of column in feet , and — a x = weight 
6 a ... 

in ounces, a representing area of section of body , similar to any in table , perpen¬ 
dicular to direction of motion, r resistance to velocity in table , and x altitude sought 
of a colum n of air, base of which is a, and pressure r. 

* This is a refutation of the popular assertion that a taper spar can be towed in water easiest when 
the base is foremost. 















648 


MOTION OF BODIES IN FLUIDS. 


2 IS 

When a = — of a foot, as in all figures in table, x becomes — r when r = re- 
9 4 

sistance in table to similar body. 

Illustration. —Assume convex face of hemisphere resistance = .634 oz. at a ve¬ 
locity of 16 feet per second. 

Then r =. 634, and x = — r — 2.3775 feet — altitude of column of air, pressure of 
4 

which — resistance to a spherical surface at a velocity of 16 feet. 

To Compute when Pressure of Air in rear of a IProjectile 
is Inferior to Pressure due to its 'Velocity. 

Assume height of barometer ==2.5 feet, and weight of atmosphere = 14.7 lbs. 

Weight of cube inch of mercury = = .49 lbs., and weight of cube inch of air 

3 ° 

= .00004357 lbs.; hence, .49 - 4 - .000043 57 = n 246, which X 2.5 feet = 28115 feet. 


Then V16.08 : V 28 I1C 5 3 2 -i6 : x, andx: 


32.16 X V 28 ri 5 
V16 : 


1341.6 feet. 


To Compute 'Velocity with, which a Plane Surface must 
he projected, to generate a Resistance just equal to 
Pressure of Atmosphere upon it. 

By table, resistance on a circle with an area of .222 sq. foot (2 = 9) = .051 oz., at a 
velocit}^ of 3 feet per second. Hence f 2 : i 2 ;; .051 : .0056 oz. at a velocity of 1 foot, 

and 1X 144X 14-7 X 16X 2=9= 7526.4 oz. Hence, -^.0056 : -f 7526.4 ;; 1 : 1160 feet. 

To Compiite "Velocity lost by a Rrojectil^. 

If a body is projected with any velocity in a medium of same density with itself, 
and it describes a space == 3 of its diameters, 

Then x = 3 d, and b — ° n — 3 


Hence, b x ■ 


J-, and 


b x — 1 


8 N d 8 d 


2.08 

-j— ~ = —g = velocity lost nearly . 66 of projectile velocity. 


c = base of Nap. system of log.; hence = number corresponding to Nap. log. 
bx. Hence, if b x X -4343j result == com. log. of cbx. 

b x =| = 1-125, which x -4343 = .488 587 5, and number to this com. log. = 3.0803. 


Hence, velocity lost = 


3.0803 ■ 


2.08 
3-08' 


3.0803 

Illustration. —If an iron ball 2 ins. diam. were projected with a velocity of 1200 
feet per second, what would be velocity lost after moving through 500 feet of space? 


d = 


-t , x = 500, N = 7 A, and n = .0012. 

D o 


„ T 3 nx 3X12X500X3X6 81 , 1200 „ , 

Hence, b x = g ^ " « v OT v - == ~ > and v = nru = 99 8 M V*r 


8 X 21 X 10000 


440 


„ 81 
"4 4 A 


second, having lost 202 feet, or nearly ~ of its initial velocity. 

12 3 6 22 1 . 

-- — .0012, — ana-and — inverted, because N and n are in denominator. 

10 000 22 • 3 6 

To Compute Time and Velocity. 


1 f 1 J \ v 3 n , j a 

T (-) — tvne, = b, and 

b \v a J ’ Q m w > 


. v. 


8 N d ’ cbx' 

Illustration. —If an iron ball 2 ins. in diameter were projected in air with a ve¬ 
locity of 1200 feet per second, in what time would it pass over 1500 feet, and what 
its velocity at end of that time? 


3 X 12 X 3 X 6 


, and 


- — , and bx — - 5 -°° ; hence 4 = ; — = 

8 X 22 X 10000 2716 2716 b 1 a 

1 cbx x.7372 1 . , . . / 1 i\ 

~r — —- —-= 7— nearly. :.v — 6go and t— 2716 X (7 - 1 =1.67 sec. 

b a 1200 690 7 ^690 1200/ ' 
















NAVAL ARCHITECTURE. 


649 


NATAL ARCHITECTURE. 

Hes-ults of Experiments upon Form of Vessels. 

( Wm. Bland.) 

Cubical Models. Head Resistance. —Increases directly with area 
of its surface. Weight Resistance. —Increases directly as weight. 

Vessels’ Models. Lateral Resistance. —About one twelfth of 
length of body immersed, varying with speed. 

Order of Superiority of Amidship Section. —Rectangle, Semicircular, 
Ellipse, and Triangle. 

Centre of lateral resistance moves forward as model progresses. 
Centre of gravity has no influence upon centre of lateral resistance. 


Relative Speeds. 

Length. —Increased length gives increased speed or less resistance. 

Depth of Flotation. —Less depth of immersion of a vessel, less the resistance. 
Amidship Section. —Curved sections give higher speed than angled. 

Sides. —Slight horizontal curves present less resistance than right lines. 
Curved sides with one fourth more beam give equal speeds with straight 
sides of less beam. Keel. —Length of keel has greater effect than depth. 
Stern. —Parallel-sided after bodies give greater speed than taper-sided. 


Form of Bow. 

Isosceles triangle , sides slightly convex. 

“ “ “ right lines. 

“ “ “ slightly concave at entrance and running) 

out convex.) 


Order of Speed. 


Spherical equilateral triangle compared to Equilateral triangle , speed is 
as ix to 12. Equilateral triangle, with its isosceles sides bevelled off at an 
angle of 45 0 , compared to bow with vertical sides, is as 5 to 4. 

When bow has an angle of 14° with plane of keel, compared with one of 
7 0 , its speed is greater. 


Bodies Inclined Upwards from Amidship Section. 

1. Model with bow inclined from £ 3 , has less resistance than model with¬ 
out any inclination. 

2. Model with stern inclined from £ 3 , has less resistance than model with¬ 
out any inclination. 

Model 1 had less resistance than model 2. Model with both bow and 
stern inclined from S3, has less resistance than either 1 or 2. 

Stability. 

Results of Experiments upon Stability of Rectangular 
Blocks of Wood, of TJniform Length and Depth, but 
of Different Breadths. (Wm. Bland.) 

Length 15, Depth 2, and Depression 1 inch. 


Width. 

Weight. 

As Observed. 

Ratio 0 
With like 
Weights. 

' Stability. 

By Squares of 
Breadth. 

By Cubes of 
Breadth. 

Ins. 

Oz. 





3 

24 

I 

I 

I 

I 

4-5 

35 

2.5 

2.4 

• 2.25 

3-375 

6 

45 

7 

3-7 

4 

8 

7 

55 

II 

4.8 

6.25 

15-625 

















650 


NAVAL ARCHITECTURE, 


Hence it appears that rectangular and homogeneous bodies of a uniform 
length, depth, weight, and immersion in a fluid, but of different breadths, have 
stability for uniform depressions at their sides (heeling) nearly as squares 
of their breadth; and that, when weights are directly as their breadths, 
their stability under like circumstances is nearly as cubes of their breadth. 

With equal lengths, ratio of stability is at its limit of rapid increase when 
width is one third of length, being nearly in cube ratio; afterwards it ap¬ 
proaches to arithmetic ratio. 

Results of Experiments upon Stability' and. Speed of 
NEodels having Amidship Sections of different Forms, 
font Uniform Length, Breadth, and Weights. (TV. Bland.) 

Immersion different , depending upon Form of Section. 


Form of Immersed Section. 

Stability. 

Speed. 

Half-depth triangle, other half rectangle. 

12 

4 

Rectangle. 

14 


Right-angled triangle*. 

7 

J 

3 

Semicircle. 

9 

2 


* Draught of water or immersion double that of rectangle. 


Statical Stability is moment of force which a body in flotation exerts to 
attain its normal position or that of equilibrium, it having been deflected 
from it, and it is equal to product of weight of fluid displaced and horizontal 
distances between the two centres of gravity of body and of displacement, or 
it is product of weight of displacement, height of Meta-centre , and Sine of 
angle of inclination. 

Dynamical Stability is amount of mechanical work necessary to deflect a 
body in flotation from its normal position or that of equilibrium, and it is 
equal to product of sum of vertical distances through which centre of grav¬ 
ity of body ascends and centre of buoyancy descends, in moving from ver¬ 
tical to inclined position by weight of body or displacement. 

To Determine Measure of Stability of Hull of a "Vessel 
or Floating Body.—Fig. 1 . 

Measure of stability of a floating body depends essentially upon horizontal dis¬ 
tance, G s, of meta-centre of body from centre 
of gravity of body; and it is product of force 
of the water, or resistance to displacement of 
it, acting upward, and distance of G s, or P x 
G s. If distance, c M, represented by r , and 
angle of rolling, c M r, by M°, measure of sta¬ 
bility, or S is determined by Pr, sin. M° = .S; 
and this is therefore greater, the greater the 
weight of body, the greater distance of meta¬ 
centre from centre of gravity of body, and the 
greater the angle of inclination of this or of 
c M r. 

Assume figure to represent transverse section of hull of a vessel, G centre of 
gravity of hull, w l water-line, and c centre of buoyancy or of displacement of im¬ 
mersed hull in position of equilibrium. Conceive vessel to be heeled or inclined 
over, so that ef becomes water-line, and s centre of buoyancy; produce s M, and 
point M is meta-centre of hull of vessel. 

Transverse meta-centre depends upon position of centre of buoyancy, for it is that point where a 
vertical line drawn from centre intersects a line passing through centre of gravity of hull of vessel 
perpendicular to plane of keel. 

Point of meta-centre may be the same, or it may differ slightly for different angles of heeling. Angle 
of direction adopted to ascertain position of meta-centre should be greatest which, under ordinary cir¬ 
cumstances, is of probable occurrence; in different vessels this angle ranges from 20° to 60°. 

If meta-centre is above centre of gravity, equilibrium is Stable ; if it coincides with it, equilibrium is 
Indifferent; and if it is below it, equilibrium is Unstable. 














NAVAL ARCHITECTURE. 


651 


Comparative Stability of different hulls of vessels is proportionate to the distance 
of G M for same angles of heeling, or of distance G s. Oscillations of hull of a ves¬ 
sel may be resolved into a rolling about its longitudinal axis, pitching about its 
transverse axis, and vertical pitching, consisting in rising and sinking below and 
above position of equilibrium. 

If transverse section of hull of a vessel is such that, when vessel heels, level of 
centre of gravity is not altered, then its rolling will be about a permanent longi¬ 
tudinal axTs traversing its centre of gravity, and it will not be accompanied by any 
vertical oscillations or pitchings, and moment of its inertia will be constant while 
it rolls. But if, when hull heels, level of its centre of gravity is altered, then axis 
about which it rolls becomes an instantaneous one, and moment of its inertia will 
vary as it rolls; and rolling must then necessarily be accompanied by vertical os¬ 
cillations. 


Such oscillations tend to strain a vessel and her spars, and it is desirable, therefore, 
that transverse section of hull should be such that centre of its gravity should not 
alter as it rolls, a condition which is always secured if all water-lines, as w l and ef 
are tangents to a common sphere described about G; or, in other words, if point ot 
their intersections, 0, with vertical plane of keel, is always equidistant from centre 
of gravity of hull. 

To Compute Statical Stability. 

D c M sin. M = S. D representing displacement , M angle of inclination , and S 
stability. 

Illustration i.— Assume a ship weighing 6000 tons is heeled to an angle of 9 0 , 
distance c M = 3 feet, 

Sin. 9 0 = .1564. Then 6000 X 3 X -1564 = 2815.2 foot-tons. 

2.—Weight of a floating body is 5515 lbs., distance between its centre of gravity 
and meta-centre is 11.32 feet, and angle M r= 20 0 . 

Sin. M = .342 02. Hence 5515 X 11.32 x -342 02 = 21 352.24 foot-lbs. 


Statical Surface Stability. 

Moment of Statical surface stability at any angle is c z D. Assuming 
centre of gravity of vessel coincided with c; coefficient of a vessel’s stability 
at any angle of heel is expressed when the displacement is multiplied by 
vertical height of the meta-centre for given angle of heel above centre of 
gravity, or D c M. 

Approximately. Rule. —Divide moment of inertia of plane of flotation 
for upright position, relatively to middle line by volume of displacement; 
and quotient multiplied by sine of angle of heel will give result. 


Per Foot of Length of Vessel t — (B 3 sin. M). 


B representing half breadth. 


Dynamical Surface Stability. 

Moment of Dynamical surface stability is expressed by product of weight 
of vessel or displacement and depression of centre of buoyancy during the 
inclination, that is, for angle M. 

To Compute Dynamical Stability of a 'Vessel. 


Approximately. Rule. —Multiply displacement by height of meta-centre 
above centre of gravity, and product by versed sine of angle of heel. 

Or multiply statical stability for given angle by tangent of .5 angle of heel. 


To Compute Elements of Stability of a Floating Body. 

A ' Q (* 

a = s, ——— = r, — —— = q. and sin. Mr — c. A representing area of 

A sin. M sin. M 

immersed section; A' section immersed by careening of body, as fo l; s horizontal 
distance , c r, between centres of buoyancy ; a horizontal distance between centres of 
gravity , i i, of areas immersed and emerged by careening; g distance , c M, between 
centre of buoyancy or of water displaced and meta-centre ; r distance , G M, between 
centre of gravity and meta-centre ; c horizontal distance, G s, between centre of grav¬ 
ity and of line of displacement of it ivhen careened; e vertical distance between centres 
of gravity and buoyancy , all in feet; and M angle of careening. 




NAVAL ARCHITECTURE. 


652 


Note.—W hen centre of gravity, G, is below that of displacement, c, then e is -f; 
when it is above c it is —; and when it coincides with c it is o; or e is — when 

<s; and a body will roll over when e sin. M = or f>s. 

Assumed elements of figure illustrated are A = 86, A' = 21.5, fi = 21.5, and e — .5. 
The deduced arc s = 3.7, 0 = 3.87, <7 = 10.82, a= 14.9, and r==. 11.32. b repre¬ 
senting breadth at water-tine or beam in feet, and P weight or displacement in lbs. 
or tons. 

3-87 


Then « = X H -9 = 3-7 f eet > 


.34202 


11.32 feet, e — r—g, g 


3 -7 

.34202 


= xo. 82 feet, c = . 342 02 X 11.32 = 3.87 feet. 

Of Hull of a Vessel. ( - ^ ,, , ± <A P, sin. M = S; d cos. .5 M = 

\10.7 to 13* A / 


b 3 


10.7 to 13 (11.93) A 


■ff, 


1 

sin. xVl 


p-)- 


± e; 


b a 


-|- e sin 


fin. M j = S; 


and 


P (s zh e sin. M) = S. d representing depth of centre of gravity of displacement un¬ 
der water in equilibrium, and d' depth when out of equilibrium, both in feet. 

Illustration i. —Displacement of a vessel is 10000000 lbs.; breadth of beam, 50 
feet; area of immersed section, 800 sq. feet; vertical distance from centre of grav¬ 
ity of hull up to centre of buoyancy or displacement, 1.9 feet, and horizontal dis¬ 
tance a between centres of gravity of areas immersed and emerged, when careened 
to an angle of 9 0 10' = 33.4 feet, immersed area being 50 sq. feet. 


Sin. 9 0 iq / = .1593 


Thensxrr-^- X 33-4 : 

800 * 


2-39 


15 feet. g 


5 ° J 


.1593 - ' 11.93X800 

10000000 X -1593 = 23905 396 lbs. , and e 


2.0875 feet, 800X2.0875 == 50X33-4? 

13.1 feet, S = ( - 5 ° + i-9 X 

\ii .93 X 800/ 


— f 

•1593 \ 


23 9°5 39 6 


IO OOO OOO 


2.0875^ = 1.9 feet. 


2.—Assume a ship having a displacement of 5000 tons, and a height of meta-centre 
of 3.25 feet, to be careened to 6° 12'. What is her statical stability? 

Sin. 6° 12' = . 1079. Then 5000 X 3.25 X -1079 — 1753-37 foot-tons. 


3. —Assume a weight, W, of 50 tons to be placed upon her spar deck, having a 
common centre of gravity of 15 feet above her load-line, 

Then 5000 X 3-25 — 5° + 15 X .1079 = 1745.29 foot-tons. 

4. —Assume 100 tons of w r ater ballast to be admitted to her tanks at a common 
centre of gravity of 15 feet below her load-line, 

Then 5000 X 3-25 + 100 X 15 X -1079 = 19x5.22 foot-tons. 

5. —Assume her masts, weighing 6 tons, to be cut down 20 feet, 

10 X 20 2 / 2 \ 

Then - = — foot = fall of centre of gravity, and 5000 X (3.25 -J-) X • 1079 

50OO 50 \ 50/ 

= 1774.95 tons. 

To Compute Elements of Power, etc., req/uirecl to 
Careen a Body or 'V'essel. 


- - b 3 IP 

Sin. M (h — n sin. M) 4 - n sec. M— s = l. - -- 3/ -- 

10.7 to 13* A 'V 64.125 LA 

W l r — P c, and W l = S. W representing weight or power exerted and l distance 
at which weight or power acts to careen body, taken from centre of gravity of displace¬ 
ment perpendicular to careening force, h vertical height from centre of gravity of dis¬ 
placement to centre of weight or power to careen 'body when it is in equilibrium , 
n horizontal distance from centre of vessel to centre of weight or power, L length of 
vessel, m meta-centre, and S as in preceding case, all in feet. 


* Unit for section of a parallelogram is 10.7 ; of a semicircle 12, and of a triangle 12.8. 





















NAYAL ARCHITECTURE. 


653 

Illustration.—A weight is placed upon deck of a vessel at a mean height of 3.87 
feet from centre line of hull; height at which it is placed is 11.32, and other ele¬ 
ments as in first case given. 

Sec. 2o° = .342. Then h — 11.32, 11 = 3.87, and Z = . 34 2 (11.3 — 3.87 x -342) + 
3.87 X 1.0642 - 3.7 = .342 x 104-4.12 —3.7=;3.84/eeL 

Assume W = 5515. Then 5515 x 3-84 = 21187.6 foot-lbs. 

Or P (w cos. M 4 -h sin. M) = S. w representing distance of weight from centre of 
vessel, and k height of w above water-line, both in feet. 

Illustration.—I f a weight of 30 tons placed at 20 feet from centre of hull or 
deck, 10 feet above water-line, careens it to an angle of 2 0 9', what is its stability? 
cos. 2 0 9' = .9993; sin. 2 0 9 ' = .0375. 

30 (20 X .99934-10 X -0375) = 30 X 20.361 =610.83 foot-tons. 

Bottom and Iixmaersed Surface of Hull ofYessels. 

To Compute Bottom and Side Surface of ITvill. 

Bottom and Side. Rule. —Multiply length of curve of amidship section, 
taken from top of tonnage or main deck beams upon one side to same point 
upon other (omitting width of keel), by mean of lengths of keel and be¬ 
tween perpendiculars in feet, multiply product by .85 or .9 (according to the 
capacity of vessel), and product will give surface required in sq. feet. 

Example.—L engths of a steamer are as follows: keel 201 feet, and between per¬ 
pendiculars 210 feet, curved surface of amidship section 76 feet; what is surface? 

Coefficient .87. 2104-201 = 2 = 205.5, and 76 X 205.5 X .87 = 13 587 sq. feet. 

Note.—E xact surface as measured was 13650 sq. feet. 

Bottom Surface. Rule. —Multiply length of hull at load-line by its 
breadth, and this product by depth of immersion (omitting the depth of 
keel) in feet; and this product multiplied by from .07 to .08 (according to 
capacity of vessel) will give surface required in sq. feet. 

Example.—L ength upon load-line of a vessel is 310 feet, beam 40 feet, depth of 
keel 1 foot, and draught of water 20 feet; what is bottom or wet surface? 

Coefficient assumed .073. 310 X 40 X 20 — 1 X -073 = 17 199 sq. feet. 

To Compute Resistance to Wet Surface of Hnll. 

C a v 2 m R. C representing a coefficient of resistance, a area of wet surface in sq. 
feet, and v velocity of hull in feet per second. 

ViIup? nf f f- 00 7 > cl ean copper. 1 .014, iron plate. 

Ydiuwui v, | oj smooth paint. | .019, iron plate, moderately foul. 

Power required to propel one sq. foot of immersed amidship section at £§ is .073 
that of smooth wet surface. 

To Coxnpnte Elerneruts of a YLessel. 

Displacement and. its Centre of Gfravity. 

Displacement of a vessel is volume of her body below water-line. 

Centre of Gravity , or Centre of Buoyancy of Displacement , is centre of 
gravity of water displaced by hull of vessel. 

For Displacement. Rule. —Divide vessel, on half breadth plan, into a 
number of equidistant sections, as one, two, or more frames, commencing 
at £5 and running each side of it. Add together lengths of these lines in 
both fore and aft bodies, except first and last, by Simpson’s rule for areas 
(see page 344) ; multiply sum of products by one third distance between 
sections, and product will*give area of water-line between fore and aft sections. 

Then compute areas contained in sections forward and aft of sections taken, in¬ 
cluding stern and rudder-post, rudder and stem, and add sum to area of body-sec¬ 
tions already ascertained. * 


To Compute Area of a Water-line, see Mensuration of Surfaces, page 344. 

3 I* 








NAVAL ARCHITECTURE 


654 


Compute area of remaining water-lines in like manner. Tabulate results, and 
multiply them by Simpson’s rule in like manner as for a water-line, and again by 
consecutive number of water-lines, and sum of products between water-line and 
product will give volume between load and lower water-line. 

Add area of lower water-line to area of upper surface of keel; multiply half sum 
by distance between them, and product will give volume; then compute areas con¬ 
tained in sections forward and aft of sections taken as before directed. 


If keel is not parallel to lower water-line, take average of distance between them. 

Compute volume of keel, rudder-post and rudder below water-line; add to volume 
already ascertained; multiply product by two, for full breadth, and product will 
give volume required in cube feet, all dimensions being taken in feet. 



Example. -Assume 
a vessel 100 feet in 
length by 20 feet in 
extreme breadth, on 
load-line of 8 feet 9 
inches immersion. 
Figs. 2 and 3. 

Distance between 
sections, for purpose 
of simplifying this 
example, is taken 
at 10 feet; usually 
frames are 18 to 30 


ins. apart, and two or more included in a section. Water-lines 2 feet apart. 


1 st Water-line. 


4 

5 



= 

5 

3 

7-7 

X 

4 

= 

30.8 

2 

9-5 

X 

2 

— 

*9 

I 

9.9 

X 

4 

=r= 

39-6 

O 

IO 

X 

2 

— 

20 

A 

9.6 

X 

4 


38-4 

B 

7.8 

X 

2 

= 

15.6 

C 

6.8 

X 

4 

== 

27. 2 

D 

4 



'== 

4 


199.6 


10-4-3 

3 = 

3 n 



665.3 

Abaft section 4, rud- 

der and post... 


25 

Forward section 

D 

and stem. 


20.7 



711 

4//1 Water-line. 


4 -7 

= 

•7 

3 2X4 


8 

2 4.3 X 2 

= 

8.6 

1 6.5 X 4 


26 

0 6.8 X 2 

=~ 

13.6 

A 5X4 


20 

B 3.6 X 2 


7.2 

C -9X4 

== 

3-6 

D -3 


•3 



88 

10-5-3 

= 

3 i 



293-3 

Abaft section 4, rud- 

der and post .. 
Forward section 

D 

3-2 

and stem. 


.8 


2 97-3 



2 d Water-line. 


4 

2.7 



=5 

2.7 

3 

6.9 

X 

4 


27.6 

2 

8.7 

X 

2 

r— 

17.4 

1 

9-5 

X 

4 

= 

38 

0 

9.6 

X 

2 

== 

I9.2 

A 

9 

X 

4 

= 

36 

B 

7 

X 

2 


14 

C 

5 

X 

4 

= 

20 

D 

2 




2 


176.9 

IO -r- 3 = 3J 

589-7 

Abaft section 4, rud¬ 
der and post. 13.2 

Forward section D 
and stem *.. 9.1 

612 


3 d Water-line. 


4 

i -5 




i -5 

3 

5 

X 

4 

= 

20 

2 

6.6 

X 

2 


13.2 

X 

8.7 

X 

4 

= 

34-8 

0 

8.9 

X 

2 

r== 

17.8 

A 

7.6 

X 

4 

=: 

30-4 

B 

7 

X 

2 

=: 

14 

C 

3 

X 

4 


12 

D 

1.2 




1.2 


144.9 

io-f-3 —__ 3 g_ 

483 

Abaft section 4, rud¬ 
der and post. 7 

Forward section D 
and stern. 5.4 

495-4 


Keel. 

Half breadth = .25 X length of 98 feet = 
Rudder-post and rudder... 


24-5 

_ A 

24.8 


1st water-line 711 
2d 
3 d 


4th 

Keel 


612 X 
495-4 X 
297-3 X 
24.8 


Results. 

711 

4 - -- 2448 X i “ 2448 
2 = 990.8 X 2 — 1981.6 
4 = 1189.2 X 3 = 3567.6 
24.8 X 4 = 99.2 


5363-8 

2 


8096.4 


3) 10727.6 

Displacement , 3575.9 X 2 = 715!.8 cube ft. 

































































NAVAL ARCHITECTURE. 


655 


To Compute Centre of GS-ravity of Displacement. 

Rule. —Divide sum of products obtained as above, by consecutive water- 
lines, by sum of products obtained in column of products by Simpson’s mul¬ 
tipliers, and quotient, multiplied by distance between water-lines, will give 
depth of centre below load water-line. 

Illustration i. 8096.4, from above, 4 - 5363.8 = 1.5, which x 2 = 3 feet. 
n 

Or,-—— = d. n representing draught of water exclusive of any drag of 

2 ( 2 a «) 

keel , a area of immersed surface of hull in sq. feet, and D displacement in cube feet. 

2.—Assume draught of water 8 feet, displacement 7152 cube feet, and area of im¬ 
mersed surface of hull noo sq. feet. 

8 8 r + 

= 3'37 feet. 


Then 


7152 


2X1-187 


noo X 


To Compute Displacement Approximately. 

Coefficient of Displacement of a vessel is ratio that volume of displacement 
bears to parallelopipedon circumscribing immersed body. 

V 

k ^ p; C. V representing volume of displacement in cube feet, L length at im¬ 
mersed water-line, B extreme breadth, and D draught in depth of immersion, both 
in feet. 

Coefficient of Area of A midship Section in Plane of a Water-line is ratio 
which their areas bear to that of circumscribing rectangle. 

L representing length of water-line, and D distance between water-lines, both in feet. 

Coefficients. (By S. M. Poole, Constructor U. S. Navy.) 

Rule.— Multiply length of vessel at load-line by breadth, and product by 
depth (from load-line to under side of garboard-strake) in feet, and this 
product by coefficient for vessel as follows : divide by 35 for salt water, 36 
for fresh water, and quotient will give displacement in tons. 

Amidship sections range from .7 to .9 of their circumscribing square, and mean 
of horizontal lines from . 55 to . 75 of their respective parallelograms. Hence, ranges 
for vessels of least capacity to greatest are .7 X -55 = .385, and .9 X - 75 = .675. 

Merchant ship, very full.6 to.7 

“ . “ medium.58 to .62 

River steamer, stern-wheel... ,6 to .65 


Ship of the line.5 to ,6 

Naval steamer, first class.5 to .6 

“ “ .52 to .58 

Merchant steamer, sharp.54 to .58 

Half clipper.5210.56 

Brigs, barks, etc.52 to ,56 

River steamer, tug-boat, med’m .52 to. 56 


Merchant steamer, medium... .52 to .54 

Clipper. 5 to.54 

Schooner, medium.48 to .52 

River steamer, tug-boat, sharp .45 to . c 

“ “ medium.4510.5 

“ “ sharp.42 to.45 

Schooner, sharp. ,46 to .5 

Yachts, sharp.4 to .45 

“ very sharp.3 to .4 

Ri ver steamers, very sharp... . 36 to . 42 


In steam launch Miranda , when making 16.2 knots per hour, with a displace¬ 
ment of 58 tons, her coefficient was 3. 

To Compute Change of Trim. 

X —- == d'. D representing displacement at line of draught in tons, L length 
at same line in feet, and m longitudinal meta-centre. 

Illustration. — “ Warrior ,” at draught of 25.5 feet, has L = 38o feet, m — ^$ feet, 
and D =z 8625 tons. If, then, a weight of 20 tons was shifted fore and aft 100 feet, 

20 X 100 380 , j. . 

——-X — = .1856 feet — 2.22 ms. 

8625 475 






















Illustration.— Vertical Plane at £5 and Horizontal at Load-line. 


656 


NAVAL ARCHITECTURE. 




.22 ^3 

Q * 


to h CO 

h o 
000 

M tO W 


I I 


o N coco O 0 0 0 
<N . ro m Tt- o . 00 vo h 

omvoo 00 mm 

M rj-OO H 


Feet. 

29 

1 1^ 

1 1 

1 5 - 

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D . I 

O H VO N 

r-. 0 1 

N 

0 H 

M 1 

ft M VO 

0) 


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r® n Nin ro w o n to to 

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•§§ 

= So 

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NAVAL ARCHITECTURE. 657 

To Compute Centre of GS-ravity or Buoyancy Approxi¬ 
mately. 

2 Q 

— to — of mean draught of hull, using larger coefficient for full-bodied vessels. 

5 20 

To Delineate Curve of Displacement. 

This curve is for purpose of ascertaining volume of water or tons weight, 
displaced by immersed hull of a vessel at any given or required draught; or 
weight required to depress a hull to any given or required draught. From 
results of computation for displacement of vessel, proceed as follows, Fig. 4: 

On a vertical scale of feet and ins., 
as A B, set off depths of keel and water- 
lines, draw ordinates thereto represent¬ 
ing displacement of keel, and at each 
water-line, in tons. 

Through points 1, 2, 3, 4, and 5 de¬ 
lineate curve A 5, which will represent 
displacement at any given or required 
draught. 

Draw a horizontal scale correspond¬ 
ing to weight due to displacement at 
load-line, as A C, and subdivide it into tons and decimals thereof, and a ver¬ 
tical line let fall from any point, as x , at a given draught, will indicate 
weight of displacement at depth, on scale A C, and, contrariwise, a line raised 
from any point, as 2, on A C will give draught at that weight. 

Illustration.— Displacement of hull (page 654) at load-line — 7151.8 cube feet, 
which -4- 35 for salt water = 204.3 tons, hence AC represents tons, and is to be sub¬ 
divided accordingly. 

Assume launching draught to have been 4 feet, then a vertical let fall from 4 will 
indicate weight of hull in tons on A C. 


Coefficients. (By C. Maclcrow, M. I. N. A .) 





Mean 


Coefficient. 


Description of Vessel. 

Length. 

Breadth. 

Displace- 

Amidship 

Water- 



Draught. 

ment. 

Section. 

lines. 

Iron-Clads. j 

225 

45 

15 

•715 

•932 

•755 

3 2 5 

59 

24-75 

.64 

.8l 

■ 7 i 

( 

350 

35 

21 

.687 

.85 

.84 

Mail Steamers. { 

3 8 5 

42 

22 

•659 

.88 

.8 

( 

368.27 

42.5 

18.71 

.516 

.812 

•635 

Merchant, small. j 

220 

27 

8 

.7 02 

.912 

•742 

90 

i 5 

4 

• 637 

• 9 I 4 

• 7°4 

Gunboats. j 

125 

160 

23 

3 i -3 

8 

12 

• 536 

.466 

•87 

•745 

.6x6 

.603 

Troop Ships. j 

350 

34°-5 

49.12 

46.13 

23-5 

15-75 

•47 
• 4 

.674 

.68 

•7 

.582 

Swift Naval Steamers.... j 

337-3 

50.28 

22.75 

• 483 

.787 

.614 

.628 

270 

42 

*9 

•497 

.792 

Fast Steamers, R. N. 

300 

40.27 

14 

.414 

.711 

• 7 11 


Curve of 'Weight. 

To Compute UNTnmUer of Tons required to Depress a 
Vessel One Incli at any Draught of Water Darallel 
to a Water-line. 

Rule. —Divide area of plane by 12, and again by 35 or 36, as may be 
required for salt or fresh water. 

Example. —Area of load water-line of a vessel is 1422 sq. feet; what is its ca¬ 
pacity per inch in salt water? 

1422 4 -12 = 118.5, which 4-35 — 3.38 tons. 


Fig. 4. 





























6 5 8 


NAVAL ARCHITECTURE. 


To Compute Common Centre of Grravity of Hull, Ar¬ 
mament, Engine s Boilers, etc., of a Vessel. 

Rule. —Compute moments of the several weights, relatively to assigned 
horizontal and vertical planes, by multiplying weight of each part by its 
horizontal and vertical distance from these planes. 

Add together these moments, according to their position forward or aft, or 
above or below these planes, and difference between these sums will give po¬ 
sition forward or aft, above or below, according to which are greatest. 

Divide results thus ascertained by total weight of vessel, and product will 
give horizontal and vertical distances of centre of gravity from these planes. 

It is customary to assume vertical plane at 22 , and horizontal plane at 
load-line. 

Note. —In following illustration, in order to simplify computation in table, com¬ 
mon centre of gravity of hull, machinery, etc., is taken, instead of centres of indi¬ 
vidual parts, as engine, boiler, propeller, etc. 


Illustration.— Assume half-girths as in following table, and distance between 
sections io feet. 


Sec¬ 

tion. 

Half- 

Girths. 

FORV 

Multi¬ 

pliers. 

fARD. 

Prod¬ 

uct. 

Multi¬ 

pliers. 

Mo¬ 

ments. 

Sec¬ 

tion. 

Half- 

Girths. 

Am 

Multi¬ 

pliers. 

IFT. 

Prod¬ 

uct. 

Multi¬ 

pliers. 

Mo¬ 

ments. 

No. 

&••• 

Feet. 

25 

I 

25 

_ 

_ 

No. 

1 . .. 

Feet. 

23 

4 

92 

I 

92 

A.... 

23 

4 

92 

I 

92 

2 . . . 

20 

2 

40 

2 

80 

13 .... 

21 

2 

42 

2 

84 

3 ••• 

l8 

4 

72 

3 

216 

C.... 

*9 

4 

76 

3 

228 

4 ... 

16 

2 

32 

4 

128 

D.... 

T? 

17 

15 

2 

34 

4 

136 

75 

. 6i 5 

5 ••• 

14 

I 

14 

5 

70 


I 

*5 

5 





534 

586 


Moments forward, 615 — moments abaft, 586 = 29 -f- sum of product 534 = .054, 
which x 10 feet = .54 fret forward of £g$. 

Centre of Lateral Resistance. 

Centre o f Lateral Resistance is centre of resistance of water, and as its po¬ 
sition is changed with velocity of vessel, it is variable. It is generally taken 
at centre of immersed vertical and longitudinal plane of vessel when upon 
an even keel. 

If vessel is constructed with a drag to her keel, the centre will be moved 
proportionately abaft of longitudinal centre. 

Yacht America had a drag to her keel of 2 feet, and centre of lateral re¬ 
sistance of her hull was 8.08 feet abaft of centre of her length on load-line. 

Centre of Effort. 

C ntre o f Effort is centre of pressure of wind upon sails of a vessel in a 
vertical and longitudinal plane. Its position varies with area and location 
of sails that may be spread, and it is usually taken and determined by the 
ordinary standing sails, such as can be carried with propriety in a moderately 
fresh breeze. 

In computing this position, the yards are assumed to be braced directly fore 
and aft and the sails flat. 

Note.—C entre of effort of sails, to produce greatest propelling effect, must accord 
with capacity of vessel at her load-line, compared with fullness of her immersed 
body at its extremities. Thus, a vessel with a full load-line and sharp extremities 
below, will sustain a higher centre of effort than one of dissimilar capacity and con¬ 
struction. 






























NAVAL ARCHITECTURE. 


659 


To Compute Location of Centre of Effort. 

Rule. —Multiply area of each sail in square feet by height of its centre of 
gravity above centre of lateral resistance in feet, divide sum of these prod- . 
ucts (moments) by total area of sails in square feet, and quotient will give 
height of centre in feet. 

2. Multiply area of each sail in square feet, centre of which is forward of 
a vertical plane passing through centre of lateral resistance, by direct dis¬ 
tance of its centre from that plane in feet, and add products together. 

3. Proceed in like manner for sails that are abaft of this plane, add their 
products together, and centre of effort will be on that side which has greatest 
moment of sail. 

Example.— Assume elements of yacht America as rigged when in U. S. Service. 


Sail. 

Area. 

Height of 
Cent, of Grav¬ 
ity of Sails. 

Vertical 

Moments. 

Distance o 
of Gravity 
Foreward. 

f Centre 
of Sails. 
Abaft. 

Mome 

Forevvard. 

nts. 

Abaft. 

Flying Jib. 

Sq. Feet. 
656 

Feet. 

28 

18 368 

52 


34 112 


Jib. 

1087 

26 

28 262 

32 

— 

34 7 8 4 

— 

Foresail. 

1455 

34 

49470 

— 

O 

J 

— 

4 365 

Mainsail. 

2185 

35 

76 475 

— 

40 

— 

87 400 


5383 


172 575 



68 896 

9 1 765 


Vertical moments 172 575 

Area of sails. 5383 

sistance. 

91765 68 896 

5383 


Moments 

sistance. 


— 32.06 = height of centre above centre of lateral re- 


= 4.25 = distance of centre abaft centre of lateral re- 


Relative Positions of Centre of Effort and. of Lateral 

Resistance. 

l(.75 <r+<n „ .. , ... l 


Square Rig. 
4 A 


10 ( d ' -f- d") 


E. Fore and Aft Rig. 


10 (d' -f- d") 


E, 


and —- = E'. L representing length of load-line, cl distance of centre of buoyancy 
5 u 

of vessel below it, d' distance of centre of lateral resistance abaft centre of it., d" dis¬ 
tance of centre of buoyancy before centre of it, E distance of centre of effort before 
centre of lateral resistance, and E' distance of centre of effort above centre of lateral 
resistance. 

iVI eta-Centre. 

Meta-centre of a vessel’s hull is determined by location of centre of grav¬ 
ity or buoyancy of immersed bottom of hull, for it is that point in transverse 
section of hull, where a vertical line raised from its centre of gravity or 
buoyancy intersects a line passing through centre of gravity of hull, as 
Fig. 1, page 650. 

To Compute Tleiglit of [Meta-Centre. 

By Moment of Inertia. = M. I representing moment of inertia of area 
of water-line or plane of flotation, and D volume of displacement in cube feet. 

Note.— Moment of Inertia of an area is sum of products of each clement of that 
area, by square of its distance from axis, about which moment of area is to be 
computed. 

To Ascertain Moment of Inertia approximately. 

Rectangle = CLB 3 ; C = — when L = 4 B; C = — when L = 5 B; and C — 

12 50 

-IL when L = 6B. With very fine lines and great proportionate length C = — . 
200 ’ 25 

L and B measured at load-line. 




























66 o 


NAVAL ARCHITECTURE. 


Illustration.—A ssume length of vessel 233 feet, breadth 43, draught 16, and 

21 

displacement 2700 tons. Length = 5.65 beams; hence C is taken at — . Volume 

400 

of displacement = 2700 x 35 — 92 500 cube feet. 

21 X 233 X 

Then -—-——— ==I0 e I> Exact height of moment was 10.44 feet. 

400 X 92 500 

By Ordinates. Rule. —Divide a half longitudinal section of load water¬ 
line by ordinates perpendicular to its length, of such a number that area 
between any two may be taken as a parallelogram. Multiply sum of cubes 
of ordinates by respective distances between them, and divide two thirds 
of product by volume of immersion, in cube feet. 


Illustration.— Take dimensions from Figs. 2 and 3, page 654. 



Length. 

Cube. 


Length. 

Cube. 

4 . 

•5 . 

. 125 

A .. 

....9.6.... 

.. 885 

0 

O *. 

• 7-7 . 

• 456 

B .. 

-7.8.... 

•• 475 

2 . 

• 9-5 . 

• «57 

(J .. 

....6.8.... 


I . 

• 9-9 . 

. 970 

D .. 

....4 - 

.. 64 

£3 . 

.IO . 

IOOO 



514 6 x 


Cube. 
51 460 
2 

3)102 92O 


7151.8) 34306.6 — 4.77 ft. 

If there are more ordinates, their coefficients must be taken in like manner, as 
— 4 — 2 — 4 — 2 — 4 — 1. 

For operation of this method, see Simpson's rule for areas, page 342. 

y d x _ ^ y representing ordinates of half-breadth sections at load- 


0 r ’i /' 


31/ D 

line , dx increment of length of load-line section or differential of x, and D displace¬ 
ment of i mmersed section in cube feet. 

I 


By- Areas. 


(ot3 ~p 4 Z>3 -|— 2 c3-j- 4 —|- e3) .-1- F -j— A 

D 


= M. «, b, c, d, 


and e representing ordinates of 1 st or load water-line , F area of irregular section 
between 1 st frame and stem , and A area of like section between last frame and 
stern-post , both in sq. feet, D displacement , in cube feet, and l distance between frames 
or sections of water-line, as m ay be taken , in feet. 


To Ascertain Areas of F and. A. 

—Big. r>. 

2 2 

— <x&X&c 3 4-4 = F, and — deXeg^-tr^ — A. 
*2 *2 


Elements of Capacity and Speed of Several Types of 
Steamers of It. IN’. (IF. H. White.) 


Classes. 

Length. 

Length 

to 

Breadth. 

Displacement. 

Speed. 

IFP 

Displace¬ 

ment. 

to 

Displace¬ 
ment ||. 

Iron-clads. 

Feet. 


Tons. 

Knots. 



Recent types. 

3 00 t0 330 

5 - 25 t 05.75 

7500 to 

9 000 

14 to 15 

.9 to 1 

16 to 20 

do. twin sc. 

280 to 320 

4-5 to 5 

6000 to 

9 000 

14 to 15 

•7 to -9 

15 to 19 

Unarmored. 








Swift cruisers 

270 to 340 

6.5 to 6.75 

3000 to 

5 500 

M 

Ln 

O 

H 

On 

1.3 to 1.5 

20 to 24 

Corvettes.... 

200 to 220 

6 

1800 to 

2 OOO 

12.75 to 13.25 

1 to 1. 2 

13 to 14 

Ships. 

160 

5 

850 to 

950 

II 

1 to 1. 2 

IO to II 

Gun - vessels.. 

125 to 170 

5.5 to 6.25 

420 to 

800 

9.5 to 11 

. 8 to 1.4 

7 to 11 

Gun-boats ... 

80 to 90 

3 to 3.25 

200 to 

250 

8 to 9 

.8 to 1.1 

5 to 7 

Merchant. 








Mail, large... 

400 to 500 

9 to II 

7000 to 

10000 

14 to 15 

.5 to .6 

10 to II 

“ smaller. 

300 to 400 

8 to 10 

5000 to 

7 000 

13 to 14 

- 4 to .5 

7 to IO 

Cargo, large.. 

25 ° t0 350 

7.5 to IO 

3000 to 

6 000 

II to 15 

• 3 to -5 

5 to 9 

“ smaller. 

200 tO 3OO 

7 to 9 

1500 to 

4 000 

9 to 11 

. 2 to .4 

3 to 6 


F'g- 5 - 

d e b a 















































NAVAL ARCHITECTURE. 


661 


To Compute Tower Required, in. a Steam "Vessel, capac¬ 
ity of another "Vessel feeing given. 

In vessels of similar models. ^-^ — Y: = 1 —^- = C: and -^- = R; 

J a ’ s 3 ’ r’ 2 r 

v and Y' representing product of volumes of given and required cylinders and revo¬ 
lutions in cube feet , a and A areas of immersed section of given and required 
vessel in sq. feet at like revolutions and speed of given vessel, s and S speeds of given 
and required vessel at revolutions of given vessel , both in feet per minute , r and r' 
revolutions of given and required vessel per minute, and C product of volume of com¬ 
bined cylinder and revolutions for required vessel. 

Illustration. —A steam vessel having an area of amidship section of 675 sq. feet, 
has two cylinders of a combined capacity of 533.33 cube feet, and a speed of 10.5 
knots per hour, with 15 revolutions of her engines. Required volume of steam 
cylinders, with a stroke of 10 feet, for a section of 700 feet and a speed of 13 knots 
with 14.5 revolutions. 

v=z 533.33 X 15 — 8000 cube feet, 80 00 f 7 °° —8296.3 cube feet , 13 x8 ' 2 96 - 3 _ 


15745.2 cube feet, 


15 X 15 745-2 


675 

16 288.1 cube feet, 


and 


16 288. 


10.5-5 
= 561.66 cube 


14-5 _ " 2X14-5 

feet , which -4-10 stroke of piston, 12 for ins., and X 1728 ins. in a cube foot = 

561.66 X *728 _ g Q g sc . j ns area 0 f e ach cylinder — diameter of 101.5 ins. 

10 X 12 

Approximate Rules to Compute Speed and IIP of Steam 

"Vessels. 


V 3 D§ 


= c; 3 


C IIP 


= V; and 


vs dS 


iip. 


Or, 3 


C IIP 


: V; and 


V 3 A 


TIP. 


HP ~ 7 V D § ' C V A C 

C representing coefficient of vessel, A area of immersed amidship section in sq.feet, 
Y velocity of vessel in knots per hour, and D displacement of vessel in tons. 

Note.—W hen there exists rig. an unusual surface in free board,deck-houses, etc., 
or any element that effects coefficient for class of vessel given, a corresponding ad¬ 
dition to, or decrease of, following units is to be made: 

Range of Coefficients as deduced from observation is as follows: 


SIDE-WHEEL. 




PPOPELLER. 







c 

1 





c 

Vessel. 

A 

D 

V 

V 3 A 

v 3 d! 

Vessel. 

A 

D 

V 

V 3 A 

v 3 d! 





1FP 

iip 





1H? 

IH? 

Steamboat. 

Sq.F. 

T’s. 

K’ts. 



Steamboat. 






Medium lines .... 

43 

73 

10 

470 

212 

Medium lines.. 

.C “ 

45 

— 

12 

— 

500 

16 66 

150 

465 

13 

570 

219 


— 

— 

— 

— 

— 


136 

300 

19 

540 

200 

Fine lines. 

150 

— 

15 

— 

53 ° 

Steam,er. 


Steamer. 




Medium full lines* 

675 

3600 

10 

650 

214 

Medium full... 

55 ° 

2532 

9 

194 

57 ° 

_ 



— 

— 

— 


390 

U 75 

10 

180 

470 

Fine lines!. 

880 

5 2 33 

15 

650 

211 

— 

— 

3600 

13 

210 

— 


— 



— 

— 

Torpedo boat.. 

— 

27 

20 

170 

500 


* Full rigged. 


Coefficients as Determined by Several Steamers 

(C. Mackrow, M. I. N. A.) 


t Bark rigged. 

of II. B. M. Service. 


Length. 


Feet. 

185 

212 

360 

27O 

38° 

400 

362 

400 


Length 

Beam. 

Area of 
Section at 

Displace¬ 

ment. 


Sq. Feet. 

Tons. 

6-53 

236 

775 

5-89 

377 

1554 

7-33 

814 

5898 

6-43 

632 

3057 

6.52 

1308 

9487 

6 -73 

1198 

9*52 

7-33 

778 

5600 

6-73 

1185 

9 ° 7 I 

3 K 


IH? 

Speed. 

v 3 a 

I FP U 

782 

Knots. 

10.34 

333 

1070 

10.89 

456 

2084 

n -5 

59 8 

2046 

12.3 

574 

3205 

12.05 

7 i 4 

5971 

13-88 

53 6 

3945 

14.06 

548 

6867 

15-43 

634 
























































662 


NAVAL ARCHITECTURE. 


Approximate Bmle for Speed, of Screw Bropellers. 

(Molesworth.) 


: N; 


P N 




ioi V 


P; 


88 v 
~F r ' 


:N; 


PN 


= v: and 


88 v 


P. 


P 7 ioi ' 7 N " 7 P 7 88 ' 7 ~" N 

V and v representing velocities in knots and miles per hour, P pitch of propeller in 
feet , and N number of revolutions per minute. 

This does not include slip, which ranges from io to 30 per cent. 

BitcL. of Screw ^Propeller. 

Pitch ranges with area of circle described by diameter of screw to that of 
amidship section. 

Area of screw circle to amidship) 


4-5 


section = 1 to.j 

Two Blades. 

Pitch to diameter of screw = 1 to 
Four Blades. 

Length =. 166 diameter. 

Slip of Side-'wlieels. 


.8 

1.08 


1.02 | 1.11 
1-38 1 i -5 


1.2 

1.62 


3-5 

1.27 

I- 7 1 


3 2 -5 

1.31 | 1.4 
1.77 | 1.89 


1.47 

1.98 


Radial Blades. 


2 (A — c) 


= S. 


Feathering. 


1.5 (A —c) 


S. A representing 


A A 

length of arc of immersed circumference of blades, c length of chord of immersed arc , 
and S slip, all in feet. 

Area of Blades. 

.75 HP . „ „ . IIP 


River Service. 


: A. Sea Service. 


• = A. D representing diameter 


D D 

of wheel in feet, and A area of each blade in square feet. 

Length of Blades. .7 in River service and 6 in Sea service. 

Distances between Radial Blades. 2.25 in River service and 3 feet in Sea service; 
between Feathering blades, 4 to 6 feet. 

Proportion of Power Utilized in a Steam "Vessel. 

P — z 

C. P representing gross TTP , z loss of 


Side Wlxeel. 


.000002 59 cZ 3 r 

effect by slip and oblique action of wheels, d diameter of wheels at centre of effect, 
r revolutions per minute, and C coefficient for vessel. 

Illustration.—IIP of engines of a side-wheel steamer is 1120; slip of wheels 
and loss by oblique action, 33.37 per cent.; diameter of centre of effect of wheels is 
29.5 feet, and number of revolutions 13.5 per minute; what is coefficient, and what 
power applied to propel vessel ? 

Note.— Slip of wheels from their centre of effect in this case is 15.37 P er cent., 
and loss by oblique action 18 per cent. Hence, representing total power by 100, 
100 — (18 -j- 15.37) =66.63 per cent, of power applied to wheels. 

As assumed power that operates upon wheels in this case is taken at 86 12 per 
cent, of power exerted by engines, 86.12 x 33-37 = 28.74 P er cent. f° r sum of loss 


by wheels. 


1120—(1120X28.74-4-100) 798.11 


65.63 coefficient. 


.00000259 X 29.53 x 13.5 2 12.16 

Speed of vessel being 10 knots per hour = 17.05 feet per second, power applied 


19076.13 X 17-05 X 60 


= 59 i - 36 - 


33000 

Friction of engines 1.5 lbs. upon 3848 sq. ins. x 13-5 revolu- 


Friction of load 6 per cent, upon pressure of steam, less 2 lbs. 
for friction of engine, as above. 


and IP 

exerted = 

IP 

Per cent, 
of Power. 

94 - 45 ] 

I- 18.83 

60.45 j 


201.6 

18 

172.14 

15-37 

59 x -36 

52.8 


1120 


100 



















NAVAL ARCHITECTURE. 


663 


IP. 


Screw Propeller. Friction of engines. 96.06) 

Friction of load. 81.48 } 

“ of screw surface and resistance of edges of blades. 53-44 

Slip of propeller. 205.55 

Absorbed by propulsion of vessel. 375-92 


782-45 


Per cent, 
of Power. 

18,83 

6.83 

26.27 

48.04 

100 


Note.— From experiments of Mr. Froude, he deduced that, as a rule, only 37 to 
40 per cent, of whole power exerted was usefully employed. 

With an auxiliary propeller, essential differences are in friction of surfaces and 
edges of blades of propeller and slip of propeller, being as 12 to 6 83 in excess in first 
case, and as 13.7 to 26.27 second case, or 50 per cent. less. 


Resistance of Bottoms of Hulls at a Speed of one Knot per Hour. 


Smooth wood or painted.ox lb. 

Smooth plank. 016 “ 

Iron bottom, painted.014 “ 


Copper....007 lb. 

Moderately foul.019 “ 

Grass and small barnacles.06 “ 


Sailing. 

Patio of Effective Area of Sails an cl of "Vessel’s Speed, 
cinder Sail to "Velocity- of Wind. 


COURSE. 

Ratio of 
Effective 
Area 
of Sails. 

Ratio of 
Speed of 
Vessel 
to Wind. 

Course. 

Ratio of 
Effective 
Area 
of Sails. 

Ratio of 
Speed of 
Vessel 
to Wind. 

5 points of wind. 

•59 

•33 

Wind abeam. 

.82 

.6 

2 “ abaft beam.. 

.91 

•5 

“ astern. 

I 

• 5 

6 “ of wind. 

.68 

•5 

“ on quarter. 

.96 

.66 


Propulsion and Area of Sails. 

Plain sails of a vessel are standing sails, excluding royals and gaff topsails. 


Resistance of vessels of similar models but of different dimensions for equal 
2 

speeds = D 3 

D ^ 

Hence . a and a' representing areas of sails of known and given ves¬ 

sels , and D and D' their displacements in tons. 

Illustration.— Assume D and D' ±= 2400 and 1600. 


( . Al. 

) = 3/1.5 2 = 1 139, hence area of sails a' — —— = 878 per centum. 
1600/ 1.139 

Tn Vessels of Dissimilar Models. —Plain sail area should be a multiple 

of D§. 

Multiples for Different Classes of Vessels, R. N. 


Sailing. 

Ships of Line. 

Frigates. 

Sloops. 

Brigs. 


100 to 120 
120 to 160 


Steamers. 

Ships, iron-clad. 

Frigates. 

Sloopsi. 

Brigs. 


60 to 80 
80 to 120 


English Yachts , designed for high speed, have multiples from 180 to 200, 
and when designed for ordinary speed from 130 to 180. 


When Area of Sail to Wet Surface of Hull is taken .—American yacht Sappho had a 
ratio of 2.7 to 1, and several English yachts nearly the same, while in some others 
it was but 2 to 1. 





































664 


NAVAL ARCHITECTURE. 


Location of ALasts, etc. Load-line = 100. 


Vessel. 

D 

Fore. 

[stance from Ste: 

Main. 

H. 

Mizaen. 

Foot of Sail.* 

Height of Centre 
of Effect above 
Water-line = 
Breadth.* 

Ship. 

10 to 20 

53 to 58 

54 to 60 

64 to 65 

55 to 61 

36 to 42 

80 to 90 

81 to 91 

125 to 160 
130 to 160 
160 to 165 
160 to 170 
170 to 190 

1.5 to 2 

Bark. 

12 to 20 

1.5 to 1.95 
1.5 to 1.75 

1.5 to 1.75 
1.25 to 1.75 

Brig. 

17 to 20 

16 to 22 

Schooner .... 
Sloop. 

— 


* Measured from Tack of Jib to Clew of Spanker or Mainsail. 


Rake of Masts. 

Ships.—Foremast o to .28 of length from heel, Main and Mizzen o to .25. 
Schooners.—Foremast .1 to .25, Mainmast .63 to .77. Sloops.—.08 to .11. 


Area of Sails. 


Sails. 

3 Yards upon 
each Mast. 

4 Yards upon 
each Mast. 

Sails. 

3 Yards upon 
each Mast. 

4 Yards upon 
each Mast. 

Jib. 

.08 

.08 

Mizzenmast.... 

. 12 7 

.14 

Foremast. 

•295 

■295 

Spanker or ) 


.068 

Mainmast. 

.417 

.417 

Driver... j '' 



Proportional Area of Sails upon each Mast under above Divisions. 


Sail. 

Fore. 

Main. 

Mizzen. 

Course. 

•ii 5 

.097 

. 162 

.138 

— 

— 

Topsail. 

105 

.09 

.149 

.127 

'•075 

.063 

Topgallant sail. 

•075 

.063 

. 106 

.089 

•052 

• 045 

Roval. 

— 

045 

— 

.063 

— 

.032 

Spanker or Driver. 

— 

— 

— 

— 

H 

00 

0 

.068 

Jib. 

• 08 

.08 

— 

— 

— 

— 


-375 

•375 

.417 

.417 

.208 

. 208 


Proportion to x. 


■389 

•33 

•358 

•303 

•253 

•215 

— 

.152 

I 

I 


Balance of Sails. —Effect of jib is equal to that of all sails upon main¬ 
mast, and sails upon mizzenmast balance those of foremast. 

Areas of sails upon masts of a ship should be in following proportion : 
Fore. 1-414 | Main. 2 | Mizzen. 1 


When, therefore, main yard has a breadth of sail of 100 feet, fore yard 
should have 70.71 feet, and mizzen 50 feet ■, topgallant and royal yards and 
sails being in same proportion. 


An' 

Approximately. 


;les of Heel for Different Vessels. 
D M a 


H 


S. D representing displacement of vessel in lbs., 


M height of meta-centre above centre of gravity in feet, a angle of heel of vessel in cir¬ 
cular measure ,* and H height of centre of effect above centre of lateral resistance , 
in feet. 

Moment of sail should he equal to moment of stability at a defined angle 
of heel. 


Angle. 

Frigates, etc...4° 

Corvettes. 5° 


Circular 

Measure. 

•°7 

.087 


Angle. 

Schooners, etc.6° 

Yachts. 6° to 9 0 


Circular 

Measure. 

.105 

.105 to .107 


Illustration. — Assume displacement 170 tons, height of meta-centre 6.75 feet) 
H = 36 feet, and angle of heel 9 0 ; what should be area of sails ? 

170 X 2240 = 380 800 lbs. 9 0 = . 107. 

380800X6.75 X-107 

-—- - -- = 7639-8 sq.feet. 

36 


* See rule, page 113. 







































































NAVAL ARCHITECTURE. 


665 


Trimming of Sailn. 

That a vessel’s sail may have greatest effect to propel her forward, it should 
be so set between plane of wind and that of her course, that tangent of angle 
it makes with wind may be twice tangent of angle it makes with her course. 

Or, tan. a — 2 tan. b. a representing angle of sail with wind, and b angle of sail 
and course of vessel. 

Angles of Course and. Sails witli Wind. 


Wind 

Ahead. 

Angle 

of 

Course. 

Tan¬ 

gent. 

Half 

Tan¬ 

gent. 

Angles 

with 

Wind. 

of Sail 
with 
Course. 

Wind 

Abaft. 

Angle 

of 

Course. 

Tan¬ 

gent. 

Half 

Tan¬ 

gent. 

Angles 

witli 

Wind. 

of Sail 
witli 
Course, 

Points. 

4 

45 ° 

.562 

. 281 

29 0 18' 

15 0 42' 

Points. 

2 

1X2° 30' 

2.166 

1.082 

65° 13' 

47 ° W 

5 

56° 15' 

•732 

■ 365 

36° 12' 

2°0 3' 

3 

I2 3 ° 45 ' 

2 -737 

1.368 

69° 56' 

53 ° 49 ' 

6 

67° 30' 

•9 2 3 

.461 

42 ° 43 ' 

2 4 ° 45 

4 

135 ° 

3 - 562 

1.781 

74 ° 17 ' 

6o° 43' 

Abeam 

90 0 

i- 4 i 5 

• 7°7 

54 ° 45 ' 

35° *6' 

6 

* 57 ° 3 °' 

7 • 5 11 

3-754 

82° 25' 

75 ° 5 ' 



Effective Impulse of Wind. 

Let P 0, Fig. 6 , represent direction by com¬ 
pass and force of wind on sail, AB; from P 
draw P C parallel to A B, from 0 draw 0 C per¬ 
pendicular to AB; o C is effective pressure 
of wind on sail A B, and r C, perpendicular to 
plane of vessel, is component of 0 0, which pro¬ 
duces lateral motion, as heel and leeway, and 
r 0 is component of 0 C, which propels vessel. 

Isin.a=l J ; Pcos.a? = L; and Psin.a; = 10 . 
I representing direct impact and P effective 
pressure of wind on sail , L effective impact 
producing leeway , and E effective impact which 
p propels vessel. 

Notb.— The law as usually given is sin. 2 . This is manifestly incorrect, as it gives 
results less than normal pressure for angles of small incidence. At an angle of in¬ 
cidence of wind of 25 0 , the law of sin. is exact. Hence, although it may not bo 
exact at all angles, it is sufficiently so for practical purposes. 

Illustration i. — Assume wind 5 points ahead, and I == 100 lbs. 

By preceding table angle of course with wind 56° 15'; lienco angle of sail a, with 
wind 36° 12', as tan 36° 12' = 2 tan. 20 0 3', and angle x 56° 15' —36° 12' = 20° 3'. 

Then, 100 X sin. 36° 12' = 100 X -5906 = 59.06; 59.06 X cos. 20 0 3' = 59.06 X 

.9394 = 55.48, and 59.06 X sin. 20° 3' — 59.06 X -3426 = 20.23 lbs. 

2. —Assume wind 4 points abaft, and I =; 100 lbs. 

Then, iooXsin. 2 74 0 17' = 100X ■ 9626 s = 92.66 ; 92.66 X cos. 180 0 — 74 0 i7'-j-45° 

= 6o° 43' = 92.66 X .49 = 45.41, and 92.66 X sin. 6o° 43' = 92.66 x .8722 — 80.82 lbs. 

To Compute Sailing Power of a ‘V'eHsel. 

F/sin. w, sin. s = P. 

To Compute Careening IPoxver of a Sailing Vessel. 
F/sin. w, cos. s = P F representing area of sails in sq. feet, f force of wind in 
lbs. per sq. foot, w angle of wind to sails, and s angle of sails to course of vessel. 

To Compute Angle of Steady IT eel. 

Within a Range of 8°. 


a PE 
I) M~ 


= sin. H. a representing area of plain sail in sq. feet, V pressure of wind 


in lbs. per sq. foot, E height of centre of effect above mid-draught, in feet , 1 ) displace¬ 
ment of hull, in lbs., and M height of meta-centre in feet. 

P assumed at 1 lb. per sq. foot, or that due to a brisk wind. 

20 

Illustration. — Assume a = i56oo, draugbt = 2o, and M = 62; hence 62 -\ -= 

IO 

72, D = 6 800 000, and M = 3. 

15600X1X72 1 123200 n , 

Thcn = = -0555 = 3° 10 • 

6 800 000 X 3 20 400 000 

3 K* 




























666 


NAVAL ARCHITECTURE. 


Course and. Apparent Course of Wind. 


Apparent course of a wind against sails of a vessel is resultant of normal 
course of wind and a course equal and directly opposite to that of vessel. 



Illustration. — If P, Fig. 7, repre¬ 
sent direction by compass and force of 
wind, and ab direction and velocity of 
vessel, from P draw P c parallel and 
equal to a b , join c a and it will repre¬ 
sent direction and force of apparent 
wind. 


Or, 


a c 
cT 


= ratio of velocity of apparent 


wind to that of vessel, 


a P 
c“P 


= ratio of velocity oj wind to that of vessel. 


Resistance of Air. (Mr. Froude.) 


Resistance of wind to a vessel is estimated as equivalent to square of its 
velocity. 

In a calm, resistance of air to a steamer = one thirty-fourth part of resist¬ 
ance of -water, and when a steamer’s course is head-to, and combined veloc¬ 
ity of vessel and wind = 15 knots, resistance is one ninth of that of the water. 

Resistance of air to a sq. foot of surface at right angles to course of a ves¬ 
sel is about .33 lb., and when surface is inclined to direction of wind, press¬ 
ure varies as sine of angle of incidence. 

Mean of angles of surface of a steamer exposed to wind may be taken at 
45 0 ; hence their resistance is about .25 lb. per sq. foot when wind has a ve¬ 
locity of 10 knots per hour. 

If sectional area of a steamer’s hull above water is 750 sq. feet, resistance 
to air at a speed of 10 knots in a calm would be 750 X .25 = 187.5 lbs., and 
resistance to smoke-pipe, spars, and rigging (brig rigged) would be 201 lbs. 

Leeway. 

Angle of Leeway in good sailing vessels, close hauled, varies from 8° to 
12 0 , and in inferior vessels it is much greater. 

Ardency is tendency of vessel to fly to the wind, a consequence of the 
centre of effort being abaft centre of lateral resistance. 

Slackness is tendency of vessel to fall off from the wind, a consequence of 
the centre of effort being forward centre of lateral resistance. 


Results of Experiments upon Resistance of Screw-propellers , at High Velocities 
and Immersed at Varying Depths of Water. 


Immersion of 
Screw. 

Resistance. 

Immersion of 
Screw. 

Resistance. 

Immersion of 
Screw. 

Resistance. 

Surface. 

I 

2 feet. 

7 

4 feet. 

7.8 

1 foot. 

5 

O 4 4 

O 

7-5 

5 “ 

8 


Slip of Propeller , 15 per cent.; of Side-wheel (feathering blades ), and tak¬ 
ing axes of blades as the centre of pressure, 23 per cent. 

R'ree'boarcL. 

Measured from, Spar-deck stringer to surface of water. Depth of Hold from under¬ 
side of spar deck to top of ceiling. 


Hold. 


Hold. 


Hold. 

Ins. l 

Hold. 


Hold. 


| Hold. 


Feet. 

Ills. 

Feet. 

Ins, 

Feet. 


Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

8 

i-5 

12 

2.25 

16 

2-75 ! 

20 

3-125 

24 

3-375 

28 

3-625 

IO 

2 

h 

2-5 

l8 

3 1 

22 

3-25 

26 

3-5 

30 

3-75 










































NAVAL ARCHITECTURE. 


667 


_ IPlating Iron Hulls. 

D L t 

800 b d ~ ' D ’ •(.presenting displacement in tons, L length of hull, b breadth, and 

d depth. Or, .05 f -Jd — T. f representing distance between centres of frames, and 
d depth of plate below load-line, all in feet, and T thickness of plate in ins. 


IVIasts and. Spars. 

Lower masts.at spar deck. 

Bowsprit. “ stem. 

Topmasts.. “ lower cap. 

Topgallant masts. “ topmast cap. 

Fore and main masts, when of pieces, 
length. Mizzenmast .66 diameter of mainmast. 

3-5 to 3-75 feet of whole length. 

Bowsprit, depth, equal diameter of mainmast; width, diameter equal to foremast. 


Diameter for Dimensions. 

Jib-boom.at bowsprit cap. 

Yards.in middle. 

Gaffs...at inner end. 

Main and Spanker booms at taffrail. 

1 inch for each 3 to 3.25 feet of whole 
Masts of one piece 1 inch for each 


Main and fore topmasts , 

Mizzen topmast. 

Topgallant masts. 

Royal masts. 

Topgallant poles. 

Jib-boom. 


inch for each 3 

(< ll 


Fore and main yards.1 

Topsail yards.875 

Cross-jack, Topgallant, and) 

Royal yards.j 1 

Main and Spanker booms.1 

Gaffs.1 

Studding-sail yards and booms. 1 

R-xxdder 


U 

u 

u 

u 

u 

u 


u 

u 

<( 


■ feet of whole length. 


to 3.25] 

3-25 “ 3-33 ' 

3-25 “ 3-33 
3.66 -I 

2.87 j 

2 ft. of length beyond bowsprit cap. 
4 

4 

5 


3-5 

3.5 to 4 

4.5 to 4.75 


► feet of whole length. 


PJ = T; .i 9 6 CD 3 = M; 3 


T 


Head. 

D: and 


(Mackrow.) 

A V 2 


. n -, - = P. P representing press- 

.196 C 2400 

ure on rudder when hard over, in tons, d distance of geometrical centre of rudder from 
axis of motion, in ins., T stress on head, and M moment of resistance of head, both in 
inch-tons, A immersed area of rudder in sq. feet, v velocity of water passing rudder 
in knots per hour, and C coefficient = 3.5 per sq. inch for Iron, and . 123 for Oak. 

Illustration. —Assume area of wooden rudder 24 sq. feet, distance of its geomet¬ 
rical centre from centre of pintles 2 feet, and velocity of water 10 knots. 

24 X 10 2 , . T , _ / 24 

— 1 ton. 1 X 2 X 12 = 24 inch-tons. 3 j ^ — 9.93 ms. 


2400 


.196 X .125 


Memoranda. 


Weights. —A man requires in a vessel a displacement ot 488 lbs. per month, for 
baggage, stores, water, fuel, etc., in addition to his own weight, which is estimated 
at 175 lbs. A man and his baggage alone averages 225 lbs. 

A ship, 150 feet in length, 32 beam, and 22.83 in depth, or 664 tons, C. H. ( 0 . M.), 
has stowed 2540 square and 484 round bales of cotton. Total weight of cargo 
1 254 448 lbs., equal to 4.57 bales, weighing 1889 lbs., per ton of vessel. 

A full built ship of 1625 tons, N. M., can carry 1800 tons’ weight of cargo, or stow 
4500 bales of pressed cotton. 

Hull of iron steamboat John Stevens — length 245 feet, beam 31 feet, and hold 
ix feet; weight of iron 239440 lbs. And of one other—length 175 feet, beam 24 
feet, and 8 feet deep; weight of iron 159 190 lbs. 

Weight of hull of a vessel with an iron frame and oak planking (composite), com¬ 
pared with a hull entirely of wood, is as 8 to 15. 

An iron hull weighs about 45 per cent, less than a wooden hull. 

Iron ship, 254 feet in length, 42 beam, and 23.5 hold, 1800 tons register, has a stow¬ 
age of 3200 tons cargo at a draught of 22 feet. Weight of hull in service 1450 tons. 

Loss by Weight per Sq. Foot per Month of Metalling of a Vessel's Bottom in Service. 

Copper .0061 lb.; Muntz metal .0045 lb.; Zinc .007 lb.; and Iron .0204 lb. 

Comparison between Iron and Steel plated Steamers. —In a vessel of 5000 tons 
displacement, hull of steel-plated will weigh 320 tons less = 6.66 per centum less. 



























668 


OPTICS. 


OPTICS. 


Mirrors, in Optics, are either Plane or Spherical. A plane mirror is a 
plane reflecting surface, and a spherical mirror is one the reflecting surface 
of which is a portion of surface of a sphere. It is concave or convex, ac¬ 
cording as inside or outside of surface is reflected from. Centre of the 
sphere is termed Centre of curvature. 

Focus —Point in which a number of rays meet, or would meet if produced. 
Fig. i. Principal Focal Distance is half radius 

of curvature, and is generally termed the 
focal distance. Line a c is termed the 
principal axis , and any other right line 
through c which meets the mirror is termed 
a Secondary axis. When the incident 
rays are parallel to the principal axis , the 
reflected rays converge to a point, F. 





Fig. 2. 


Conjugate Foci are the, foci of the rays proceeding from any given point 
in a spherical concave mirror, and which are reflected so as to meet in an¬ 
other point, on a line passing through centre 
of sphere. Hence, their relation being mu¬ 
tual, they are termed conjugate. 

Let P be a luminous point on principal axis, 
Fig. 2, and Pt a ray; draw the normal line c i, 
which is a radius of the sphere; then c i P is an¬ 
gle of incidence, and ci 0 the angle of reflection, 
equal to it; hence ci bisects an angle of triangle 
i P c P 

P i 0 , and therefore, —- = 

i 0 c 0 



When conjugate focus is behind a mirror, and reflected rays diverge, as 
if emanating from that point, such focus is termed Virtual , and a focus in 
which they actually meet is termed Real. 



As a luminous point, as P, Fig. 3, is 
moved to the mirror, the conjugate focus 
moves up from an indefinite distance at 
back, and meets it at surface of mirror. 

If an incident ray converges to a point 
s, at back of mirror, it will be reflected 
to a point P in front. The conjugate 
foci P s having changed places. 


Pencil. —Rays which meet in a focus and are taken collectively. 

Objects. —As regards comparative dimensions or volumes, it follows, from 
similar triangles, that their linear dimensions are directly as their distances 
from centre of curvature. 


To Compute Dimension or ‘V'olmxie of an Image. 

When Dimensions and Position of Object are Given , and for either Convex 
or Concave Mirrors. 

-b — —- , or — = - - . L and l representing lengths of image and object, F focal 
l t L r 

length , and D and d respectively , distances of image and object from principal focus. 


Hefraotion. 

Deviation. —Angle at which a ray is diverted from its original or normal 
course when subjected to refraction is thus termed. 

Indices of Refraction. —Ratio of sine of angle of incidence to sine of angle 
of refraction, when a ray is diverted from one medium into another, is termed 
relative index of ref raction from former to latter. 











OPTICS. 


669 


When a ray is diverted from vacuum into any medium, the ratio is greater 
than unity, and is termed absolute index or index of refraction. 


Mean Indices of Refraction. 


Eye, vitreous humor. 

“ crystalline lens, under, 
“ “ “ central 

Diamond,. 

Glass, flint. 


i -339 

!-379 

1.4 

2.6 

i -57 


Glass, lead, 3 flint.. 
“ lead 2, sand 1 
“ “ 1, flint 1 

Ice. 

Quartz. 


For indices of other substances, see page 584. 


Heat increases refractive power of fluids and glass. 


2.03 

1.99 

1.78 

I - 3 I 

i -54 


Critical Angle .—Its sine is reciprocal of index of refraction, the incident 
ray being in the less refractive medium. 

Thus, = sin. of angle. 


Visual Angle is measure of length of image of a straight line on the retina. 

Total Reflection is when rays are incident in the more refractive medium, 
at an angle greater than the critical angle. 


Mirage. —An appearance as of water, over a sandy soil when highly heated 
by the sun. 

Caustic Curves or Lines are the luminous intersections from curve lines, as 
shown on any reflective surface in a circular vessel. 


To Compute Index of Refraction. 

= Index. I representing angle of incidence, and R that of refraction. 

Sin. Iv 

To Compute Refraction. 

Concave-Convex and Meniscus. —Effect of a concave-convex in refracting 
light is same as that of a convex lens of same focal distance, and that of a 
meniscus is same as a concave lens of same focal distance. 

2 R 7 * 

Meniscus , icith parallel rays ■ — F. 

Magnifying Power. —In Telescopes the comparison is the ratio in which it 
apparently increases length. In Microscopes the comparison is between the 
object as seen in the instrument and by the eye, at the least distance of 
vision, which is assumed at 10 ins., and the magnifying power of a micro¬ 
scope is equal to the distance at which an object can be most distinctly ex¬ 
amined, divided by the focal length of the lens or sphere. 

Linear power is number of times it is magnified in length, and Super¬ 
ficial, number of times it is magnified in surface. 

Magnifying power of microscopes varies, according to object and eye¬ 
glass, from 40 to 350 times the linear dimensions of object, or from 1600 to 
122 500 times its superficial dimensions. 

Apparent A rea. —As areas of like figures are as the squares of their linear 
dimensions, the apparent area of an object varies as square of visual angle 
subtended by its diameter. 

The number expressing Magnification of Apparent Area is therefore 
square of magnifying power as above described. 

Illustration.— If diameter of a sphere subtends i° as seen by the eye, and io° 
as seen through a telescope, the telescope is said to have a power of 10 diameters. 
















OPTICS. 


67O 


To Compute Elements of HVIirrors and. Lenses. 

0 i* l “i* 

Mirrors. Spherical Concave.* -— = D ; -— = L. 


Spherical Convex .f 


Or 


D; 


Lr 


2 L + r * ’ 
2 B T 

Unequally Convex, i 


r- 

= 1 . 


•2 l 


■ 2 l 


Hyperbolic Concave. | 


R + r 
Elliptic Concave .If 


2 L + r 
F. Plano-Convex. § 


d 2 

Parabolic Concave. — — r: 

16 h 


:F. 


2 B 


Sphere. 


■.66 t. 
T" 


F. 


2 i — x 

0 representing objects 1, r radius of convexity, l and L length or distance of object 
from vertex of curve , and from external vertex , D dimension of object , d diameter of 
base , ¥ focal distance , and h depth of mirror in like dimensions,! index of ref'action, 
and t thickness of lens. 

Illustration i. —Before a concave mirror of 5 feet radius is set an object at 1.5 
feet from vertex of curve; what is ratio of apparent dimension of image, and what 
is length of and distance of object from external vertex ? Object = 1. 

— 1 X -- -- - = 2.5 feet, and * 1- 5 * 5 -- — 3.75 feet. 

5 —2 X 1 5 5~2 X 1.5 

2.—If object is set at 4.5 feet from vertex of a like mirror, what is length of and 
distance of inverted object from internal vertex? 

1 X 5 _^ 4 - 5 X 5 


2X45—5 


= 1.25 feet, and 


2 X 4.5 —5 


5.625 feet. 


3.—Before a convex mirror of 3.5 feet radius is set an object at 3 feet from ver¬ 
tex of curve; w'hat is length of and distance of object from external curve? 

1X30 = .368 foot, and — — 5 — = 1.105 feet. 


2 X 3 + 3-5 


2 X 3 + 3-5 


4.—A parabolic reflector has a depth of 1.25 feet and a diameter of 2 feet; what 
is its focal distance from vertex of internal curve? 


Lenses. 
o F 


16 X 1 25 
Double Convex. — 


= .2 feet or 2.4 ins. 
R r 


F. When R = r 


D; 


l F 


S + F 


m— 1xR+r 


2 to- 


= F; 


F — l “» F- 
Double Concave. 


= P: 


0 F 
FTTq 


R r 


0 F 


TO - 


— F • ~ ~ - — D • 

i X R + r ’ F + L- ’ 


Y; and 
Fo-D 


S F 
S + F 


= 0 . 


; L; and 


L F 


— l. 


D ’ L + F' 

Plano - Convex and Plano - Concave. 


TO 


Optical centres are in centres of lens. 
r 

— = P• Optical centres are respectively centres of convex and concave sur¬ 


faces. Convex Concave ( Meniscus) and Concavo-Convex. 


R r 


: F. 


TO - 


I x R —r 


Optical Centres. Convex Concave. Delineate lens in half section, draw R from 
its centre to circumference of lens (intersection of radii), draw r parallel thereto 
and extending to its circumference, connect R and r at these external points of 
contact with circumference and external curve, extend line to axis of lens, and point 
of contact is centre required. Concavo-Convex. Proceed in like manner, but in 
this case r extends to, or delineates, the inner surface of the lens, and point of con¬ 
tact with axis is centre required. 


and 


* D or image disappears when l = .5 r 

Lr 


= l. 


2 D-f r 

and when parallel rays fall upon plane side, F = 2 E. || Rays of light, heat, or sound, reflected from 

focus of a hyperbola, will diverge from its concave surface, and — 1 — c - 

will be refracted by surface of the other 


t When O is beyond F, it will be inverted, as 
J When equally convex F = R. 


O r 


= D, 


2 L — r 

§ When convex side is exposed to parallel rays 


. when from the focus of an ellipse, 

































OPTICS.-PILE-DRIVING. 


67I 


OF L F 

When object is beyond focal distance (F), its image (D) will be inverted, as j—— = D, and- 

l — F L — F 


= /. 


P representing magnifying power of lens, S limit of normal sight , 10 to 12 ins. for 
far-sighted eyes and 6 to 8 for near-sighted , ordinarily 10 ins. , V limit of distinct 
vision , 0 extreme distance of object from optical centre at distinct vision , and m index 
of refraction. 

Illustration i.— If a double convex lens of flint glass has radii of 6 and 6.25 ins., 
what is its focal distance? Index of refraction = 1.57, see page 584. 


6 x 6.25 


5.37 ms. 


1.57 —1 x 64-6.25 

2.—If a double concave lens has a focal distance of 2 ins., and object is 6 ins. from 
vertex of curve, what is its dimension and what is its distance from vertex of inner 
curve ? 


6X2 


= 2 ins. , and 


4X2 


1.33 ms. 


24-4 44“ 2 

3.—If focal distance of a single microscope is 4 ins., what is its limit of distinct 
vision, and what its magnifying power? 0 = 2.857 ins. 


2.857 x 4 
4 — 2.857' 


10 ins., and 


10 4 -4 


3.5 times. 


Telescopes, Opera-glasses, etc. 

D: 0 = F; o/-y F = D, and ^ = l\ ± = F +f frepresent- 

ing length of focal distance from object lens. 

Illustration. —Principal focal distance of ocular lens of a telescope is .9 in., of 
objective lens 90 ins.; what is its magnifying power? 

go-i- .9 — 100 times the object. 


PILE-DRIVING. 

Effect of blow of a ram, or monkey, of a pile-driver, is as square of 
its velocity; but the impact is not to be estimated directly by this rule, 
as the degree and extent of the yielding of the pile materially affects it. 
The rule, therefore, in application, is of value only as a means of com¬ 
parison. 

By my experiments in 1852, to determine the dynamical effect of a fall¬ 
ing body, it appeared that while the effect was directly as the velocity, it 

was far greater than that estimated by the usual formula V s 2 g, which, for 
a weight of 1 lb. falling 2 feet, would be 11.34 lbs., giving a momentum of 
11.34 foot-lbs.; whereas, by the effect shown by the record of actual obser¬ 
vations, it would be W v 4.426 = 50 lbs. 

Piles are distinguished according to their position and purposethus, 
Gauge Piles are driven to define limit of area to be enclosed, or as guides to 
the permanent piling. 

Sheet or Close Piles are driven between gauge piles to form a compact and 
continuous enclosure of the work. 

Weight which each pile is required to sustain should be computed as if the 
pile stood unsupported by any surrounding earth. 

A heavy ram and a low fall is most effective condition of operation of a 
pile-driver, provided height is such that force of blow will not be expended 
in merely overcoming friction of leader and inertia of pile, and at same time 
not from such a height as to generate a velocity wdiich will be essentially 
expended in crushing fibres of head of pile. 


* -f for telescopes and — for opera-glasses, etc. 














6/2 


PILE-DRIVING. 


Refusal of a pile intended to support a weight of 13.5 tons can be safely 
taken at 10 blows of a ram of 1350 lbs., falling 12 feet, and depressing the 
pile .8 of an inch at each stroke. 

Pneumatic Piles .—A hollow pile of cast iron, 2.5 feet in diameter, was depressed 
into the Goodwin Sands 33 feet 7 ins. in 5.5 hours. 

Nasmyth's Steam Pile-hammer has driven a pile 14 ins. square, and 18 feet in 
length, 15 feet into a coarse ground, imbedded in a strong clay, in 17 seconds, with 
20 blows of monkey, making 70 strokes per minute. 

Morin computed work of a ram in foot-lbs., in raising a monkey for 8 hours per 
day, as follows: Tread-wheel 3900, Winch 2600. 

French engineers estimate the safe load for a pile, when driven to refusal of .4 
inch under 30 blows, to be 25 tons. 

Shaw's Gunpowder Pile-driver is operated by cartridges of powder on head 
of pile, which are ignited by fall of the ram. 30 to 40 blows per minute 
have been made under a fall of 5 and 10 feet. 27 piles have been driven in 
rough gravel and clay 7.2 feet in one day. 

To Compute Safe Load tliat in ay "be Borne Toy a Bile. 

(Maj. John Sanders, U. S. E.) 


Approximately. -— : — = W. R representing weight of ram in lbs., h height of 

fall , and d distance pile is depressed by blow , both in feet. 

Illustration. —A ram weighing 3500 lbs., falling 3.5 feet, depressed a pile 4.2 ins. 

Then 35 ° 0 - ^ - ^ 2 i — 35 °°° — 4375 lbs., weight ivhich pile would bear with 

O O 

safety. > 

Molesworth gives this, but with a variation in symbols and their expression. 


To Compute Coefficient of Resistance of the Barth.. 

R h 

—— = C. R representing resistance of the earth , and d as preceding. 

Weisbach gives following formula: Resistance of bed of earth being con- 

R 2 ^ 

stant, mechanical effect expended in penetration of pile will be ... — = W. 


P representing weight of pile in lbs. 


P + R d 


Illustration.— Assuming elements of preceding case, with addition of weight of 
pile at 1500 lbs., 

3500 2 x 3.5 42 875 000 

-;-;- r —- = 24 SOO IbS. 

1500 + 3500 X (4-2 — 12) 1750 

To Compute Weight of Ram. ( Molesworth.) 

P ( „ A p ~~ = R. P representing weight of pile in lbs., h height of fall and L 

length of pile, both in feet, and A area of section of pile in sq. ins. 


Theoretical Force of Blow of Ram. 


Fall. 

IOOO 

Wei 

1200 

ght. 

1500 

2000 

Fall. 

IOOO 

We 

1200 

ght. 

1500 

2000 

Feet. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

8 000 

9 600 

12 OOO 

16 000 

15 

31 060 

37 272 

46590 

62 120 

5 

17 920 

21 504 

26 880 

35840 

20 

35 860 

43032 

53 79 ° 

71 720 

IO 

25 360 

30 432 

38040 

50 720 

25 

40 IOO 

48 720 

60 150 

80 200 


Sheet Riling. 

Bevelling.. I20 ° | Shoeing. 25 o 


Ringing Engine 

Requires i man to each 40 lbs. weight of ram, which varies from 450 to 
900 lbs. 



























PILE-DRIVING. 


-PNEUMATICS.-AEROMETRY. 6/3 


File-sin. king. 

Mitchell's Screw Piles are constructed of a wrought-iron sliaft of suitable 
diameter, usually from 3 to 8 ins., with 1.5 turns of a cast-iron thread of 
from 1.5 to 3 feet diameter. 

Hydraulic Process is effected by the direction of a stream of water under 
pressure, within a tube or around the base of a pile, by which the sand or 
earth is removed. 

Pneumatic and Plenum Process— For illustration and details, see Traut- 
wine’s Engineer’s Pocket-book, page 326. 

Dr. Whewell deduced the following results: 

1. A slight increase in hardness of a pile or in weight of a ram will con¬ 
siderably increase distance a pile may be driven. 

2. Resistance being great, the lighter a pile the faster it may be driven. 

3. Distance driven varies as cube of the weight of ram. 

Relative Resistance of Formations to Driving a Pile. 

Coral. 100 I Hard clay.60 I Light- clay and sand... 35 

Clay and gravel. 83 | Clay and sand. 45 | River silt. 25 


PNEUMATICS.—AEROMETRY. 

Motion of gases by operation of gravity is same as that for liquids. 
Force or effect of wind increases as square of its velocity. 

If a volume of air represented by 1, and of 32 0 , is heated t degrees without 
assuming a different tension, the volume becomes (1 + .002088 t) =V; and 
if it requires a temperature in excess of t 32 0 , it will then assume volume 
(1 + .002088 t‘ — 32 0 ). All aeriform fluids follow this law of dilatation as 
well as that of compression proportional to weight. 

When air passes into a medium of less density, its velocity is determined 
by difference of its densities. Under like conditions, a conduit will discharge 
30.55 times more air than water. 

To Compute tlie Degree of Rarefaction th.at may "be ef¬ 
fected. in a Vessel. 

Let quantity of air in vessel, tube, and pump be represented by 1, and 
proportion of capacity of pump to vessel and tube by .33; consequently, it 
contains .25 of the air in united apparatus. 

Upon the first stroke of piston this .25 will be expelled, and .75 of original 
quantity will remain; .25 of this will be expelled upon second stroke, which 
is equal to .1875 of original quantity; and consequently there remains in 
apparatus .5625 of original quantity. Proceeding in this manner, following 
Table is deduced: 


No. of Strokes. 

Air Expelled at each Stroke. 

Air Remaining in Vessel. 

I 

.25 = .25 

' -75 = -75 


3 _ 3 

9 _3 X 3 

2 

16 4X4 

16 4X4 


9 3 X 3 

27_ 3X3X3 

3 

64 4X4X4 

64 4X4X4 


And so on, multiplying air expelled at preceding stroke by 3, and dividing 
it by 4; and air remaining after each stroke is ascertained by multiplying 
air remaining after preceding stroke by 3, and dividing it by 4. 

3 L 














PNEUMATICS.-AEROMETRY. 


674 


Distances at which. Different Sounds are AAudihle. 


Feet. 


A full human voice speaking in open air, calm. 

In an observable breeze, a powerful human voice with the 

wind can be heard.... 

Report of a musket. 

Drum. 

Music, strong brass band. 

Cannonading, very heavy. 


.. 460 

| 15 840 

.. 16 000 

.. io 560 
.. 15840 

.. 575000 


Miles. 

.087 

3 


3.02 

2 

3 
90 


In Arctic Ocean, conversation has been maintained over water a distance 
of 6696 feet. 

In a conduit in Paris, the human voice has been heard 3300 feet. 


For an echo to be distinctly produced, there must be a distance of 55 feet. 


Coefficients of Efflux of Discharge of Air. (D' Aubuisson.) 


Orifice in a thin plate.65 .751 

Cylindrical ajutage.93 .958 

Slight conical ajutage.94 1.09 


To Compute Volume of ATr Discharged, through, an Open¬ 
ing into a Vacuum, per Second. 

a C V 2 g li = V in cube feet, a representing area of opening in square feet , C co¬ 
efficient of efflux , and V2 g h = 1347.4, as shown at page 428. 


Illustration. —Area of opening 1 foot square, and C = .707. 

Then 1 x -707 X 1347.4 == 952.61 cube feet. 
Inversely , V - 4 - a = velocity in feet per second. 


"Velocity and Pressure of Wind. 

Pressure varies as square of velocity, or P cc V 2 . 

V 2 X-oo5r=P; y/2ooV — Y\ v 2 X .0023 = P; and .0023 v 2 sin. x = P. 

V representing velocity in miles per hour , v in feet per second , P pressure in lbs. 
per sq.foot, and x angle of incidence of wind with plane of surface. 


Table deduced from above Formulas. 


Velocity 

Pressure 


Velocity 

Pressure 

Character of the 

T> er 

per 

on a 

Character of the Wind. 

per 

per 

on a 

Hour. 

Minute. 

Sq. Foot. 


Hour. 

Minute. 

Sq. Foot. 

Wind. 

Miles. 

Feet. 

Lbs. 


Miles. 

Feet. 

Lbs. 


I 

88 

.005 

Barely observable. 

25 

2200 

3-125 

Very brisk. 

2 

3 

176 

264 

.02 ) 
•045 J 

Just perceptible. 

3 ° 

35 

2640 

3080 

4-5 \ 

6.125) 

High wind. 

4 

352 

.08 

Light breeze. 

40 

3520 

8 

Very high wind. 

5 

6 

440 

528 

•125) 

.18 

Gentle, pleasant 
wind. 

45 

50 

396° 

44OO 

10.125 

12.5 

Gale. 

Storm. 

8 

7°4 

• 32 ) 

60 

5280 

18 

Great storm. 

IO 

880 

• 5 

Fresh breeze. 

80 

7040 

32 

Hurricane. 

15 

1320 

1.125 

Brisk blow. 

90 

7920 

40.5 \ 

Tornado. 

20 

1760 

2 

Stiff breeze. 

IOO 

8800 

50 \ " 


Illustration. —What is pressure per sq. foot, when wind has a velocity of 18 
miles per hour? l8 » x .005 = i bs . 


To Compute Force of Wind upon a Surface. 

- — a S ' IL —— = P. v representing velocity of wind in feet per second, a area of 
440 

surface in sq.feet, and A angle of incidence of wind. 

At Mount Washington wind has been observed to have had a velocity of 150 miles 
per hour. 

Extreme pressure of wind at Greenwich Observatory for a period of 20 years was 
41 lbs. per sq. foot. 






























PNEUMATICS.-AEROMETRY. 


675 


Force of wind upon a surface, perpendicular to its direction, has been ob¬ 
served as high as 57.75 lbs. per sq. foot; velocity = 159 feet per second. 

Dr. Hutton deduced that resistance of air varied as square of velocity 
nearly, and to an inclined surface as 1.84 power of sine X cosine. 

Figure of a plane makes no appreciable difference in resistance, but con¬ 
vex surface of a hemisphere, with a surface double the base, has only half 
the resistance. 

At high velocities, experiments upon railways show that the resistance 
becomes nearly a constant quantity. 


Course of* "Wind. 


Direction in 
Northern Hemisphere. 


Cyclones. 


Direction in 
Southern Hemisphere. 



Wind has its direction nearly at 
right angles to line between points of 
highest and lowest pressure of air, or 
barometer readings, and its course is 
with the point of lowest pressure at 
its left, and its velocity is directly as 
difference of the pressures. 




In Northern Temperate zone, winds course around an area of low pressure 
in reverse direction to course of hands of a watch, and they flow away from 
a location of high pressure, and cause an apparent course of the winds in di¬ 
rection of course of the hands. 


To Compute Resistance of a Plane Surface to Air. 

.0022 a v 2 = P in lbs. a representing area of plane in sq.feet, v velocity in direc¬ 
tion of wind in feet per second, -j- when it moves opposite, and — when with the wind. 


To Compute Resistance of a Plane Surface when moving 
at an Angle to Air. 


v 2 a sin. 2 x 


= P in lbs. 


45o 


x representing angle of incidence. 


To Compute Height of a Column of Mercury to induce 
an iCfflux of Air through, a given Nozzle. 

Barometer assumed at 2.46 feet = 29.52 ins., and Temperature 52 0 . 


—, - : —- — H, and 48.073 d 2 fH. = P. d representing diameter of nozzle ancl H 

48.073 2 d 4 

height of column of mercury, both in feet, and P volume of air in lbs. per one second. 

Illustration. — Assume d = .19, and P = .7 lbs. 

_ 2 

= .1511 foot. 48.073 X .ig 2 -if.1511 — 


48.073 2 X -i 9 4 


To Compute Pressure or AVeight of Air under a given 
Height of Barometer and Temperature, Discharged in 
One Second. 

30.787 d 2 \J& =.pressure in lbs. Or, 48.073 d 2 fB = lbs. b representing 

height of barometer in external air, B manometer or pressure of air in reservoir in 
mercury, both in feet, and t temperature of air or gas in degrees. 


Illustration. —Assume 6 = 2.5 feet; d = . 25 foot; B = .1 foot; and < = 1.055 fed. 

Then 30.787 X .0625 ./.iX = 1.924 X V- 2 4^5 = -9543 M s ■ 

V 1.055 


1-055 









676 


PNEUMATICS.-AEROMETRY. 


To Compute Temperature for a given Latitude and. Ele¬ 
vation. 


82.8 cos. I — .001 981 E — .4 =zt. E representing elevation in feet. 
Illustration. —Assume Z = 45°; cos. =.707; and E = 656 feet. 


Then 82.8 X .707 
57.641. 


■.001 981 X 656 — .4 = 58.54 — 1.299 — -4 = 5 8 -54 — -899 = 


To Compute Volume of -A_ir or Gras Discharged through 
an Opening and under a Pressure above that of Ex¬ 
ternal ^Air. 

__ 

A ir. 1347.4 C — VB (b' + B) T = V in cube feet per second. 

T = 1 -|- .002 22 (t — 32 0 ), and b' — 2.5 — .00009 elevation. 

Or, 621.28 d 2 -^/B = V. 

Illustration. —What would be volume of air that would flow through a nozzle 
.246 foot in diam. from a reservoir under a pressure of .098 foot of mercury, into 
air under a barometric pressure of 2.477 feet, temperature of air 55.4 0 , location 45 0 
of latitude, and at an elevation of 650 feet above level of sea? 

C =.75; b' = 2.5 — .00009 X 650 = 2.4415 (2.44); and T = 1.0502. 

Then 1347.4 X .75 - 24 -— A/.098 (2.44 -f- -098) X 1.0502 = 24.689 X ^.2617 = 12.63 
2.477 

cube feet. 

When Densities of External Air and that in Reservoir are Equal. 

,d 2 


1347.4 C VB (6 -f- B) T = V. b' representing height of mercury in reservoir. 


Gas. 


4231 

VP 


Bds 


— V. p representing specific gravity of gas compared 


E -f- 42 X d 

with air, and L length of pipe or conduit in feet. 

Illustration. —If a pipe .05 feet in diameter and 420 feet in length, communi¬ 
cates with a gasometer charged with carburetted hydrogen (illuminating gas), under 
a water pressure as indicated by a manometer of .1088 foot, what would be the dis¬ 
charge per second ? 

1088 

d = .o$ foot; L = 420 feet; and B = -—— = .008 foot. Specific gravity of gas 
•5625- 

.008 X .05 s 4231 /.000000002 5000 

— ' .01371 cube foot. 


4231 

V • 5625 


4231 /■_ 

’ -75 V 


420 -j- 2.1 


420+ 42 X -05 

Resistance of Curves and Angles. —Curves and angles increase resistance 
to discharge of air or gas very materially. By experiment of D’Aubuisson 
7 angles of 45 0 reduced discharge of gas one fourth. 

To Compute Diameter of Discharge-pipe or TVozzle. 

When Length and Diameter of Pipe , Volume , and Pressure are given. 

42 V 2 , 

—- —- — d in feet. 

4230 2 B d b — I. v 2 

Illustration.—I f a pipe 1000 feet in length, and .4 foot in diameter, leads to a 
reservoir of air, under a mercurial manometric pressure of .18 foot, what diameter 
must be given to a nozzle to discharge 4 cube feet per second? 

42 X 4 2 X -45 . / 6.88128 

:= Z /-o- 1 ~ V • 000 4°5 2 — 

t 32980.19 — 16000 


Then 4 / — 

v 4230 2 X 18 X - 4 5 ■ 


1000 X 4 2 


. 1418 foot — 1.703 ins. 

Volumes of two gases flowing through equal orifices, and under equal pressures, 
are in inverse ratio of square roots of their respective densities. 


Specific gravity of mercury compared with water. 


















RAILWAYS. 


67; 


RAILWAYS. 

To Define a Curve.-Fig. 1 . 

1719 c 



(Molesworth.) 
or t tan. x = R; R (cotan. x) = t; 


1719 c 


= 


-, R (cosec. x — 1) = d) 

R (cosin. x) == s; R (coversin. x) = V; 

5400 — 


= w, and (5400 — x) .000 582 R = l. 

c representing any chord, £ length of tangent, d distance of centre of curve from, in¬ 
tersection of tangents, s half chord of curve, and l length of curve, all in like dimensions, 
a tangential angle of c in minutes, n number of chords in curve, and x half angle of 
intersection, but in formulas for number of chords and length of curve to be expressed 
in minutes. 

Illustration. —Assume radius 900 and chord 400 feet; angle of intersection = 
12 0 44' = 764 minutes, and x = 56° 15' 5". 

Tangent of 56° 15' 5" = 1.496 73. Cotangent = .66814. 


1719 x 400 
764 


R = 900 feet ; 


1719 X 400 
900 


601 .33 feet ; 900X1-20269 — 1 ==.cZ — 182.42 feet) 900 X -555 55 — s — 500 feet; 

5400 — 3379 


900 X • 168 33 = Y = 161.5 feet; 


764 


764 minutes ; 900 X • 66814 = t = 

5 55 = s = 500 feet; 
2.645 times, and .000582 X 900 X 


5400 — 3379 = 1058 .6 feet. 

Tangential Angles for Chords of One Chain. 


Radius of 
Curve. 

Tangential 

Angle. 

Radius of 
Curve. 

Tangential 

Angle. 

Radius of 
Curve. 

Tangential 

Angle. 

Radius of 
Curve. 

Tangential 

Angle. 

Chains. 


Chains. 


Chains. 




5 

5° 43-8' 

15 

i° 54-6' 

40 

42.97' 

1 mile 

21.48' 

8 

3° 34-87' 

20 

i° 25.95 

45 

38.2' 

1.25 mil’s 

17.19' 

9 

3° xi' 

25 

i° 8.76' 

50 

34-38' 

1.5 miles 

14-33 

IO 

2° 51.9' 

30 

57-3', 

60 

28.65' 

i-75 “ 

12.28 

12 

2° 23.25' 

35 

49. IX 

70 

24-55' 

2 “ 

IO - 74' 


Note. —Angle for 2 chain chords is double angle for 1 chain chords. Angle for .5 
chain chords is . 5 the angle for 1 chain chords. 

Curves of less than 20 chains radius should be set out in . 5 chain chords. Curves 
of more than 1 mile radius may be set out in 2 chain chords. 

Angles in above Table are in degrees, minutes, and decimals of minutes. 

Sidings. 



2 V d R — (.5 d) z — l. R representing radius of 
curve, l length of curve over points, and d distance 
between tracks, 

all in feet. * '£>• 3 * 

'i\ t 


Turn-out of* TJneqnal Radii. 
r x 


■ y, x—y = z ; a + 6 = Z; r- 


R + r 

Vy (r 4 - A) = a; R — z = B] Vz(R + B) 

R and r representing radii of the curves re¬ 
spectively as to length, x distance between outer 
rails of tracks and other symbols as shown , all 











































678 


RAILWAYS. 


Points and. Crossings. 


Fig. 4. 


V(R + x ) G = l] 


l 


:sin. a: 


G 


R. R repre- 


R ' ver. sin. a 

senting radius of curves, G gauge of road, a angle of crossing, 
and x = R — G, all in feet. 

In horizontal curves, width required for clearance of 
flange of wheel, and for width of rail at heel of switch, 
render it necessary to make an allowance in length of l, 
as ascertained by formula. 

For other diagrams and formulas, see Moleswortlfs Pocket- 
book, pp. 208-18, 21 st edition. 

1719 c 

To Compute Tangential Angle for Curves. ——— = a. c 

XV 

representing chord in feet, and a angle in minutes. 

Illustration. —What is angle for a curve with a radius of 900 feet, and a chord 
of 400 feet? 

I 7 I 9 X 400 _ ^ minutes. 



90Q 

Carving of Rails. 


1.56 l 2 


R 


: v. I representing length of rail in feet , v versed sine at centre, when 
curved, in ins. 

Illustration _What is curve for a rail 20 leet jn length, with a radius of 900 feet ? 

1.5 X 20^ 


900 


=.666 ins. 


Carves Toy Offsets in Equal Chords. 

Chord 2 



Chord 2 

— = 0 offset. 


— 2,0 offset. 


2 R R 

Illustration.—A ssume chords 150, andra- 


22 500 22 500 ,. , 

12.5 feet; - _ = 25 feet. 


2 X 900 900 

To Compate Versed Sines and Ordinates of Carves. 
F J g‘ 7 R — \/R 2 — ( .5 C) 2 = a - ( ‘ sC)2 


■ -j-v = D : and 

k x >\o \ _ ’ ' v 

vR 2 — x 2 — (R— v) = o. D representing diameter of 
• v \ circle, and v versed sine of curve. 

\ Illustration.—A ssume radius 900, and chord 400 feet. 



D 


900 — V 810000 — 40 000 = 900 — 877.5 = 12.5 feet. 


Relation of Rase of Driving or Rigid Wheels to Carve. 

R 

rqr —■ B. R representing minimum radius of curve, G gauge of road, and B base , 
all in feet. 

To Compate Elevation of Oater Rail. 

For any Radius or Combination of Curve with Straight Line. 

•5 F ffG — c. V representing velocity of train in feet per second, G gauge of road, 
and c length of a chord, both in feet, the versed sine of which = elevation in ins. 


V 2 

1.25 R 


On Carves. 

G = E. E representing elevation of outer rail in ins. 
























RAILWAYS. 


679 


Radii of Curves set cut in. Tangential Angles. 


Angle for 
Chord of 
100 Feet. 

Radius 

of 

Curve. 

Angle for 
Chord of 
100 Feet. 

Radius 

of 

Curve. 

Angle for 
Chord of 
100 Feet. 

Radi us 
of 

Curve. 

Angle for 
Chord of 
100 Feet. 

Radius 

of 

Curve. 

0 ' 

Feet. 

O ' 

Feet. 

O ' 

Feet. 

O ' 

Feet. 

3 ° 

57 2 9 - 6 

2 30 

H 45-9 

4 30 

636.6 

6 30 

440.7 

I 

2864.8 

3 

954-9 

5 

573 

7 

4 ° 9-3 

1 30 

1909.9 

3 30 

818.5 

5 3 ° 

520.9 

7 30 

382 

2 

1432.4 

4 

716.2 

6 

447-5 

8 

358 -1 


Note. —If chords of less length are used, radius will be proportional thereto. 

To Ascertain Radius of Curve in Inches for Scale, in Feet per Inch. 
Divide radius of curve in feet by scale of feet per inch. 

To Compute Required "Weight of Rail. 

Rule.— Multiply extreme load upon one driving-wheel in lbs. by .005, 
and product will give weight of rail in lbs. per yard. 


To Compute Radius of Curve aiad "Wheel Base. 

J> 

9 B G = R. —— = P>. B representing maximum rigid wlieel base of cars , and G 

9 G 

gauge of way , both in feet. 

To Determine Elevation of Outer Rail. 

For any Radius or Construction of Curve with Straight. —Fig. 7. 

Fig. 7, V .5 fG — c. V representing speed of train in feet per sec¬ 

ond, G gauge of rails in feet, and c length of chord, versed sine 
v of which will give at its centre the elevation required. 

Thus, determine chord c, align it on inner 
side of rail, and distance of rail from it at 
centre of its length will give elevation re¬ 
quired, whatever the radius of rail. 

V s 



For Curves, 


N D R 


= E - ° r ' W ^lR = E ’ 


D representing 


diameter of wheels, W width of gauge, P lateral play between flange and rail, and 
R radius of curve, all in feet, i 4 -N ratio of inclination of tire, V velocity of train in 
miles per hour, and E elevation of outer rail in ins. (Molesworth.) 

WC (d + I) _ resigtance due t0 curve> an d tV representing weight of body, both in 


2 R 


lbs., C coefficient of friction of wheels upon rails = .1 to .27, according to condition of 
weather, d distance of rails apart, l length of rigid wheel base, and R radius of curve, 
all in feet. (Morrison.) 


Illustration.— Assume weight of locomotive 30 tons, radius of curve 1000 feet, 
distance of rails apart 4 feet 8.75 ins., length of base 10 feet, and rails, dry, C = i. 

494.93 lbs. 


30 X 2240 X ■ 1 X (4- 73 4 ~ 10) . 
2 X 1000 


To Compute Resistance due to Gravity upon an In¬ 
clination. 


2240 

gradient 


— lbs. per ton of train. 


Rise per Male, and Resistance to Gravity, in Lbs. per 

Ton. 


Gradient of 1 inch.. 

20 

25 

30 

35 

40 

45 

Rise in feet. 

264 

211 

176 

151 

132 

117 

Resistance. 

112 

89.6 

74-7 

64 

56 

50 


50 

60 

70 

80 i 90 

IOO 

106 

88 

75 

66 59 

53 

44.8 

37-3 

32 

28 124.8 

22.4 
















































68 o 


RAILWAYS. 


To Compute Load, which. a Locomotive will Draw up 

an Inclination. 

T- 4 -r + r'— W = L. T representing tractive power of locomotive in lbs., r re¬ 
sistance due to gravity, and r' resistance due to assumed velocity of train in lbs. per 
ton, W weight of locomotive and tender, and L load locomotive can draw, in tons, ex¬ 
clusive of its own weight and tender. 

Coefficients of Traction of Locomotives. —Railroads in good order, etc., 4 to 6 lbs.; 
in ordinary condition, 8 lbs. 

In coupled engines adhesion is due to load upon wheels coupled to drivers. 


To Compute Traction, Retraction, and Adhesive Rower 
of a Locomotive or Train. 

When upon a Level. asP-rD = T. a representing area of one cylinder in 
sq. ins., s stroke of piston and D diameter of driving-wheels, both in feet, P mean 
pressure of steam in lbs. per sq. inch, and T traction, in lbs. 

When upon an Inclination. asP-PD — r w h — T. r representing resistance 
per ton, w weight of locomotive upon driving-ivheels, in tons, h height of rise in feet 
per 100 of road, and R = r w h — retraction, in lbs. 

C iv b ~ 100 = A. b representing base of inclination in feet per 100 of road. 

C w = A. C = coefficient in lbs. per ton, and A adhesion, in lbs. 

When Velocity of a Train is considered. 

When upon a Level, W (C + 3/V) = R. When upon an Inclination, 
W (r h + C + V V) — R. V representing velocity of train in miles per hour. 

Illustration. —A train weighing 200 tons is to be driven up a grade of 52.8 feet 
per mile, with a velocity of 16 miles per hour; required the retractive power? 

52.8 per mile = 1 in 100 feet = r — 22.4 lbs. 0 = 5. 

200 (22.4X 1 —|— 5 —j— 1:6) = 200 X 22. 4 -f- 9 = 6280 lbs. 

Velocity of Trains. 


Miles per hour. 


IO 

15 

20 

30 

40 

50 

60 

70 

Resistance upon straight 
line per ton. 


Lbs. 

8-5 

Lbs. 

9- 2 5 

Lbs. 

10.25 

Lbs. 

13-25 

Lbs. 

i 7- 2 5 

Lbs. 

22.5 

Lbs. 

29 

Lbs. 

36-5 

Do., with sharp curves 
and strong wind*_ 


13 

14 

15-5 

20 

26 

34 

43-5 

55 


* Equal to 50 per cent, added to resistance upon a straight line. 

Friction of locomotive engines is about 9 per cent., or 2 lbs. per ton of weight. 
Case-hardening of wheel-tires reduces their friction from .14 to .08 part of load. 


To Compute Maximum Load tliat can "be drawn by an 
Engine, up the Maximum Grade tliat it can Attain, 
"Weight and Gfrade being given. (Maj. McClellan, U.S.A.) 


-~ 2 ———^ L, and - 2 ^ = G. A representing adhesive weight of engine, 

in lbs., G grade in feet per mile, and L load, in tons. 

Note r.—When rails are out of order, and slippery, etc., for .2 A, put .143 A. 

2. —With an engine of 4 drivers, put .6 as weight resting upon drivers; with 6 
drivers the entire weight rests upon them. 

Illustration.— An engine weighing 30 tons has 6 drivers; what are the maximum 
loads it can draw upon a level, and upon a grade of 250 feet, and what is its maxi¬ 
mum grade for that load ? 


.2X2240X3° 13 44 ° 


; 1595.4 tons upon a level. 


.2X2240X30 13440 


.4242-j-8 8.4242 

117.8 tons up a grade of 250 feet. 

Adhesion of a 4-wheeled locomotive, compared with one of 6 wheels, is as 5 to 8. 


4252X250 + 8 114.05 

.2X2240X30 — 8X117.8 12497 . , 

-—-5 - = - -= 250.1 Jeet. 

.4242X117-0 49-97 



























RAILWAYS. 


681 


OPERATION OF LOCOMOTIVES. ( 0 . Chanute, Am. Soc. C. E.) 

.A. cohesion. 

Adhesion of a locomotive is friction of its driving-wheels upon the rails, 
varying with condition of the surface, and must exceed traction of the engine 
upon them, otherwise the wheels will slip. 

Improvements heretofore made in the construction of locomotives and 
tracks have gradually increased the proportion which the adhesion bears to 
the insistent weight upon the driving-wheels. 

The first accurate experiments were those of Mr. Wood upon the early English 
coal railways. He deduced the adhesion to be as follows: 

Upon perfectly dry rails.14 of weight on drivers. 

“ damp or muddy rails.08 “ “ “ “ 

“ very greasy rails.04 “ “ “ “ 

In 1838, B. H. Latrobe indicated .13 as a safe working adhesion, while modern 

European practice assumes about .2 of weight as maximum, and .11 as a minimum, 
except perhaps in some mountainous regions, subject to mists. Thus, on the Soem¬ 
mering line, adhesion is generally .16, and between Pontedecimo and Busalla, in 
Italy, it never exceeds .12 in open cuttings, or .1 in tuunels. 

Extensive experiments made upon French railways, 1862-67, by Messrs. Vuille- 
min, Guebhard, and Dieudonne gave following coefficients in actual working: dry 
weather, extreme, .105 to .2; damp, .132 to .139; wet, .078 to .164; light rain , .09; 
extreme rain, .109 to .2, mean, .13; rain and fog, .115 to .14; heavy rain, 16. 

Materially better results are obtained in United States, partly, perhaps, in con¬ 
sequence of greater dryness of the weather, and certainly because of the American 
method of construction and equalizing the weight between the drivers, and of mak¬ 
ing the locomotive so flexible as to adapt itself to inequalities in the track. 

Modern engines in America can safely be relied upon to operate up to an adhesion 
equal to .222 in summer and .2 in winter, of weight upon the driving wheels. 

From these data the following tables have been computed: 


Coefficients of Adhesion upon Driving Wheels per Ton. 


Condition of Rails. 

European 

Practice. 

American 

Practice. 

Condition of Rails. 

European 

Practice. 

American 

Practice. 


C. 

Lbs. 

C. 

Lbs. 


C. 

Lbs. 

C. 

Lbs. 

Rails very dry. 

•3 

670 

•33 

667 

In misty weather . 

.015 

350 

.2 

400 

Rails very wet. 

.27 

600 

•25 

500 

In frost and snow. 

.09 

200 

. l6 

333 

Ordinary working.. 

.2 

450 

.222 

444 







Adhesion of Locomotives , in Lbs. (.222 in Summer and .2 in Winter). 


Type of Locomotive. 

No. of Drivers. 

Wei 

Locomotive. 

?bt. 

On Drivers. 

Adhc 

Summer. 

sion. 

Winter. 




Lbs. 

Lbs. 

Lbs. 

Lbs. 

American. 

4 

wheels coupled.... 

64000 

42 OOO 

9 350 

8 400 

Ten-wheeled. 

6 

“ connected.. 

78 000 

58 000 

13 OOO 

11 600 

Mogul. 

6 

U U 

88 000 

72 OOO 

16 OOO 

14 OOO 

Consolidation. 

8 

u a 

IOO OOO 

88 000 

19550 

17 600 

Tank switching.... 

6 

u u 

68 000 

68 000 

15 IOO 

13 600 

u u 

4 

u u 

48 000 

48 000 

10650 

9 600 


Tractive Power. 

Traction of a locomotive is the horizontal resultant on the track of the 
pressure of the steam, as applied in the cylinders. 

D 2 PL-f-W = T. D representing diameter of cylinder, L length of stroke, and W 
diameter of driving wheels, all in ins. , P mean pressure in cylinder , in lbs. per sq. 
inch, and T tractive force on rails, in lbs. 

Illustration. —Assume a locomotive, cylinders 18 ins. in diam., 22 ins. stroke, 
wheels 68 ins. in diam., and average steam pressure in cylinders 50 lbs. per sq. inch. 

Then 18 X 18 X 50 X 22 - 4 - 68 = 5241 lbs. » 







































682 


RAILWAYS. 


Train Resistances. 

Usual formula for train resistances, on a level and straight line , is 
V 2 V 2 

2 _L 8 = R per ton of train, and- 1 - 6 = R per ton of train alone. V repre- 

171 240 

senting velocity in miles per hour , and 8 constant axle friction. (D. K. Clark.) 

Note.— To meet the unfavorable conditions of quick curves, strong winds, and 
imperfection of road, Mr. Clark estimates results as obtained by above formula 
should be increased 50 per cent. 

Illustration.— At 20 miles per hour, the resistance would be: 

20 2 - 4 - 171 + 8 = 10.3 lbs. per ton of train. 

This formula, however, is empirical. It gives results which are too large for 
freight trains at moderate speeds, and too small for passenger trains at high speeds. 

Engineers are not agreed as to exact measure and value of each of the elements 
of train resistances, but following approximations are sufficient for practical use: 

Analysis of Train Resistances. 

Resistance of trains to traction may be divided into four principal ele¬ 
ments : 1st. Grades; 2d. Curves • 3d. Wheel friction; 4th. Atmosphere. 

1st. Grades. — Gradients generally oppose largest element of resistance 
to trains. Their influence is entirely independent of speed. The meas¬ 
ure of this resistance is equal to weight of train multiplied by rate of in¬ 
clination or per cent, of grade. Thus, a gradient of .5 per 100 feet (26.4 

feet per mile) offers a resistance of 5 x - 2 — = 11.2 lbs. per ton, or 10 lbs. 

per 2000 lbs., which is to be multiplied by weight in tons of entire train. 

Following table shows resistance, due to gravity alone, for the most usual grades, 
in lbs. per ton of train: 


1st. Resistance due to Grades. 


Rate per 100 feet. 

. I 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

Lbs. per ton of 2240 lbs... 

2.24 

4.48 

6.72 

8.96 

II .2 

13'44 

15.68 

17.92 

Rate per mile. 

5 

II 

l6 

21 

26 

32 

37 

42 

Lbs. per ton of 2000 lbs... 

2 

4 

6 

8 

IO 

12 

14 

l6 

Rate per 100 feet. 

•9 

I 

I. I 

1.2 

i -3 

1.4 

i-5 

1.6 

Lbs. per ton of 2240 lbs... 

20.16 

22.4 

24.64 

26.88 

29.12 

31-36 

33-6 

35-84 

Rate per mile. 

47 

53 

58 

63 

68 

74 

79 

85 

Lbs. per ton of 2000 lbs... 

18 

20 

22 

24 

26 

28 

30 

32 


2d. Curves .—Recent European formula is that given by Baron von Weber. 

.6504 -4- R — 55 = W. R representing radius of curve in metres. 

This formula assumes that resistance due to curve increases faster than radius 
diminishes. It gives results varying from a resistance of .8 lb. per 2000 lbs. per 
degree for a curve of 1000 metres radius (3310 feet, or i° 44') to a resistance of 1.67 
lbs. per 2000 lbs. per degree for curves of 100 metres radius (331 feet, or 17 0 20'). 

Messrs. Yuillemin, Guebhard, and Dieudonn6 found curve-resistance to European 
rolling-stock to be from .8 to 1 lb. per 2000 lbs. per degree, on a gauge of 4 feet 8.5 
ins., while Mr. B. H. Latrobe, in 1844, found that with American cars resistance on 
a curve of 400 feet radius did not exceed .56 lb. per 2000 lbs. per degree. 

Resistance of same curve varies with coning given tires of wheels, elevation of 
outer rail, and speed of train running over it, but both reasoning and experiment 
indicate that the general resistance of curves increases very nearly in direct pro¬ 
portion to degree of curvature, or inversely to the radius. 

Recent American experiments show that a safe allowance for curve resistance 
may be estimated at .125 of a lb. per 2000 lbs. for each foot in width of gauge. 
Thus, for 3 feet gauge resistance would be .375 lb. per degree of curve; for standard 
gauge of 4 feet 8.5 ins. .589, say .60, and for 6 feet gauge .75 lb. per degree. 

For standard gauge, when radius is given in feet, resistance due to this element is: 

.60 X 5730 - 4 - R = C in lbs. per ton of train. 


























RAILWAYS. 



This is somewhat reduced when curve coincides with that for which wheels are 
coned (generally about 3 0 ), and when train runs over it, at precise speed for which 
outer rail is elevated, an allowance of .5 lb. per ton per degree is found to give good 
results in practice. 

2d. Resistance on Curves. 

It follows from above estimate of curve resistance that, in order to have the same 
resistance on a curve as on a straight line, the gradient should be diminished by 
.03 per 100 feet of each degree of curve. Thus a 3 0 curve requires an easing of the 
grade by .09 per 100 feet, a io° curve an easing of .3 per 100, etc. 

This, however, need only be done upon the limiting gradients, and when sum of 
grade and curve resistances exceeds resistance which has been assumed as limiting 
the trains. 

3d. Resistance due to Wheel Friction. 

Experimenters are not agreed whether friction of wheels increases simply with 
weight which they carry, but also in some ratio with the speed. Originally as¬ 
sumed as a constant at 8 lbs. per ton, improvements in condition of track (steel 
rails, etc.) and in construction and lubrication of rolling-stock have reduced it to 
3.5 and 4 lbs. per ton for well-oiled trains. Under ordinary circumstances, in sum¬ 
mer, it will be safe to estimate it at 5 lbs. per ton on first-class tracks, and 6 lbs. 
per ton on fair tracks. It may run up to 7 or 8 lbs. per ton on bad tracks (iron 
rails) in summer, and all these amounts should be increased from 25 to 50 per cent, 
in cold climates in winter, to allow for inferior lubrication. 

4th. Resistance due to Atmosphere. 


Atmospheric resistance to trains, complicated as it is by the wind which may be 
prevailing, has not been accurately ascertained by experiment. It consists of— 
1st. Head resistance of first car of train, which is presumably equal to its exposed 
area, in sq. feet, multiplied by air pressure due to speed. 

2d. Head resistance of each subsequent car. This varies with distance they are 
coupled apart, and so shield each other from end air pressure due to speed. 

3d. Friction of air against sides of each car depending upon the speed. This is 
generally so small that it may be neglected altogether. 

4th. Effect due to prevailing wind, which modifies above three items of resistance. 
A head wind retards the train, a rear wind aids it, while a side wind increases re¬ 
sistance by pressing flanges of wheels against one rail, and, in consequence of curves, 
a train may assume all of these positions to same wind. 

Recent experiments on Erie Railway seem to indicate that in a dead calm re¬ 
sistance of first car of a freight train may be assumed at an exposed surface of 63 
sq. feet,* multiplied by air pressure due to speed, and that each subsequent car may 
be assumed to offer a resistance of 20 per cent, of that of first car, while in a pas¬ 
senger train first car may be assumed at an area of 90 sq. feet,t multiplied by air 
pressure due to speed, and that each subsequent car adds an increment equal to 40 
per cent, that of first car, in consequence of greater distance they are coupled apart. 

This resistance is, of course, entirely independent of cars being loaded or empty. 
In practice it has been found that an allowance of 1.5 to 2 lbs. per ton of weight of 
a freight train covers atmospheric resistance, except in very high winds. 

In consequence of complexity of elements above enumerated, exact formulas can¬ 
not probably be now given for train resistances, but following, if applied with judg¬ 
ment (and modified to fit circumstances), will be found to give fairly accurate results 
in practice. They are for standard gauge, and in making them, curve resistance has 
been assumed at .5 lb. per degree, wheel friction at 5 lbs., exposed end area of first 
car at 90 sq. feet for passenger cars and 63 feet for freight cars, and increment for 
succeeding cars at .4 for passenger trains and .2 for freight trains. 

Passenger Train. W -f- — — b 5^ * 4 * 4 9° ^ 

Freight Train. W -f" — + 5^ + H “ ^ 63 P = R. 


* This is less than area of car, which generally measures about 71 sq. feet; but part is shielded by 
tender, and parts being convex, as wheels, bolts, etc., offer less resistance than a flat plane. 

+ Not only is end area of passenger cars greater than that of freight cars, but in consequence of the 
projecting roof the end forms a hood in nature of a concave surface, and so opposes greater resistance 
than a flat plane. 







RAILWAYS. 


684 


W representing weight of train, without engine, in tons (2000 lbs.), G resistance of 
gradient per ton (2000 lbs.; see table, page 683), C 0 curve in degrees, n number of cars 
in train, ¥ pressure per sq. foot due to speed, to which an allowance must be made for 
wind, if existing, R resistance of train, and 5, wheel friction, both in lbs. 

Illustration i. —Assume a passenger train of 5 cars, weighing 136 tons (2000 lbs.), 
ascending a grade .5 per 100 (26.4 feet per mile), with curves of 4°, at a speed of 60 
miles per hour (for which the pressure is 18 lbs. per sq. foot), resistance will be: 

136 (10 + 2 + 5) + ^1 + —^ (90 X 18) = 6524 lbs., of which 2312 lbs. are due to 

grade, curve, and wheels, and 4212 lbs. to atmospheric resistance. 

2.—Assume a freight train of 31 cars, weighing 620 tons (2000 lbs.), turning a curve 
of 3 0 , up a grade of 52.8 feet per mile (1 foot per ioo), at a speed of 21 miles per hour 
(pressure 2 lbs. per sq. foot), resistance will be: 

620 (20-)- 1.5 + 5) + -j- yj (63 x 2) = 17 312 lbs., requiring a “ Consolidation ” 

engine to haul it, allowance being made for possible winds, etc. 

Assume conversely, it is desired to know how many tons an American engine, 
with an adhesion of 10650 lbs., will draw up a grade of .9 per 100 (47 feet per mile), 
with curves of 4 0 , assuming atmospheric resistance between 1.5 to 2 lbs. per ton of 
train. 

Resistance from grade .9 X 2000-=-100.=18 lbs.) 

“ “ curve 4-4-2. - 2 “ [27 lbs. 

“ “ wheel friction 5, atmosphere 2.— 7 “ ) 

Hence, 10650-4-27 = 395 tons, or about 20 cars, and in winter same engine will 
haul 960027 = 355 tons (2000 lbs.), or about 18 cars. 

Following table approximates to best modern practice. Tor freight trains it gives 
aggregate resistance, in lbs. per ton (2000 lbs ), for various grades and curves. In 
using it, it is sufficient to divide the adhesion in lbs. of locomotive used by number 
found in table, in order to obtain number of tons of train that it will haul at or¬ 
dinary speeds on gradient and curve selected. Of course, if grade has been equated 
for curves, only number found in first column (for straight lines) is to be used in 
computing tons of train on limiting gradient. 

Approximate ZEFreiglit-train. Resistances. 

Gauge 4 feet 8.5 ins. 

In Lbs. per 2000 lbs. at Ordinary Speeds. 

Curve Resistance assumed at .5 lbs. per °, Wheel Friction at 5 lbs., Atmospheric Re¬ 
sistance at 2 lbs. per Ton. 


Gra 

Per 

Cent. 

DE. 

Per 

Mile. 

Straight. 


2 ° 

3 ° 

4 ° 

5 ° 

6 ° 

c 

7 ° 

UR V] 

8 ° 

2 . 

9 ° 

10 ° 

11 0 

12 ° 

i 3 ° 

14° 

15* 



lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

Level. 

Feet. 

7 

7-5 

8 

8-5 

9 

9-5 

IO 

10.5 

II 

n -5 

12 

12.5 

I 3 

x 3-5 

*4 

14-5 

.1 

5 

9 

9-5 

IO 

10.5 

II 

n -5 

12 

12.5 

x 3 

I 3-5 

14 

x 4-5 

i 5 

x 5-5 

l6 

16.5 

.2 

II 

II 

n -5 

12 

12.5 

I 3 

x 3-5 

r 4 

x 4-5 

x 5 

15-5 

l6 

16.5 

I 7 

x 7-5 

18 

18.5 

• 3 

l6 

13 

i 3-5 

x 4 

* 4-5 

15 

15-5 

l6 

16.5 

17 

* 7-5 

18 

18.5 

J 9 

* 9-5 

20 

20.5 

•4 

21 

15 

i 5-5 

l6 

16.5 

x 7 

* 7-5 

l8 

18.5 

x 9 

* 9-5 

20 

20.5 

21 

21.5 

22 

22.5 

• 5 

26 

17 

* 7-5 

18 

18.5 

r 9 

* 9-5 

20 

20.5 

21 

21.5 

22 

22.5 

23 

23-5 

24 

24-5 

.6 

32 

*9 

19-5 

20 

20.5 

21 

21.5 

22 

22.5 

23 

23-5 

24 

24-5 

25 

25-5 

26 

26.5 

•7 

37 

21 

21-5 

22 

22.5 

23 

23'5 

24 

24-5 

25 

25-5 

26 

26.5 

27 

27-5 

28 

28.5 

.8 

42 

23 

23-5 

24 

2 4-5 

25 

25-5 

26 

26.5 

27 

27-5 

28 

28.5 

29 

29-5 

30 

3°-5 

■9 

47 

25 

25-5 

26 

26.5 

27 

275 

28 

28.5 

29 

29-5 

30 

30-5 

3 X 

3 r -5 

32 

32.5 

I 

S 3 

27 

27-5 

28 

28.5 

29 

29-5 

3 ° 

30-5 

3 X 

3 J -5 

32 

3 2 -5 

33 

33-5 

34 

34-5 

I. I 

58 

29 

29-5 

30 

30-5 

31 

3 i -5 

32 

32-5 

33 

33-5 

34 

34-5 

35 

35-5 

36 

36-5 

1.2 

63 

31 

3 i -5 

32 

32-5 

33 

33-5 

34 

34-5 

35 

35-5 

36 

3 6 -5 

37 

37-5 

38 

38.5 

i -3 

68 

33 

33-5 

34 

34-3 

35 

35-5 

36 

36-5 

37 

37-5 

38 

38-5 

39 

39-5 

40 

40-5 

I -4 

74 

35 

35-5 

36 

36-5 

37 

37-5 

38 

38.5 

39 

39-5 

40 

40-5 

4 1 

4 x -5 

42 

42-5 

i -5 

79 

37 

37-5 

38 

38-5 

39 

39-5 

40 

40-5 

4 i 

4 x -5 

42 

42-5 

43 

43-5 

44 

44-5 

1.6 

85 

39 

39-5 

40 

4°-5 

4 i 

4 i -5 

42 

42-5 

43 

43-5 

44 

44-5 

45 

45-5 

46 

46-5 


Illustration.—A ssume a “Mogul 
what weight will it haul up a grade of 

16 OOO -r- 39.5 


” engine to have an adhesion of 16 
74 feet per mile, combined with a cur 

= 405 tons (2000 lbs .). 


000 lbs.; 
ve of 9 0 ? 






































RAILWAYS. 685 

Hence, To Compute Adhesion on a Given Grade and Curve, having Weight 
of Train. 

Rule.— Multiply tabular number by weight of train in tons (2000 lbs.), 
and product will give adhesion, in lbs. 

Example.— Assume preceding elements. Then 39.5 X 405 = 16000 lbs. 

Note.— A “Consolidation” engine, by its superior adhesion (19550 lbs.) would 
haul up a like grade and curve 495 tons. 


Memoranda 011 English Railways. 

Regulations (Board of Trade). 

Cast-iron girders to have a breaking weight = 3 times permanent load, added to 
6 times moving load. 

Wrought-iron bridges not to be strained to more than 5 tons per sq. inch. 
Minimum distance of standing work from outer edge of rail at level of carriage 
steps, 3.5 feet in England and 4 feet in Ireland. 

Minimum distance between lines of railway, 6 feet. 


Stations. —Minimum width of platform, 6 feet, and 12 at important stations. 
Minimum distance of columns from edge of platform, 6 feet. Steepest gradient for 
stations, 1 in 260. Ends of platforms to be ramped (not stepped). Signals and dis¬ 
tant signals in both directions. 

Carriages. —Minimum space per passenger 20 cube feet. Minimum area of glass 
per passenger, 60 sq. ins. Minimum width of seats, 15 ins. Minimum breadth of 
seat per passenger, 18 ins. Minimum number of lamps per carriage, 2. 

Requirements.— Joints of rails to be fished. Chairs to be secured by iron spikes. 
Fang bolts to be used at the joints of fiat-bottomed rails. 


Construction. 

Width, single line... 

“ double line. 

“ top of ballast, single line. 

“ “ “ double line. 


Narrow. 

Broad. 

Feet. Ins. 

Feet. Ins. 

18 

24 6 

30 

38 

13 6 

15 6 

24 6 

29 


Slope of cuttings from centre, 1 in 30. Width of land beyond bottom of slope, 
o to 12 feet. Ditch with slopes, 1 foot at bottom, 1 to 1. Quick mound, 18 ins. in 
height. Post and rail-fence posts, 7 feet 6 ins. x 6 ins. X 3.5 ins., 9 feet apart, 3 feet 
in ground. Intermediate posts, 5 feet 6 ins. x 4 ins. x 1.5 ins., 3 feet apart. F.ails 
4 of 4 X 1.5 ins. 


Parliamentary Regulations for? Crossing Roads. 



Turnpike 

Road. 

Public 

Road. 

Occupation 

Road. 

. 

Feet. Ins. 

Feet. Ins. 

Feet. Ins. 

Clear width of under bridge, or approach .... 

35 — 

25 — 

12 - 

Clear height of under bridge for a width of 12 ft. 

16 — 

— — 

- - 

44 44 44 44 44 IO ^ 

— — 

15 — 

- - 

a u a u n q u 

— — 


14 — 

“ “ “ at springing. 

12 — 

12 - 


Over bridge, height of parapets. 

4 ~ 

4 — 

4 — 

Approaches, inclination. 

1 in 30 

1 in 20 

1 in 16 

“ height of fencing . 

3 — 

3 — 

3 — 


Limits of Deviation. —In towns, 10 yards each side of centre line. In 
country, 100 yards, or 5 chains nearly. 

Level. —In towns, 2 feet. In country, 5 feet. 

Gradient. — Gradients flatter than 1 in 100, deviation 10 feet per mile 
steeper. Do., steeper, 3 feet per mile. 

Curve. —Curves upwards of .5 a mile radius, may be sharpened to .5 mile 
radius. Curves of less than .5 mile radius may not be sharpened. 

3 M 



















686 ROADS, STREETS, AND PAVEMENTS. 


ROADS, STREETS, AND PAVEMENTS. 

Classification of Roads. 

i. Earth. 2. Corduroy. 3. Plank. 4. Gravel. 5. Broken stone (Mac¬ 
adam). 6. Stone sub-pavement with surface of broken stone (Telford). 
7. Stone sub-pavement with surface of broken stone and gravel, or gravel 
alone. 8. Rubble stone bottom with surface of broken stone or gravel, or 
both. 9. Concrete bottom with surface of broken stone or gravel, or both. 

Grracle of Ttoarls. 

Limit of practicable grade varies with character of road and friction of ve¬ 
hicle. For best carriages on best roads, limit is 1 in 35, or 15 feet in a mile. 

Maximum grade of a turnpike road is 1 in 30 feet. An ascent is easier 
for draught if taken in alternate ascents and levels, than in one continuous 
rise, although the ascents may be steeper than in a uniform grade. 

Ordinary angle of repose is 1 in 40 if roads are bad, and 1 in 30, to 1 in 20. 

When roads have a greater grade than 1 in 35, time is lost in descending, 
in order to avoid unsafe speed. Grade of a road should be less than its angle 
of repose. Minimum grade of a road to secure effective drainage should be 
1 in 80. In France it is 1 in 125. 

In construction of roads the advantage of a level road over that of an in¬ 
clined one, in reduction of labor, is superior to cost of an increased length 
of road in the avoiding of a hill. 

Alpine roads over the Simplon Pass average 1 in 17 on Swiss side, 1 in 22 
on Italian side, and in one instance 1 in 13. 

In deciding upon a grade, the motive power available of ascent and avoid¬ 
able of waste of power in descending are to be first considered. 

When traffic is heavier in one direction than the other, the grade in as¬ 
cent of lighter traffic may be greatest. 

When axis of a road is upon side of a hill, and road is made in parts by 
excavation and by embankment, the side surface should be cut into steps, 
in order to afford a secure footing to embankment, and in extreme cases, 
sustaining walls should be erected. 


Construction. 

Estimate of Labor in Construction of Hoads. (M. Ancelin.) 
A day’s work of 10 hours of an average laborer is estimated as follows: 


In Cube Yards. 


■Work. 


Picking and digging. 

Excavation and pitching { 

6 to i2 feet.J 

Loading in barrows. 

Wheeling in barrows perl 

100 feet.j 

Loading in carts. 

Spreading and levelling... 


Ordinary 
Earth. 

Loose 

Earth. 

Mud. 

Clay and 
Earth. 

Gravel. 

Blasting 

Rock. 

W 

00 ! 

O ' 

to 

GO 

l6 

— 

9 

7 to 11 

2.4 

8 to 12 

8 

7 to 16 

4 

— 

2.2 

22 

— 

8 

— 

*9 

— 

20 to 33 

— 

— 

— 

to 

<“♦- 

O 

to 

00 

— 

16 to 48 

■ -e—r - • 


— 

17 to 27 

_ 

44 to 88 

— 

25 

— 

30 to 80 

— 


Time of pitching from a shovel is one third of that of digging. 


Ditches .—All ditches should lead to a natural water-course, and their min¬ 
imum inclination should be 1 in 125. 

Depressions and elevations in surface of a roadway involve a material loss of 
power. Thus, if elevation is 1 inch, under a wheel 4 feet in diameter, an inclined 
plane of 1 in 7 has to be surmounted, and, as a consequence, one seventh of weight 
has to be raised 1 inch. 

















ROADS, STREETS, AND PAVEMENTS. 


68 ; 

An unyielding foundation and surface are indispensable for a perfect roadway. 

Earth in embankment occupies an average of one tenth less space than in natural 
bank, and rock about one third more. 

Rufs. — Surface of a roadway should be maintained as intact as prac¬ 
ticable, as the rutting of it not only tends to a rapid destruction of it, but 
involves increased traction. 

The general practice of rutting a road displays a degree of ignorance of 
physical laws and mechanical effects that is as inexplicable as it is injurious 
and expensive. 

On compressible roadways, as earth, sand, etc., resistance of a wheel decreases as 
breadth of tire increases. 

Depressing of axles at their ends increases friction. Long and pliant springs de¬ 
crease effect of shock in passing over obstacles in a very great degree. 

Transverse Section .—Best profile of section of roadway is held to be one 
formed by two inclined planes meeting in centre of road and slightly 
rounded off at point of junction. 

Roads having a rough surface or of broken stone should have a rise of 
i in 24, equal to a rise on crown of 6 ins., and on a smooth surface, as a 
block-stone or wood pavement, the rise may be reduced to 1 in 48. 

On roads, when longitudinal inclination is great, the rise of transverse 
section should be increased, in order that surface water may more readily 
run off to sides of roadway, instead of down its length, and consequently 
gullying it. 

Stone Breaking. A steam stone-breaking machine will break a cube yard 
of stone into cubes of 1.5 ins. side, at rate of 1 to 1.5 IP per hour. 

IVEacaclainized. Roads. 

In construction of a Macadamized road, the stones (road metal) used 
should be hard and rough, and cubical in form, the longest diameter of which 
exceed 2.5 ins., but when they are very hard this may be reduced to 1.25 
and 1.5 ins. 

The best stones are such as are difficult of fracture, as basaltic and trap, 
and especially when they are combined with hornblende. Flint and sili¬ 
ceous stone are rendered unfit for use by being too brittle. Light granites 
are objectionable, in consequence of their being brittle and liable to disinte¬ 
gration ; dark granites, possessing hornblende, are less objectionable. Lime¬ 
stones, sandstones, and slate are too weak and friable. 

Dimensions of a hammer for breaking the stone should be, head 6 ins. in 
length, weighing 1 lb., handle 18 ins. in length; and an average laborer can 
break from 1.5 to 2 cube yards per day. 

Stones broken up in this manner have a volume twice as great as in their 
original form. 100 cube feet of rock will make 190 of 1.5 ins. dimension, 
182 of 2 ins., and 170 of 2.5 ins. 

A ton of hard metal has a volume of 1.185 cube yards. 

Construction of a Roadway .—Excavate and level to a depth of 1 foot, 
then lay a “bottom” 12 ins. deep of brick or stone spalls or chips, clinker 
or old concrete, etc., roll down to 9 ins, then add a layer of coarse gravel or 
small ballast 5 ins. deep, roll down to 3 ins., and then metal in 2 equal lay¬ 
ers of 3 ins., laid at an interval, enabling first layer to be fully consolidated 
before second is laid on and rolled to a depth of 4 ins.; a surface or “blind” 
of .75 inch of sharp sand should be laid over last layer of metal and rolled 
in with a free supply of water. 


688 ROADS, STREETS, AND PAVEMENTS. 


Proportion of Getters , Fitters, and Wheelers in different Soils. Wheelers computed 
at a Run of 50 Yards. (Molesworth.) 


| Getters. 

Fillers. 

Wheelers. 


Getters. 

Fillers. 

Wheelers. 

Loose earth, 1 




Hard clay. 

I 

1.25 

1.25 

Sand, etc. j 

I 

I 

I 

Compact 1 




Compact earth ... 

I 

2 

2 

gravel j ' * * ’ 

1 

2 

1 

Marl. 

I 

2 

2 

Rock. 

3 

I 

I 


Telford. Hoads. 

In construction of a Telford road, metalling is set upon a bottom course of 
stones, set by hand, in the manner of an ordinary block stone pavement, 
which course is composed of stones running progressively from 3 inches in 
depth at sides of road to 4, 5, and 7 inches to centre, and set upon their 
broadest edge, free from irregularities in their upper surface, and their in¬ 
terstices filled with stone spalls or chips, firmly wedged in. 

Centre portion of road to be metalled first to a depth of 4 ins., to which, 
after being used for a brief period, 2 ins. more are to be added, and entire 
surface to be covered, “ blinded,” with clean gravel 1.5 ins. in depth. 

Telford assigned a load not to exceed 1 ton upon each wheel of a vehicle, 
with a tire 4 ins. in breadth. 

Grravel or Eartlv Roads. 

In construction of a gravel or earth road, selection should be made between 
clean round gravel that will not pack, and sharp gravel intermixed with 
earth or clay, that will bind or compact when submitted to the pressure of 
traffic or a roll. 

Surface of an ordinary gravel roadway should be excavated to a depth of 
from 8 to 12 ins. for full width of road, the surface of excavation conforming 
to that of road to be constructed. 

The gravel should then be spread in layers, and each layer compacted by 
the gradual pressure due to travel over it, or by a roller, the weight of it in¬ 
creasing with each layer. One of 6 tons will suffice for limit of weight. 

If gravel is dry and will not readily pack, it should be wet, and mixed 
with a binding material, or covered with a thin layer of it, as clay or loam. 

I11 rolling, the sides of road should be first rolled, in order to arrest the 
gravel, when the centre is being rolled, from spreading at the side. 

To re-form a mile of gravel or earth road, 30 feet in width between gutters, 
material cast up from sides, there will be required 1640 hours’ labor of men, 
and 20 of a double team. 

Corduroy Roads. 

A Corduroy road is one in which timber logs are laid transversely to its plane. 

Plank Roads. 

A single plank road should not exceed 8 feet in width, as any greater width 
involves an expenditure of material, without any equivalent advantage. 

If a double track is required it should consist of two single and independ¬ 
ent tracks, as with one wide track the wear would be mostly in the centre, 
and consequently, wear would be restricted to one portion of its surface. 

Materials. —Sleepers should be as long as practicable of attainment, in depth 3 or 
4 ins., according to requirements of the soil, and they should have a width of 3 ins. 
for each foot of width of road. 

Pine, oak, maple, or beech are best adapted for economy and wear. 

Planks should be from 3 to 3.5 ins. thick, and not less than 9 ins. in width, or 
more than 12 if of hard wood, or 15 if of soft. 

A plank road will wear from 7 to 12 years, according to service, material, 
and location, and its traction, compared with an ordinary Macadamized road, 
is 2.5 to 3 times less, and with a common country road in bad order 7 times. 

For other elements, see Earth-work, page 466. 
















ROADS, STREETS, AND PAVEMENTS. 


689 


Asphalt. 

Asphalt is a bituminous limestone, and is synonymous with bitumen; it 
consists of from 90 to 94 per cent, of carbonate of lime and 6 to 10 per cent, 
of bitumen. 

In forming a pavement the powder is heated to from 212 0 to 250 0 , and its par¬ 
ticles caused to adhere by pressure, or it is applied as a liquid asphalt or asphaltic 
mastic, which is thus manufactured. The powder is heated with from 5 to 8 per 
cent, of free bitumen for a flux, and the mixture when melted is run into molds. 
To be remelted, additional bitumen must be mixed with it, without which it would 
only become soft. 

For paving 60 per cent, of sand or gravel must be mixed with it. No chemical 
union takes place between the mastic and the sand or gravel, but cohesion is so 
complete that gravel will fracture with the mastic, and the admixture increases the 
resistance of the mass to heat of the sun. The roadway should have a convexity 
of .01 of its breadth. 

Artificial Asphalt .—Heated limestone and gas tar, when mixed, possess 
some of the proportions of alphalt mastic, but it is very inferior for the 
purposes of a pavement. 

To repair surface of roadway, dissolve bitumen 1 part in 3 of pitch oil or 
resin oil, apply 10 oz. of mixture over each sq. yard of roadway, sprinkle on 
it 2 lbs. of asphalt powder, and then cover surface with sand. 

Wood. Pavement. 

Close-grained and hard woods only are suitable, such as oak, elm, ash, 
beech, and yellow pine, and they should be laid on a foundation of concrete. 

Block Stoive Pavement. 

Paving-blocks, as the Belgian, etc., where crest of street or area of pave¬ 
ment does not exceed 1 inch in 7.5 feet, should taper slightly toward the 
top, and the joints be well filled, “ blinded,” with gravel. The common 
practice of tapering them downward is erroneous. 

The foundation or bottoming of a stone pavement for street travel should 
consist either of hydraulic concrete or rubble masonry in hydraulic mortar. 
The practice in this country of setting the stones in sand alone is at variance 
with endurance and ultimate economy, but when resorted to, there should be 
a bed of 12 ins. of gravel, rammed in three layers, covered with an inch of 
sand. Granite or Trap blocks should be 4 x 9 X 12 ins. 

Rubble Stone Pavement. 

Bowlders or Beach stone of irregular volumes and forms, set in a bed of 
sand, involves great resistance to vehicles and frequent repairs; it is wholly 
at variance with requirements of heavy traffic or city use. 

Concrete Roads. 

Concrete roads are constructed of broken stones (road metal) 4 volumes, 
clean sharp sand 1.25 to .33 volumes, and hydraulic cement 1 volume. The 
mass is laid down in a layer of 3 or 4 ins. in depth, and left to harden during 
a period of 3 days, when a second and like layer is laid on and well rolled, 
and then left to" harden for a period of from 10 to 20 days, according to 
temperature and moisture of the weather. 

Roads. (Moles worth.) 

Ordinary turnpike roads .— 30 feet wide, centre 6 ins. higher than sides ; 
4 feet from centre, .5 inch below centre; 9 feet from centre, 2 ins. below 
centre; 15 feet from centre, 6 ins. below centre. 

Foot-paths —6 feet wide, inclined 1 inch towards road, of fine gravel, or 
sifted quarry chippings, 3 ins. thick. 

Cross-roads —20 feet wide. Foot-paths —5 feet. 

Side drains —3 feet below surface of road. 

Road material —bottom layer gravel, burned clay or chalk, 8 ins. deep. 
Top layer, broken granite not larger than 1.5 cube ins., 6 ins. deep. 

3 M* 


ROADS, STREETS, AND PAVEMENTS. 


Miscellaneous ISTotes. 

Metalling should be from 6 ins. to i foot in depth, and in cubes of 1.5 to 1.75 ins. 

One layer of material of a road should be spread and submitted to traffic or roll¬ 
ing before next is laid down, and this process should be repeated in 2 or 3 layers 
of 3 ins. each. 

When new metal is laid on old, the surface of the old should be loosened with a 
pick. Patching is termed darning. 

Sand and Gravel, Blinding , should not be spread over a new surface, as they tend 
to arrest binding of metal. Mud should be scraped off of surface. 

Hoggin is application of a binding of surface of a metal road, composed of loam, 
fine gravel, and coarse sand. 

Metalled Roads should be swept wet. 

Rolling. —Steam rolls are most effective and economical. 1000 sq. yards of metal¬ 
ling will require 24 hours’ rolling at 1.5 miles per hour. A roller of 15 tons’ weight 
will roll 1000 sq. yards of Telford or Macadam pavement in from 30 to 40 hours, at 
a speed of 1.5 miles per hour, equal .675 and .9 ton mile per sq. yard. 

Sprinkling. — Go cube feet of water with one cart will cover 850 sq. yards. 100 
cube feet per day will cover 1000 sq. yards; ordinarily two sprinklings are necessary. 

Granite Pavement. —The wear of granite pavement of London Bridge was .22 inch 
per year, and from an average of several streets in London, the wear per 100 vehicles 
per foot of width per day is equal to one sixteenth of an inch per year. 

Sweeping and Watering of granite pavement and Macadam road, for equal areas 
and under alike conditions in every respect, costs as 1 for former to 7 of latter. 

By men, with cart, horse, and driver, costs 3.25 times more than by a machine, 
one of which will sweep 16000 sq. yards of street per period of 6 hours. 

Asphalt Pavement. —Average cost per sq. yard in London: foundation, 50 cents; 
surface, $3.25; cost of maintenance per sq. yard per year, 40 cents. Wear varies 
from .2 to .42 near curb, and .17 to .34 inch on general surface per year. 

Washing. —Surface cleaning of stone or asphalt pavement by a jet can be effected 
at from 1 to 2 gallons per sq. yard. 

Wood Pavement. —Wear of wood pavement in London, per 100 vehicles per day 
per foot of width, .083 inch per year. 

Macadamized Roads. — Annual cost of maintenance of several such roads in 
London was 62 cents per sq. yard. 

Block Stone Pavement. —Stones should be set with their tapered or least ends up¬ 
wards, with surface joints of 1 inch. 

Fascines , when used, should be in two layers, laid crosswise to each other and 
picketed down. 

Bituminous road may be made by breaking up asphalt, laying it 2 ins. thick, 
covering with coal tar, and ramming it with a heavy beetle. To repair a bitumi¬ 
nous surface, dissolve one part of bitumen (minei'al tar) in three of pitch oil or resin 
oil, spread .625 of a lb. of solution over each sq. yard of road, sprinkle 2 lbs. pow¬ 
dered asphalt (bituminous limestone) and then sand, and sweep off the surplus. 

Slipping. —Granite safest when wet, and asphalt and wood when dry. 

Gravel , alike to that of Roa Hook, from its uniformity, will bear an admixture 
of from .2 to .25 of ordinary gravel or coarse sand. 

Annual cost of a Telford pavement 4.2 cents per sq. yard, including sprinkling, 
repairs, and supervision. 

Voids in a Cube Yard of Stone. 

Broken to a gauge of 2.5 ins.10 cube feet. Shingle.9 cube feet. 

“ “ 2 “.10.66 “ “ Thames ballast.... 4.5 “ “ 

CC 7- £ ^ T T OO ^ 


For further and full information, see Law and Clarke on Roads and Streets, New 
York, 1867; Weale’s Series, London, *86i and 1877; Roads, Streets, and Pavements, 
by Brev. Maj.-Geu. Q. A. Gilmore, U. S. A., New York, 1876; Engineering Notes, by 
F. Robertson, London and New York, 1873; and Construction and Maintenance of 
Roads, by Ed. P. North, C. E., see Transactions Am. Soc. of C. E., vol. viii., May. 1879. 







SEWERS. 


69I 


SEWERS. 


Sewers are the courses from a series of locations, and are classed as 
Drains, Sewers, and Culverts. 

Drains are small courses, from one or more points leading to a sewer. 
Culverts are courses that receive the discharge of sewers. 

Greatest fall of rain is 2 ins. per hour = 54 308.6 galls, per acre. 
Inclination of sewers should not be less than 1 foot in 240, and for 
Rouse or short lateral service it should be 1 inch in 5 feet. 




1. Circular. 55 Vx 2 f— v, and v a = V. 

____ r Egg. ^ = w, 2 -^ = w ', and D = r. x representing 

area of se wer - 4 - wetted perimeter, f inclination of sewer 
per mile, and v velocity of flow of contents in feet per 
minute; a area of flow, in sq. feet, Y volume of discharge, 
in cube feet per minute ; D height of sewer, w and w' 
width at bottom and top, and r radius of sides, in feet. 

For diameter of sewer exceeding 6 feet. (T. Ilawksley.) 

D —- = iv'. D diameter of a circular sewer of area required. 

9 

Elliptic. —Top and bottom internal should be of equal diam¬ 
eters. Diameter .66 depth of culvert; intersections of top 
and bottom circles form centres for striking courses connect¬ 
ing top and bottom circles. 


Pipes or Small Servers. — Height of section =?i; diameter 
of arch=z .66; of invert = .33, and radius of sides = x. 

In culverts less than 6 feet internal depth, brickwork should be 9 ins. thick ; 
when they are above 6 feet and less than 9 feet, it should be 14 ins. thick. 

If diameter of top arch = 1, diameter of inverted arch = .5, and total 
depth = sum of the two diameters, or 1.5 ; then radius of the arcs which are 
tangential to the top, and inverted, will be 1.5. 

From this any two of the elements can be deduced, one being known. 


Drainage of Lands "by Pipes. 


Soils. 

Depth 
of Pipes. 

Distance 

apart. 

Soils. 

Depth 
of Pipes. 

Distance 

apart. 

Coarse gravel sand .... 

Ft. Ins. 

4 6 

Feet. 

60 

Loam with gravel ... 

Ft. Ins. 

3 3 

Feet. 

27 

Light sand with gravel 

4 

50 

Sandy loam. 

3 9 

40 

T.iglit. loam 

3 6 

3 2 


Soft clay. 

2 9 

2 6 


Loam with clay. 

jj 

21 

Stiff clay. 

15 


Minimum Velocity and Grade of Sewers and Drains 
in Cities. (Wicks teed.) 


Diam. 

Vel. 

per 

Minute. 

Grade, 

1 in 

Grade 

per 

Mile. 

Diam. 

Vel. 

per 

Minute. 

Grade, 

1 in 

Grade 

per 

Mile. 

Diam. 

Vel. 

per 

Minute. 

Grade, 

1 in 

Grade 

per 

Mile. 

Ins. 

Feet. 


Feet. 

Ins. 

Feet. 


Feet. 

Ins. 

Feet. 


Feet. 

4 

240 

3 6 

146.7 

15 

180 

244 

21.6 

42 

180 

686 

7-7 

6 

220 

65 

81.2 

18 

180 

294 

18 

48 

180 

784 

6.8 

8 

220 

87 

60.7 

24 

180 

39 2 

13-5 

54 

180 

882 

6 

IO 

210 

09 

44.4 

30 

180 

490 

10.8 

60 

180 

980 

5-4 

12 

I90 

X 7 S 

30.2 | 

36 

180 

538 

9 






Area 0/ Servers or Pipes .—An area of 20 acres, miles, etc., will not re¬ 
quire 20 times capacity of pipes for one acre, mile, etc., as the discharge from 
the 19 acres, etc., will not flow into the main simultaneously with that from 
one acre, etc. Ordinarily in this country an area of sewer or pipe that will 
discharge a rainfall of 1 inch per hour (3630 cube feet per acre) is sufficient. 




































692 


SEWERS. 


Sewage .—The excreta per annum of 100 individuals of both sexes and 
all ages is estimated at 7250 lbs. solid matter and 94 700 fluid, equal to 1020 
lbs. per capita , and in volume 16 cube feet, to which is to be added the 
volume of water used for domestic purposes. A velocity of flow of from 2.5 
to 3 feet per second will discharge a sewer of its sewage matter and prevent 
deposits. The minimum velocity should not be less than 1.3 feet per second. 


Surface from, ■which Circular Sewers with proper Curves 
will discharge Water equal in Volume to One incli in 
Depth per Hour, including City Drainage. [John Roe.) 


Inclination in Feet. 


Diameter of Sewers in Feet. 



2 

2-5 

3 

4 

5 

6 

7 

8 


Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

None. 

38.75 

67.25 

120 

2 77 

570 

1020 

1725 

2850 

1 in 480. 

48 

75 

x 35 

3°8 

630 

III7 

x 9 2 5 

3025 

1 in 240. 

50 

87 

155 

355 

735 

1318 

2225 

3500 

1 in 160. 

6 3 

IT 3 

203 

460 

95° 

1692 

2875 

4500 

x in 120. 

78 

*43 

257 

59° 

1200 

2180 

3700 

5825 

1 in 80. 

90 

165 

295 

570 

1388 

2486 

4225 

6625 

1 in 60. 

*25 

182 

3 l8 

730 

1500 

2675 

4550 

7 I2 5 


Surface of a Town from which small Circular Drains 
-will discharge Water equal in Volume to Two Inches 
in Depth per Hour. (John Roe.) 


Inclination. 

Diameter of 

Drain in Ins. 

Inclination. 

Diameter of Drain in Ins. 

Fall of 1 Inch. 

3 

4 

5 

6 

7 

8 

Fall of 1 Inch. 

9 

12 

15 

18 

Acres. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Acres. 

Feet. 

Feet. 

Feet. 

Feet. 

•125 

120 

— 

— 

— 

— 

— 

2.1 

120 

— 

— 

— 

•25 

20 

120 

— 

— 

— 

— 

2.5 

80 

— 

— 

— 

•4375 

— 

40 

— 

— 

— 

— 

2-75 

60 

— 

— 

— 

•5 

— 

30 

80 

— 

— 

— 

4-5 

— 

120 

— 

— 

.6 

— 

20 

60 

— 

— 

— 

5-3 

— 

80 

— 

— 

I 

— 

— 

20 

60 

— 

— 

5-8 

— 

60 

240 

— 

1.2 

— 

— 

— 

40 

20 

— 

7.8 

— 

— 

120 

— 

i -5 

— 

— 

— 

20 

60 

120 

9 

— 

— 

80 

— 

1.8 

— 

— 

— 

— 

— 

80 

IO 

— 

— 

60 

240 

2. I 

— 

— 

— 

— 

— 

60 

x 7 

— 

— 


120 


Dimensions, Areas, and Volume of ^Material per Lineal 
Foot of Egg-shaped Sewers of different Dimensions. 



Internal Dimensions. 


Volume of Brick-work. 

Depth. 

Diam. of 

Diam. of 


4.5 IllS. 

9 Ins. 

13.5 Ins. 

Top Arch. 

Invert. 

Area. 

thick. 

thick. 

thick. 

Feet. 

Feet. 

Feet. 

Sq. Feet. 

Cube Feet. 

Cube Feet. 

Cube Feet 

2.25 

i -5 

•75 

2-53 

2. 81 

— 

— 

3 

2 

I 

4-5 

3-56 

— 

— 

3-75 

2-5 

1.25 

7-03 

4 - 3 x 

9-56 

— 

4-5 

3 

x -5 

IO. 12 

5.06 

10.87 

— 

5-5 

3-5 

x -75 

13-78 

5 -8 1 

12.75 

— 

6 

4 

2 

18 

6.56 

14.25 

— 

6-75 

4-5 

2.25 

22.78 

7 - 3 i 

15-75 

24-75 

7-5 

5 

2.5 

28.12 

— 

17.06 

27 

8.25 

5-5 

2.75 

34-03 

— 

18 

28.41 

9 

6 

3 

40-5 

— 

19.69 

30-94 


Area = product of mean diameter x height. 


Sewer Pipes should have a uniform thickness and be uniformly glazed, 
both internally and externally. 

Fire-clay pipes should be thicker than those of stone-clav. 

























































STABILITY. 


6 93 


STABILITY. 

Stability, Strength , and Stiffness are necessary to permanence of a 
structure, under aU variations or distributions of load or stress to which 
it may be subjected. 

Stability of a Fixed Body —Is power of remaining in equilibria without 
sensible deviation of position, notwithstanding load or stress to which it 
may be submitted may have certain directions. 

Stability of a Floating Body .—A body in a fluid floats, or is balanced, 
when it displaces a volume of the fluid, weight of which is equal to weight 
of body, and when centre of gravity of body and that of volume of fluid dis¬ 
placed are in same vertical plane. 

When a body in equilibria is free to move, and is caused to deviate in a 
small degree from its position of equilibrium, if it tends to return to its 
original position, its equilibrium is termed Stable; if it does not tend to de¬ 
viate further, or to recover its original position, its equilibrium is termed 
Indifferent; and when it tends to deviate further from its original position, 
its equilibrium is Unstable. 

A body in equilibrio may be stable for one direction of stress, and unstable 
for another. 

Moment of Stability of a body or structure resting upon a plane is mo¬ 
ment or couple of forces, which must be applied in a plane vertically inclined 
to the body in addition to its weight, in order to remove centre of resistance 
of body upon plane, or of the joint, to its extreme position consistent with 
stability. The couple generally consists of the thrust of an adjoining struct¬ 
ure, or an arch and pressure of water, or of a mass of earth against the 
structure, together with the equal and parallel, but not directly opposed, re¬ 
sistance of plane of foundation or joint of structure to that lateral thrust. 
It may differ according to position of axis of applied couple. 

Couple .—Two forces of equal magnitude applied to same body or struct¬ 
ure in parallel and opposite directions, but not in same line of action, consti¬ 
tute a couple. 

Note.— For Statical and Dynamical Stability, see Naval Architecture, page 649. 


To Ascertain Stability of a Body 01a a Horizontal Blane. 

— Big. 1. 



Illustration. — Stability of a body, A, Fig. 1, when a 
thrust is applied as at 0 , to turn it on a, is ascertained by 
multiplying its weight by distance as, from fulcrum a to 
line of centre of gravity, cs. 

Hence, if cubical block weighed 10 tons and its base is 

6 feet, its moment would be 10 X — = 30 tons. 

2 

If upper part, abdc, was removed, remainder, a e d, 


2 

would weigh but 5 tons, but its centre of gravity . would be — a e = 4 feet. Hence 


its moment would be 5 x 4 = 20 tons, although it is but half the weight. 


To Compute Weiglxt of a Given Body to Snstain a 
Given Thrust. 

F h 

— — w. F representing thrust in lbs ., h height of centre of gravity of body — c s, 

and l distance of fulcrum from centre of gravity — as. 

Illustration.— Assume figure to be extended to a height of 20 feet, and required 
to be capable of resisting the extreme pressure of wind. 







STABILITY.-REVETMENT WALLS. 


694 


Pressure estimated at 50 lbs. F = 6 X 20 X 50 = 6000 lbs. at centre of gravity of 
surface of body. 

6000 X 10 

Then -- — 20 000 lbs. 

3 

Note i. —This result is to be increased proportionately with the factor of safety 
due to character of its material and structure. 

2.—If form of body has a cylindrical section, as a round tower, the thrust of wind 
would be but one half of that of a plane surface. 

When the Body is Tapered , as Frustum of Pyramid or Cone. — Ascertain 
centres of gravity of surface for pressure or thrust, and of body for its sta¬ 
bility, and proceed as before. 



To A.scertain Stability of a Body 011 
an Inclination.-—Fig. S. 

Illustration. — Stability of body, Fig. 2, when thrust 
is applied at c, is ascertained by multiplying its weight 
by distance a b from fulcrum, b, to line of centre of 
gravity, a g. 

If thrust was applied at 0, stability would be ascer¬ 
tained by distance s r from fulcrum r. 


Angles of Fqnilitorinm at 'wlaicli various Substances will 
Iriepose, as determined L>y a Clinometer. 

Angle measured from a Horizontal Plane , and failing from a spout. 



Degrees. 

Degrees. 

Lime-dust.... 



Dry sand.. 


37 

Moist sand_ 


■••• 37 


Degrees. 

Common mold... 37 
Common gravel.. 35 to 36 
Stones or Coal... 43 


Weiglit of a Cnloe Foot of ^Materials of Embankments, 
Walls, anti Dams. 


Concrete in cement... 137 

Stone masonry. 130 

Brick “ . 112 


Gravel. 125 

Loam. 126 

Sand. 120 


Clay... 120 

Marl. 100 


Revetment Walls. 

When a wall sustains a pressure of earth, sand, or any loose material, it 
is termed a Revetment wall, and when erected to arrest the fall or subsidence 
of a natural bank of earth, it is termed a Face wall. 

When earth or banking is level with top of wall, it is termed a Scarp re¬ 
vetment, and when it is above it, or surcharged, a Counterscarp revetment. 

When face of wall is battered, it is termed Sloping, and when back is bat¬ 
tered, Countersloping. 

Thrust of earth, etc., upon a wall is caused by a certain portion, in shape 
of a wedge, tending to break away from the general mass. The pressure 
thus caused is similar to that of water, but weight of the material must be 
reduced by a particular ratio dependent upon angle of natural slope, which 
varies from 45 0 to 6o° (measured from vertical) in earth of mean density. 

Or, natural slope of earth or like material lessens the thrust, as the cosine 
of the slope. 

Angle which line of rupture makes with vertical is .5 of angle which line 
of natural slope, or angle of repose , makes with same vertical line. When 
earth is level at top, its pressure may be ascertained by considering it as a 
fluid, weight of a cube foot of which is equal to weight of a cube foot of the 
earth, multiplied by square of tangent of .5 angle included between natural 
slope and vertical. 



















STABILITY.-REVETMENT WALLS. 


695 


Therefore squares of the tangents of .5 of 45 0 and .5 of 6o° = .i7i6 and 
.3333, which are the multipliers to be used in ordinary cases to reduce a 
cube foot of material to a cube foot of equivalent fluid, which will have 
same effect as earth by its pressure upon a wall. 

Pressure of Earth, against Revetment Walls. 

Fig. 3. Let A B C D, Fig. 3, be vertical section of a revetment 

wall, behind which is a bank of earth, A D/e ; let D o 
represent angle of repose, line of rupture, or natural slope 
which earth would assume but for resistance of wall. 

In sandy or loose earth angle 0 D A is generally 30 0 ; 
in firmer earth it is 36°; and in some instances it is 45 0 . 

If upper surface of earth and wall which supports it are 
both in one horizontal plane, then the resultant, l n, of 
pressure of the bank, behind a vertical wall, is at a dis¬ 
tance, D n, of one third A D. 

Line of Rupture behind a wall supporting a bank of vegetable earth is at 
a distance A 0 from interior face, A D = .618 height of it. 

When bank is of sand, A o = .677 h ; when of earth and small gravel = 
.646 h ; and when of earth and large gravel = .618 h. 

The prism, vertical section of which is A D 0, has a tendency to descend 
along inclined plane, 0 D, by its gravity; but it is retained in its place by 
resistance of wall, and by its cohesion to and friction upon face 0 D. Each 
of these forces may be resolved into one -which will be perpendicular to o D, 
and into another which will be parallel to 0 D. The lines c i, i l represent 
components of the force of gravity, which is represented by vertical line c l , 
drawn from centre of gravity, c, of prism. Lines n r , l r represent compo¬ 
nents of forces of cohesion and friction, which is represented by horizontal 
line n l. Force that gives the prism a tendency to descend is i l, and that 
opposed to this is r l , together with effects of cohesion and friction. 

Thus, il — rl + cohesion -f- friction. Consequently, exact solution of prob¬ 
lems of this nature must be in a great measure experimental. 

It has been found, however, and confirmed experimentally, that angle 
formed with vertical, by prism of earth that exerts greatest horizontal stress 
against a wall, is half the angle which angle of repose or natural slope of 
earth makes with vertical. 

Memoranda. 

Natural slope of dry sand = 39 0 , moist soil = 43 0 , very fine sand = 21 0 , wet clay 
= 14 0 , and gravel = 35 0 . 

In setting or founding of retaining walls, if earth upon which wall is to rest is 
clayey or wet, coefficient of friction between wall and earth falls to .3; hence it is 
necessary, in order to meet this, that the wall should be set to such a depth in the 
earth that the passive resistance of it on outer face of wall, combined with its fric¬ 
tion on its bottom, may withstand the pressure or thrust on its inner face. 

Moment of a Retaining Wall is its weight multiplied by distance of its centre of 
gravity to vertical plane passing through outer edge of its base. 

Moment of Pressure of Earth against a retaining wall is pressure multiplied by 
distance of its centre of pressure to horizontal plane passing through base of wall. 

Equilibrium of Retaining Wall is when respective moments of wall and earth are 
equal. 

Stability of a Retaining Wall should be in excess of its equilibrium, according to 
character of thrust upon it, and the line of its resistance should be within wall and 
at a distance from vertical passing through centre of gravity of wall, at most .44 of 
distance of exterior axis of wall from this line. 

Coefficient of Stability varies with character of earth, location, exposure to vibra¬ 
tions, floods, etc.; hence thickness of base of wall will vary from 1.4 to 2 b. 

Backs of x'etaining walls should be laid rough, in order to arrest lateral subsidence 
of the filling. 










STABILITY.-KEVETMENT WALLS. 


696 


When filling is composed of bowlders and gravel, the thickness of wall must be 
increased, and contrariwise; when of earth in layers and w T ell rammed, it may be 
decreased. 

Courses of dry wall should be inclined inwards, in order to arrest the flow of 
water of subsidence in filling from running out upon face of wall. 

Less the natural slope, greater the pressure on wall. 

Sea walls should have an increased proportion of breadth, as the earth backing 
is not only subjected to being flooded, but the walls have at times to sustain the 
weight of heavy merchandise. 

Buttress .—An increased and projecting width of wall on its front, at intervals in 
its length. 

Counterfort .—An increased and projecting width of w r all at its back and at in¬ 
tervals. 

Coefficient of Friction of masonry on masonry .67, of masonry on dry clay .51, 
and on wet clay .3. 

Face of wall should not be battered to exceed 1 to 1.25 ins. in a foot of height, in 
consequence of the facility afforded by a greater inclination to the permeation of 
rain between the joints of the courses. 

Footing of a wall, projecting beyond its faces, is not included in its width. 

Pressure.— Limit of pressure on masonry 12 500 to 16 500 lbs. per sq. foot wall. 


Thickness of Walts , in Mortar, Faces vertical. For Railways or Like Stress. 

Cut stone or Ranged rubble.35 | Brick or Dressed rubble.4 

When laid dry, add one fourth. 

Friction in vegetable earths is .5; pressure in sand .4. 

When vegetable earths are well laid in courses, the thrust is reduced .5. 

When bank is liable to be saturated with water, thickness of wall should be 
doubled. 


Centre of Pressure of earthwork,.etc., coincides with centre of pressure of w T ater, 
and hence, when surface is a rectangle, it is at .33 of height from base. 

The theory of required thickness of a retaining wall, as before stated, is, that the 
lateral thrust of a bank of earth with a horizontal surface is that due to the prism 
or wedge-shaped volume, included between the vertical inner face of the wall and 
a line bisecting the angle between the wall and the angle of repose of the material. 


To Compute lillexneiats of Revetment Walls.—ITig. 4 . 

Let A D 0 represent angle of repose of material, resting 


Fig. 4. 



.492. 
10 ti¬ 


ll h ' 2 

Tan. A D n h — , or — tan. ADn = Y. 
2 


tan. AD« = W; 


2 

w 1 C 


tan. 2 A D n = P; 


w h 


w 7 i 3 


tan. 2 AD n —, or 
2 36 


tan. 2 A D= M; W h~, 
2 


W h 35 2 W ll X- wh* Whx 2 W A3 

or- - — vi] -= —— tan. 2 ADn = E; -—-tan. 2 ADn = S; 

2 2 0 23 3 

h tan. A D n — x , and h tan. AD n ~^ = x> • h representing height of 


wall in feet, V volume of section of prism of material A D n one foot in length in cube 
feet , W and w iveights of a cube foot of wall and of material,'? lateral pressure of 
prism of earth upon wall , M and m moments of pressure and weight on and of wall, 
E and S equilibrium and stability of wall, all in lbs., and x and x', C D for weights 
of wall for equilibrium and stability. 


Illustration.— A revetment wall, Fig. 4, of 125 lbs. per cube foot and 40 feet in 
height, sustains a bank of earth having a natural slope of 52 0 24', and a weight of 
89.25 lbs. per cube foot; what is pressure or thrust against it, etc ? 


















STABILITY.-REVETMENT WALLS. 


697 


Tan. 2 A I) n — .242. Then .492 X 40 X — = 393-6 cube feet. 


89.25 X 4 ° 2 


X -492 = 35 128.8 lbs. 


89.25 X 40 £ 


X • 492 s = 17 278.8 lbs. 


89 25 X 4 — x 492 2 X — = 230 384 lbs. 125 X 40 X = 230 400 lbs. 
23 2 


40 X - 49 2 -y ^x ‘ 12 5 = 9-6 feet, and 40 X • 49 2 yj 2 — *3- 5^ feet. 


For Rubble Walls in Mortar or Dry Rubble, add respectively to base as above 
obtained, .14 and .42 part. 

Note i. —When coefficient of friction is known, use it for tan. 2 A D n. 

h X C = base of wall for stability. (Molesworth.) 

2.—When either relative weights of equal volumes of wall and bank of earth or 
their specific gravities are given, S and s may be taken for W and w. 

These equations involve simply the operation of a lever, the fulcrum being at 
the outer edge of wall C. The moment of pressure of bank is product of lateral 
pressure and perpendicular distance from fulcrum to line of direction of pressure. 

The moment of weight of wall is product of weight of wall and perpendicular 
distance from fulcrum to vertical line drawn through centre of gravity of wall. 

When Weights of Embankment and Wall are equal per Cube Foot. 

C for clay =3.336, and for sand .267. 

When Weights are as 4 to 5. C for clay = .3, and for sand .239. 

When Wall has an Exterior Slope or Batter. —Fig. 5. 



W h 




D + EC 


E C 2 ' 


j = M. 


M representing 


moment of weight of wall in lbs. 

Illustration. — Assume weight of wall 120 lbs. per 
cube foot, and C D and E C respectively 10 and 2.5 feet, 
and all other elements as in preceding case. 

120 X 4 ° .. 

Hence, -— X 


^10 -f- 2.5 — —= 370000 lbs. 


Or, hyj- 


W h /— r — 2 w 2 h 2 \ w h 3 . -p. „ 

- ( x + n h - — ) —- tan. 2 AD« = S. 

2 \ 3/3 


/ Yb ^ 2 'UO 

-1 -tan. 2 A D iz — nh — x. x representing AB or C D. n ratio of 

3 3 W 

difference of widths of base and top to height. In absence of tan. 2 AD n put C, co¬ 
efficient of material. 

C = .0424 for vegetable or clayey earth, mixed with large gravel; .0464 if mixed 
with small gravel; .1528 for sand, and .166 for semi-fluid earths. 

Illustration. — Assume elements of preceding case. n=_one fortieth, and tan. 
ADii = .492. 




: + 2 X 8 9f5 x .492 2 — 1 = 12.6 feet. 


40 V 3 X 4° 2 1 3 x 125 
Hence, thickness of w r all at base = 12.6 -J- 1 (one fortieth of height) = 13.6 feet. 


Note. —If « = one twentieth, 40 


V 3X2 


2 X 89.25 


X -492 2 —2 = 11.63 feet. 


3 X 2o‘ ■ 3 X 125 

Hence, wall at base = 11.63 + 2 (one twentieth of height) = 13.63 feet. IfC was 
used, u.32 feet. 
































STABILITY.—REVETMENT WALLS 


698 

Fig. 6. 


When Wall has an Inferior Slope or Batter , B E.- 



Fig. 6. 


w h 2 


X tan. 1 


oEr 


W h 3 


X tan. : 


0E1' 


= M of 


earth 


w h / -— CEA 

/or equilibrium ; -^DCXL(J-j-CL-- — J == 


M of wall; and 
bility. 


w A 3 


X tan. 2 0 E » = M 0/ earth for sta- 


Coefficients for Batter of following Proportions. 
Base = Height x Tab. number. 

Weight of Earth to Wall. 


Weight of Earth to Wall. 


Batter of 
Wall. 

As 4 
Clay. 

to 5. 
Sand. 

As 1 
Clay. 

to 1. 
Sand. 

1 in 4. 

.083 
. 122 

.029 

.065 

.092 

•US 

•155 

.183 

•054 

.092 

.118 

1 “ c. 

I “ 6. 

.149 


Batter of 
Wall. 

As 4 
Clay. 

to 5. 
Sand. 

As 1 
Clay. 

to 1. 
Sand. 

1 in 8. 

. l84 

.125 

.16 

.218 

•153 
. 189 
.267 

T 4 4 T *? . 

. 221 

.256 

•336 

Vertical.... 

■3 

•239 


To Compute Pressure Perpendicular to Back of Wall. 

—Bis. V. 


Fig. 7. 




EE :/ 

PtT 


n 

~r~ 


yr/ 


P * =-or — and/* at right angle to bach of wall, 

3 3 

whether vertical or inclined. 

Lx Art _ . , _ ivxh 2 x tan. 2 ADn 

- , or L X tan. A D n, or - , or 

h 2 

to X A n 2 


zf *. L representing weight of triangle of em- 


-m 


X) 


bankment, as A D n. 


This is pressure independent of friction between surfaces of wall and earth. 

To Ascertain and Compute Amount and Effect of Fric¬ 
tion of Wall and Earth.—Eig. 8. 

Draw / * by scale to computed pressure at right angle 
to back of wall, draw angle/ *r = i»Do of natural slope 
of earth with horizon, draw fr at right angle to/ #, make 
r c =/ *, then c r will represent by scale effect of friction 
against back of wall. 

Assume friction to act at point *, then r * will give by 
scale resultant of the two forces of pressure and friction, 
equal to pressure in force and direction, which bears 
against wall. 

This resultant is also equal to / * X sec. m 1 ) 0. 

w X h 2 X tan. 2 m D 0 _ , _ 

-X sec. m D 0. or L X tan. AD11 



L X A n X sec. ml) 0 


h 


r *, or 


X sec. »Do. 

To Ascertain Point of Moment of Pressure of a Wall. 


Fig. 9. 



—Fig. 9 . 

By its resisting lever l a,added to its weight. 

Weight of wall as computed assumed as concentrated at its 
centre of gravity . 

Draw a vertical line • 0 through its centre of gravity, and con¬ 
tinue line of pressure P * to l , take any distance r 0 by scale rep¬ 
resenting weight of wall, and r n , by same scale, for amount of 
pressure or thrust against wall, complete parallelogram r 0 n u, 
then diagonal ru will give resultant of pressure in amount and 
direction to overturn wall. 

For stability this diagonal should fall inside of base at a point 
not less than one third of its breadth. 














































































STABILITY.-REVETMENT WALLS. 


699 


Surcharged. Revetments. 


Fig. 10. f r 

«/ r ~ 



X) c 


When the earth stands above a wall, as A B e, 
Fig. 10, with its natural slope, Ay, A B C is termed 
a Surcharged Revetment. 

If C r is line of rupture, A fr C is the part of earth 
that presses upon wall, which part must be taken into 
the computation, with exception of portion A B e, 
which rests upon wall; that is, the computation must 
be for part C efr, which must be reduced by multiply¬ 
ing weight of a cube foot of it by square of tangent of 
angle e C r = angle of line of rupture, or half angle 
e C o, which natural slope makes with vertical, and 
then proceed as in previous cases for revetments. 



h' w tan. 2 eCr 
3 *W 


breadth or C D. 


W and iv representing weights of wall and 


embankment in lbs. per cube foot, and h' height of embankment, as C e. 

Illustration. —Height of a surcharged revetment, BC, Fig. 10, is 12 feet, weight 
130 lbs. per cube foot; what is its width or base to resist pressure of earth of a weight 
of 100 lbs. per cube foot, and a height, C e, of 15 feet, angle of repose 45 0 ? 


Tan. 2 45 ° -4- 2) ==. 1716. 


Then 15 



X 100 X -171 
3 X 12 x 130 


6 


= 15 V - 0 55 — 3 - 5 2 feet- 


To Ascertain Point of jVloiueiit of Pressure of a Sur¬ 
charged. Wall.—Wig. 11. 

Fig. 11. ,f Draw a line, P *, parallel to slope, C r, through centre 

of gravity of sustained backing, BCr. 

When, as in this case, this section is that of a triangle, 
point * will be at .33 height of wall. 

When natural slope is 1.5 in length to 1 in height, as 
with gravel or sand, w x .64 = pressure P #. 

In a surcharged revetment, as/B o, at its natural slope, 
the maximum pressure is attained when the backing 
reaches to r. When slope of maximum pressure, C n r , 
intersects face of natural slope, B f so that if backing is 
raised to f or above it, there is theoretically no addi¬ 
tional stress exerted at back of or against wali, but prac¬ 
tically there is, from effect of impact of vibration of a 
passing train, proximity to percussive action, alike to that of a trip-hammer, etc. 

When backing rests on top of wall, as A B e. Fig. 10, small triangle of it is omitted 
in computations. Direction of pressure against wall is same as when wall is not 
surcharged. 

When Wall is set below Surface of Earth. —Fig. 12. 



Fig. 12. 



1.4 tan. 45° —— V -- = d - 

a representing angle of repose of earth , w and W weights 
of earth and wall per cube foot, f friction of wall on base 
A B, and V weight of wall. 

Illustration. —If a wall of masonry, Fig. 12, 8 feet in thickness 
and 13 in height, is to sustain earth level with its upper surface, 
earth Weighing 100 lbs. per cube foot, weight of wall 150 lbs. per 
cube foot=:i5 6oo lbs., and angle of repose of earth 30 0 ; what 
should be the depth of wall below surface of earth? 

Tan. 45 — 30 -f- 2 = . 5774, and /=. 3. 


/i3 2 X iooX-5774 2 " 02 X .3X 15600 Q _ 0 w 793600)5634.3 

Then 1.4 X. 577 V- - ^ - -•8o8 4 X A y- 


= 4.027 feet. 

Note.— Coefficient, of stability is assumed by French engineers for walls of forti¬ 
fications 1.4 h, and if ground is clayey or wet/W.3. 












































700 STABILITY.-EMBANKMENT WALLS AND DAMS. 



In Computing Stability of a Surcharged Wall , Fig. 13, sub¬ 
stitute d for h , as in following illustration. (Molesworth.) 
d, representing depth at distance l, = h. 

In slopes of 1 to 1, d = 1.71 A; of 1.5 to 1,= 1.55; of 2 to 1,= 
1.45; of 3 to 1,= 1.31, and 4 to x,= 1.24. 

To Determine Form of a IPier to Sustain 
equal Pressure per Unit of Surface at all 
its Horizontal Sections, or any Height. 
A nd — a, or A N = a. A and a representing areas of sections at summit of pier 
and at any depth, d, measured from summit , n a number the hyp. log. of which — 1 - 4 - 
height , H, of a column of the material of which pier is constructed, due to required 

• 4343 & 

pressure, and N the number, com. log. of which = ———. 

H 

Illustration.—H eight of a pier is 20 feet, and area of section of its summit = 
x foot; what should be its areas at 10 feet and base? 

i-t-2o = .05, and number = 1.0513; 1 X 1.0513 10 = 1.649 feet; and 1 X i.o5i3 20 = 
2.719 feet. 

Counterforts are increased thicknesses of a wall at its bach, at intervals of 
its length. 

Embankment AValls and. Danas. 

Thrust of water upon inner face of an Embankment wall or Dam is 
horizontal. 

When Both Faces are Vertical , Fig. 14. 

Assume perpendicular embankment or wall. A B C D, Fig. 14, to sustain 
pressure of water, BCe f. 

Fig. I4 . Let Jc i be a vertical line passing through 0, centre 

of gravity of wall, c centre of pressure of water, dis¬ 
tance Cc being = .33 B C. Draw cl perpendicular 
to B C ; then, since section A C of wall is rectangular, 
centre of gravity, 0 , is in its geometrical centre, and 
therefore Di = .5 DC. Now l D i is to be consid¬ 
ered as a bent lever, fulcrum of which is D, weight of 
wall acting in direction of centre of gravity, o , on arm 
D i, and pressure of water on arm D l, or a force equal 
to that pressure thrusting in direction c l. 

Then P x D l = P X ^-^ = W X —— , or P = 5 * ' P representing pressure 

3 2 " u ‘ 



2 B C 

of water. 

Note. — When this equation holds, a wall or embankment will just be on the 
point of overturning; but in order that they may have complete stability, this 
equation should give a much larger value to P than its actual amount. 

The following formulas are for walls or embankments one foot in length; 
for if they have stability for that length they wdll be stable for any other 
length. 
h 2 

P = — w, also W = Zi b W, each value being for 1 foot in length, which, being sub¬ 
stituted in the equations, there will result 

h 2 3 b X h b W' ' /o W / w 

— w = -- — -, or h 2 w = 3 b 2 W; b ^ l — = h, and h ~ = b. h rep- 

2 2 h \w y 3 W * 

resenting depth of water and wall or embankment , which are here assumed to be 
equal , 6 breadth of wall or embankment, and W and w weights of wall and water 
per cube foot in lbs. 

Which gives breadth of a wall or embankment that will just sustain 
pressure of the water. 




















STABILITY.—EMBANKMENT WALLS AND DAMS. ^01 


To Compute Ecpu.ili'briu.ixi. h 


/— = b. 
V 3 W 


Illustration i.— Height of a wall, B C, equal to depth of water, is 12 feet, and re¬ 
spective weights of water and wall are 62.5 lbs. and 120 lbs. per cube foot; required 
breadth of wall, so that it may have complete stability to sustain the pressure of 
water. 


V: 


62.5 


= 12 X .4166 — 5 feet, breadth that will just sustain pressure of the 


3 X 120 

water. 

Therefore an addition should be made to this to give the wall complete stability, 
say 2 feet; hence 54-2 — 7, required width of wall. 

2.—Width of a wall is 3 feet, and weight of a cube foot of it is 150 lbs ; required 
height of wall to resist pressure of fresh water to the top. 

V^° =8 ' o49/ “ ( ' 

/ 2 W 

~w ~ 


Illustration. —Take elements of preceding case. 

/2 X 62.5 


3 X 120 


12 X ■ 589 — i-oi feet. 


Or, Divide 1, 2, or 3, etc., according as the nature of the ground, the mate¬ 
rial, and the character of the thrust of the water requires, by .05 weight of 
material of wall, per cube foot, extract the square root of quotient, and mul¬ 
tiply result by extreme height of water. 

Example. — What should be the thickness of a vertical faced wall of masonry, 
having a weight of 125 lbs. per cube foot, to sustain a head of water of 40 feet, and 
to have stability ? 

•\/(2 -f- .05 X 125) 40 = +.32 X 40 = 22.63 feet. 

'2 w 


Or, h 


3 W 


= 4° V • 347 2 = 2 3- 56 feet. 



^ c Assume prismoidal wall, A B C D, to sustain press- 


When Dam has an Exterior Slope or Batter , as A D.—Fig. 15 

Fig. 15. A___B 

: BJ| 5 | ure of water, B C ef. 

Draw A E perpendicular to D C ; h =z B C, the top 
IjggJ breadth A B = E C = 6, and bottom breadth, D E, 

gg§ of sloping part, A E D S. 

Then weights of portions A C and A E D respec- 
- ~ : tively for one foot in length are h b W and .5 W S h, 

J these weights acting at points n and i respectively. 


i e c u C 


To Compute HVIoinent. 

hS W 2 S 


7 i 6 W X ^ = moment for A C, and —-— X ~ —moment for A E D. 

H enc e ; ^ ft -j_ u — — moment of dam, S representing batter or base E D. 

Illustration.— Height of a dam, B C, Fig. 15, is 9 feet, base C E 3, and E D 4 feet; 
what is its moment ? 

AC = 9 X 3 X 120 X (4 + ^ = 3 2 4 o X 5-5 — 17 820 l hs. 

A I) E = 9 X 4 X 12 0 ^ 2_Xj. _ x 2^ = 5760 lbs. 

23 

Hence, 17 820 + 5760 = 23 580 lbs. moment. Or, 9 ^4 + 3 —= 54 ° X 43 f 
— 23 580 lbs. moment. 

3 N* 


















702 STABILITY.-EMBANKMENT WALLS AND DAMS. 


To Compute Elements of "Walls or Dams with an 
Exterior Batter.—Eig. 1 f">. 

To Compute "Width. of Top. 


When Width of Batter is Given. 


^ + 5 !-s = ,. 

3 W 3 

Illustration. —Assume height of wall 9 and batter 3 feet, and W and w 120 and 
62.5 lbs. per cube foot. 


V 


2 X 9 2 X 62.5 3 


X --3 = V28.125 -f 3 — 3 = 2.58 feet. 

3 


3 X 120 

To Compute "Width of Base 
When Width of Batter is Given. 

j2 X 9 2 X 62.5 


/2 h 2 w S 2 

T*r+T = B - 

v , =S- 58 /ee« = S + 6. 

3 X 120 3 

To Compute "Width, of Batter. 

3 & 


When Width of Top is Given. ^/ : 


h 2 w , 3 6 2 

—- 4 - -— 
W ^ 4 


— 2 - = S. 


V : 


9 2 X 62-5 3 X 2.58-’ 


3 X 58 


= V42.18 + 4.99 — 3.87 = 3 feet. 


When Width of Bottom is Given, yj 3 B s 


h 2 w 


W 


= s. 



To Determine Stability of a Retaining Wall or Dana by 
Protraction.—Fig. 16. 

Assume A B C D, section of a wall. On horizontal 
line of centre of thrust or pressure, with a suitable 
scale, lay off, from vertical line of centre of gravity • 
of wall, line or — thrust against Avail, and on vertical 
line at centre of gravity of wall, at its intersection, 0, 
with centre of thrust, let fall 0 s = Aveight of Avail. 

Complete parallelogram, and if diagonal 0 u or its 
prolongation falls within C, the Avail is stable, and 
W' X distance from line 0 s = moment of wall. 

W representing whole weight of wall in lbs. 

To Determine Centre of Gravity of a "Wall or Dam.— 

Eig. 16. 

r,0 7- , X ( A „ . n „ A B X C D\ ,CD/2AD + CD\ 

By Ord,nates. - (a B + C D - A ^ ) = *, and — (-fB + cir) = *■ 

To Compute Base of Dam. 

When Height , Rate of Batter , and Weight of Materials are given. Rule. 
—Multiply square of width of batter by .0166 weight of material per cube 
foot, add 1, 2, or 3 times square of depth of water, according as resistance 
due to equilibrium is required, divide result by .05 Aveight of material per 
cube foot, and extract square root of quotient. 


0T 'f 


'x h 2 -f- 6 2 X .0166 W 


— b. x = n umber of times of resistance required. 


.05 W 

Example.— Assume a dam 40 feet in height, constructed of masonry weighing 
120 lbs. per cube foot, to batter 3 ins. per foot, and to have twice the resistance due 
to its equilibrium; what should be its breadth at its base, DC? 


40 X 3 


: baiter , and 




40 2 X 2 -j- 10 2 X .0166 X 120 
.05 X 120 


— 73399 __ 


23.8 feet. 




























STABILITY.-EMBANKMENT WALLS AND DAMS. 



When Section of Dam is a Triangle , Fig. 17. — As¬ 
sume dam, A B C, to sustain a head of water, ef. 

Rule. —Proceed as by Rule for Fig. 14; multiply by 
.033 instead of .05. 

Example.—A s before. 

V( 2 = .033 x 125) 40= V-4 8 5 x 40 = 27.84/^. 

Or, Formula for S (C B), Fig. 15. — 28.28 feet. 


To Determine Section of a Vertical Wall which shall have Equal Resist - 
mice of one having Section of a Triangle. (See J. C. Trautwine , Phil a., 1872.) 


To Compute Thickness of Base of a Wall 01* Dam.— 

Fig. 18 . 


Or. 


F ‘g- l8 - Rule. —Divide x, 2, or 3 times square of depth of water 

by .05 weight of material, add quotient to .5 batter on one 
face, and square root of this sum, added to half batter on 
other side, will give thickness. 

/ h 2 x / b \ 2 . h' _ 

, yf ' \—) "u ~ — Base. b and b representing 

exterior and interior batters , and x, as before , number of times 
of resistance or square of depth. 

~I> b f c Example.—A ssume a dam 40 feet in height, to batter 5 feet 
on each side, constructed of masonry weighing 120 lbs. per cube 
foot, and to have twice the resistance due to its equilibrium; what should be 
breadth of base, DC? 

\/+ (I)' +1 = V539 ' 58 + “' 5 = 35-73 feet. 



High. Masonry Hams. 

Rubble Masonry, well laid in strong cement, will bear with safety a load 
equivalent to weight of a column of it 160 feet in height. Assuming such 
Fig. 19. masonry as twice weight of water, it is equivalent 

to a pressure of 20 000 lbs. per sq. foot. 

Log. B-)-.434 294 X ~ — b. B representing width of 

oO 

wall at top, and d depth at any desired point below top , 
both in feet. 

Ordinarily, B may be taken at 18 feet, and in cases 
of extreme and exposed heights of dam at 20 and more, 
and when b is determined, .9 of it is to be on outer face 
of wall, as A B, and . 1 on inner face. 

Illustration.—D etermine section of a dam, Fig. 19, 
80 feet in height, at depths of 10, 20, 40, 60, and 80 feet. 
Log. B = 1.2553. 

Log. 1. 2553 + .4343 x ^ ==l0g ’ i - 2 553 + - 0 543 = 2o. 4 , which X .9 = 18.36. 

“ 1.2553 + .4343 X = = log- i.2553+ .1086 = 23.11, which X .9 = 20.8. 

OO 

“ 1-2553+ .4343 X ^ = log. 1-2553 + -2172 = 29.68, which X .9 = 26.81. 

OO 

“ 1-2553+ -4343 X y- = l°g- I .2553 + - 3257 = 3 8 .ii) which X -9 = 34 - 3 - 

OO 

80 

“ 1.2553+ .4343 x 8^ = lo g- 1.2553 +.4343 = 50.07, which X .9 = 45.06. 





































704 


STEAM. 


STEAM. 

Steam is generated by heating of water until it attains temperature 
of ebullition or vaporization, and elevation of its temperature is sensible 
to indications of a thermometer up to point of ebullition; it is then 
converted into steam by additional temperature, which cannot be in¬ 
dicated by a thermometer, and is termed latent. (See Heat, page 5°8-) 

Pressure and density of steam, which is generated in free contact with water, 
rises with the temperature, and reciprocally its temperature rises with the press¬ 
ure and density, and higher the temperature more rapid the pressure. There is 
but one and a corresponding pressure and density for each temperature, and steam 
generated in free contact with water is both at its maximum density and pressure 
for its temperature, and in this condition it is termed saturated , from its being in¬ 
capable of vaporizing more water unless its temperature is raised. 

Saturated Steam is the normal condition of steam generated in free contact with 
water, and same density and same pressure always exist in conjunction with same 
temperature. It therefore is both at its condensing and generating points; that 
is, it is condensed if its temperature is reduced, and more water is evaporated if 
its temperature is raised. 

If, however, the whole of the water is evaporated, or a volume of saturated steam 
is isolated from water, in a confined space, and an additional quantity of heat is 
supplied to the steam, its condition of saturation is changed, the steam becomes 
superheated , and both temperature and pressure are increased, while its density is 
not increased. Steam, when thus surcharged, approaches to condition of a gas. 

With saturated steam, pressure does not rise directly with the temperature. 

Steam, at its boiling-point, is equal to pressure of atmosphere, which is 14.723307 
lbs. (page 427), at 6o° upon a sq. inch. 

In all computations concerning steam, it is necessary to have some or all of fol¬ 
lowing elements, viz.: 

Its Pressure , which is termed its tension or elastic force, and is expressed in lbs. 
per sq. inch. Its Temperature , which is number of its degrees of heat indicated by 
a thermometer. Its Density , which is weight of a unit of its volume compared 
with that of water. Its Relative volume , which is space occupied by a given weight 
or volume of it, compared with weight or volume of water that produced it. 

Under pressure of the atmosphere alone, temperature of w T ater cannot be raised 
above its boiling-point. 

Expansive force of steam of all fluids is same at their boiling-point. 

A cube inch of water, evaporated under ordinary atmospheric pressure, is convert¬ 
ed into 1642* cube ins. of steam, or, in a unit of measure, very nearly 1 cube foot, 
and it exerts a mechanical force equal to raising of 14.723307 X 144 = 2120.156208 
lbs. 1 foot high. 

A pressure of 1 lb. upon a sq. inch will support a column of mercury at a tem¬ 
perature of 6o°, 1- 4 -. 4907769 (page 427) = 2.037 586 ins. in height; hence it will 
raise a mercurial siphon gauge one half of this, or 1.018 793 ins. 

Velocity of steam, when flowing into a vacuum, is about 1550 feet per second when 
at a pressure equal to the atmosphere; when at 10 atmospheres velocity is increased 
to but 1780 feet; and wflien flowing into the air under a similar pressure it is about 
650 feet per second, increasing to 1600 feet for a pressure of 20 atmospheres. 

Boiling-points of Water, corresponding to different heights of barometer, see 
Heat, page 517. 

Volume of a cube foot of water evaporated into steam at 212 0 is 1642 cube feet; 
hence 1 -f-1642 = .000 609 013, which represents density or specific gravity of steam 
at pressure of atmosphere. 

Elasticity of vapor of alcohol, at all temperatures, is about 2.125 times that of steam. 

Specific Gravity , compared with air, is as w r eight of a cube foot of it compared 
with equal volume of air. Thus, weight of a cube foot, of steam at 212 0 and at 
pressure of atmosphere is 266.124 grains; weight of a like volume of air at 32 0 is 
565.096 grains, and at 62° 532.679 grains. Hence 266. 124-4- 532.679 = .499 59, specific 
gravity of steam compared with air at 32 0 , and with water it is .000609013. 


* Pole’s Formula makes it 1712. 



STEAM. 


705 


Total Heat of Saturated Steam. 

1081.4 -|- .305 T = total heat. T representing initial temperature of water. 
Illustration.—W hat is total heat of steam at 212 0 ? 

1081.4 -f-. 305 X 212 = 1146.06. 

As specific heat of water is .9 greater at 212 0 than at 32°, hence the 212 0 would 
be 212.9, an d 1146.33 the result. 

Total Heat of Gaseous Steam 1074.6 -f- 475 T = total heat. 

Absorption of Heat in Generation of 1 Lb. of Water from 32° to 212 0 . 

Sensible heat, or heat to raise temperature of water Units. Force, lbs. 

from 32 0 to 212 0 ... 180.9 X 772 -139655 

Latent heat to produce steam.892.9 

“ “ to resist atmospheric pressure 14.7 lbs. 

per sq. inch. 72.3 965.2X772 = 745134 

Total or constituent heat. 1146.1 884789 

This number, 1146.1, is a Constant , and expresses units of heat in 1 lb. of steam 
from 32 0 up to temperature at which conversion takes place. 

Thus, 1 lb. water heated from 32 0 to 332°, requires as much heat as 

would raise 300 lbs. i°. Hence... 300 0 

And x lb. water converted into steam at 332 0 (= 106 lbs. pressure), ab¬ 
sorbs as much heat for its conversion as would raise 846.1 lbs. water 


i°. Hence. 846. i° 

H46. i° 

Mechanical Equivalent of Heat contained in Steam. 

1 lb. water heated from 32 0 to 212 0 requires as much heat as would raise 

180 lbs. i°. Hence. 180.9° 

1 lb. water at 212°, converted into steam at 212 0 (= 14.7 lbs. pressure), 
absorbs as much heat for its conversion as would raise 966.6 lbs. water 
i°. Hence. 965.2° 

1146. i° 


Mechanical Equivalent , or maximum theoretical duty of quantity of heat in 1 lb. 
of steam, is 772 lbs., which X 1146.1 units of heat = 884 789.2 lbs. raised 1 foot high. 

To Compute Pressure of Steam. 

When Height of Column of Mercury it will Support is given. Rule. —Di¬ 
vide height of column of mercury in ins. by 2.037 5861 and quotient will give 
pressure per sq. inch in lbs. 

Example. — Height of a column of mercury is 203.7586 ins.; what pressure per 
sq. inch will it contain ? 

203.7586 = 2.037586 = 100 lbs. 

To Compute "Weight of a Cifbe Foot of Steam. 

Rule. —Multiply its density by 62.425. 

Example. —Density of a volume of steam is .000609013; what is its weight? 

.0006090x3 X 62.425 = .038 016 825 lbs. 

Note.—S ee table, page 708. 

1 atmosphere or 14.723 307 lbs. per sq. inch = 30 ins. of mercury. 

To Compute Temperature of Steam. 

Rule. —Multiply 6th root of its force in ins. of mercury by 177.2, sub¬ 
tract 100 from product, and remainder will give temperature in degrees. 

Example.— When elastic force of steam is equal to a pressure of 64 ins. of mer¬ 
cury, what is its temperature ? 

Note. —To extract 6th root of a number, ascertain cube root of its square root. 

-^64 = 8 , and -§/8 = 2. Hence, 2 x 177-2 —100 = 254.4° t. 

Or.-2938^16-85 — t. p representing pressure in lbs. per sq. inch. 

’6.1993544 — log. p 














STEAM. 


706 

To Compute Volume of "Water contained in a given Vol¬ 
ume of* Steam. 

When its Density is given. Rule. —Multiply volume of steam in cube 
feet by its density, and product will give volume of water in cube feet. 

Example. —Density of a volume of 16420 cube feet of steam is .000609; what is 
the weight of it in lbs. ? 

16 420 X .000609 = 10 — volume of water, which X 62.425 = 624.25 lbs. 

To Compute Pressure of Steam in Ins. of Mercury, or 
DT>s. per Sq. Inch. 

When Temperature is given. Rule i.— Add 100 to temperature, divide 
sum proportionally by 177.2 for temperature of 212 0 , and by 160 for tem¬ 
peratures up to 445 0 ; or, 177.6 for sea-water, and 185.6 for sea-water sat¬ 
urated with salt, and 6th power of quotient will give pressure. 

Example. —Temperature of steam is 254 0 ; what is its pressure? 

100 -j- 254 - 4 - 177.2 = 1.998, and 1.998 s = 63.62 ins. 

When Ins. of Mercury are given. 2.—Divide ins. of mercury by 2.037586, 
and quotient will give pressure. 

When Pressure in Lbs. is given. 3.—Multiply pressure by 2.037 586. 

To Compute Specific Gravity of Steam compared witlx 

AVir. 

Rule. — Divide constant number 829.05 (1642 x .5049) by volume of 
steam at temperature of pressure at which gravity is required. 

Example. —Pressure of steam is 60 lbs., and volume 437; what its specific gravity ? 
829.05-1-437 = 1.898. 

To Compute Volume of a Cube Ifoot of Water in Steam. 

When Elastic Force and Temperature o f Steam are given. Rule. —To 
430.25 for temperature of 212 0 , and 332 for temperatures up to 445 0 , add 
temperature in degrees; multiply sum by 76.5, and divide product by elastic 
force of steam in ins. of mercury. 

Note. —When force in ins. of mercury is not given, multiply pressure in lbs. per 
sq. inch by 2.037 586. 

Example. —Temperature of a cube foot of water evaporated into steam is 386°, 
and elastic force is 427.5 ins.; what is its volume? 

Assume 369 for proportionate factor. 369 + 386X76.5-1-427.5 = 135.1 cubefeet. 

Or, for 1 lb. of steam, 2.519 — .941 log. 7? = log. V in cubefeet. 

Assume p = 14.7 lbs. 2.519 — .941 log. 14.7 = 2.519 — 1.098 = 1.421 = log. 26.34 
cube feet, which x 62.425 = 164 feet. 

Or, When Density is given. —Divide 1 by density, and quotient will give volume 
in cube feet. 

To Compute Density or Specific Gravity of Steam. 
When Volume is given. Rule. —Divide 1 by volume in cube feet. 
Example. —Volume is 210; what is density? 

1 - 4 - 210 = .004 761. Or, for 1 lb. of steam, .941 log. p — 2.519 = log. D. 

When Pressure is given. —Take temperature due to pressure, and proceed 
as by rule to compute volume, which, when obtained, proceeds as above. 

To Compute Volume of Steam required. to raise a Given 
Volume of Water to any Given Temperature. 

Rule. —Multiply water to be heated by difference of temperatures between 
it and that to which it is to be raised, for a dividend; then to temperature 
of steam add 965.2°, from that sum take required temperature of water for 
a divisor, and quotient will give volume of water. 


STEAM. 


707 


Example. —What volume of steam at 212° will raise 100 cube feet of water at 8o° 

tO 212°? 


IOO X 212 80 

-:—;-= 13.68 cube feet water; or, (13.68X1642 — 212) = 22 4630/ steam. 

212-I-965.2—2x2 1 

To Compute Volume of Water, at any Given Temper¬ 
ature, that must be IVtixeci with. Steam to Raise or Re¬ 
el vice tlie HVIixtnre to any Required Temperature. 
Rule.—F rom required temperature subtract temperature of water; then 
ascertain how often remainder is contained in required temperature sub¬ 
tracted from sum of sensible and latent heat of the steam, and quotient will 
give volume required. 

Sum of Sensible and Latent Heats for a range of temperatures will be found under 
Heat, pages 508 and 509. 

Example. —Temperature of condensing water of an engine is 8o°, and required 
temperature ioo°; what is proportion of condensing water to that evaporated at a 
pressure of 34 lbs. per sq. inch ? 

Sum of sensible and latent heats 1190.4°. 

100 — 80 = 20. Then, 1190.4 —100 - 4 - 20 = 54-52 to 1. 


When Temperature of Steam is given. 


i—(- T — t 


V. I representing latent heat, 


T and t temperatures of steam and required temperature , w temperature of condensing 
water , and V volume of condensing water in cube feet. 

Illustration.— Temperature of steam in a cylinder is 257.6°, and other elements 
same as in preceding example; required volume of injection water? Latent heat 
of steam at 230° = 932.8°. 


932.8 -j- 257.6 —100 1090.4 

100 — 80 20 


54.52 volumes. 


To Compute Temperature of Water in Condenser or 
Reservoir of a Steam-engine. 


1 +T+VX w 


; t. Illustration. —Assume elements as preceding. 


932.8 + 257. 6 + 54.52 X 80 _ 5552 _ 0 

54.52 + 1 55-52 

To Compute Latent Heat of Saturated Steam. 
1115.2—-.708 t = l. Illustration.— Assume temperature 257.6° as preceding. 
1115.2 —.708 X 257.6 = 932.8°. 

To Compute Total Heat of Saturated Steam. 

305 t + 1081.4 = H. Illustration.— Assume temperature as preceding. 

.305 X 257.6 + 1081.4 = 1160. 

Elastic Eorce and. Temperature of Vapors of gAlcoliol, 
Etlier, Sulphuret of Carbon, Petroleum, and Tur¬ 
pentine. 

Force in Ins. of Mercury. 

o | Ins. 
Petroleum. 


0 

Ins. 

0 

Ins. 

0 

Ins. 

0 

Ins. 

Alcohol. 

Alcohol. 

Ether. 

Sulphuret of 

32 

•4 

140 

13-9 

34 

6.2 

Carbon. 

5 ° 

.86 

160 

22.6 

54 

15-3 

53-5 

7-4 

60 

1.23 

173 

3 ° 

74 

16.2 

7 2 - 5 

12.55 

70 

1.76 

180 

34-73 

94 

24.7 

no 

3 ° 

80 

2-45 

200 

53 

96) 


212 

126 

9° 

3-4 

212 

67-5 

104 j 


279.5 

3 °° 

IOO 

4-5 

220 

7 8 -5 

120 

39-47 

347 

606 

120 

8.1 

24O 

in.24 

15° 

67.6 



130 

10.6 

264 

166.1 

212 

178 




316 

345 

375 


3 ° 

44.1 

64 


Oil of 
Turpentine. 


315 

357 

37 ° 


3 ° 

47.78 

62.4 





























708 


STEAM, 


Saturated. Steam. 

Pressure, Temperature, Volume , and Density. 


Pri 

per 
Sq. 
Inch. 

ASSURE 

in 

Mer¬ 

cury. 

Temperature. 

Total Heat 
from Water 
at 32°. 

Volume of 

x Lb. 

Density, 

or Weight of 

one Cube Foot. 

Pr 

per 

Sq. 

Inch. 

ESSURE 

in 

Mer¬ 

cury. 

Temperature. 

Total Heat 

from Water 

at 32 0 . 

Volume of 

x Lb. 

Density, 

or Weight of 

one Cube Foot. 

Lbs. 

| Ins. 

0 

O 

Cub. ft. 

Lb. 

Lbs. 

Ins. 

O 

I 0 

Cub. ft 

Lb. 

I 

2.04 

102. I 

1112.5 

330 - 36 

•003 

58 

118.08 

290.4 

11 70 

7.24 

■138 

2 

4.07 

126.3 

m 9-7 

172.08 

.005 8 

59 

120. 12 

291.6 

1170.4 

7. 12 

•1403 

3 

6.11 

141.6 

1124.6 

117.52 

.0085 

60 

122.16 

2Q2. 7 

II7O.7 

7.OI 

.1425 

4 

8.14 

I 53 -I 

1128.1 

89.62 

.Oil 2 

61 

I24. 19 

293.8 

1171.1 

6.9 

.1447 

5 

10.18 

162.3 

1130-9 

72.66 

• 013 8 

62 

126.23 

294.8 

II 7 I -4 

6.81 

.1469 

6 

12.22 

170.2 

H 33-3 

61.21 

.0163 

63 

128.26 

295-9 

II 7 I -7 

6.7 

-1493 

7 

14.25 

176.9 

H 35-3 

52-94 

.018 9 

64 

130-3 

296.9 

1172 

6.6 

.1516 

8 

16.29 

182.9 

H 37-2 

46. 6q 

.021 4 

65 

132-34 

298 

1172-3 

6.49 

•1538 

9 

18.32 

188.3 

1138.8 

41.79 

.0239 

66 

x 34 - 37 

299 

1172.6 

6.41 

.156 

IO 

20.36 

193-3 

1140.3 

37-84 

.026 4 

67 

136-4 

3 °o 

1172.9 

6.32 

•1583 

II 

22.39 

197.8 

1x41.7 

34-63 

.028 9 

68 

138.44 

300.9 

1 X 73-2 

6.23 

.1605 

12 

24.43 

202 

H 43 

31.88 

.031 4 

69 

140.48 

301.9 

1 X 73-5 

6.15 

. 1627 

13 

26.46 

205.9 

1144.2 

2 9-57 

• 0338 

70 

142.52 

302.9 

1173.8 

6.07 

. 1648 

14 

28.51 

209.6 

II 45-3 

27.61 

.036 2 

7 i 

144-55 

303-9 

1174.x 

5-99 

. 167 

14.7 

2 Q.Q 2 

212 

1146.1 

26.36 

.038 02 

72 

146.59 

304.8 

IX 74-3 

5 - 9 i 

. 1692 

is 

30-54 

213-1 

1146.4 

25.85 

.0387 

73 

148.62 

305-7 

1174.6 

5-83 

.1714 

16 

32-57 

216.3 

H47.4 

24.32 

.O4I I 

74 

150.66 

306.6 

H 74-9 

5-76 

•1736 

17 

34.61 

219.6 

1148.3 

22.96 

•043 5 

75 

152.69 

307-5 

1 X 75-2 

5.68 

•1759 

18 

36.65 

222.4 

II 49.2 

21.78 

•045 9 

76 

r 54-73 

308.4 

1 X 75-4 

5.61 

.1782 

19 

38.68 

225.3 

1150-1 

20.7 

.0483 

77 

156.77 

309-3 

H 75-7 

5-54 

. 1804 

20 

40.72 

228 

1150-9 

19.72 

.0507 

78 

158.8 

310.2 

1176 

5-48 

. 1826 

21 

42-75 

230.6 

ii 5 i -7 

18.84 

•0531 

79 

160.84 

311-1 

1176.3 

5 - 4 i 

. 1848 

22 

44-79 

233-1 

1152-5 

18.05 

•055 5 

80 

162.87 

312 

1176.5 

5-35 

.1869 

23 

46.83 

235-5 

H 53-2 

17.26 

• 058 

81 

164.91 

312.8 

1176.8 

5-29 

.1891 

24 

48.86 

237.8 

” 53-9 

16.64 

.0601 

82 

166.95 

313-6 

1177.x 

5-23 

• 19 X 3 

25 

50.9 

240.1 

1154.6 

15-99 

.0625 

83 

168.98 

314-5 

1 X 77-4 

5 -i 7 

•1935 

26 

52-93 

242.3 

H 55-3 

15-38 

■ 065 

84 

171.02 

3x5-3 

1177.6 

5- xi 

•1957 

27 

54-97 

244.4 

1155.8 

14.86 

.0673 

85 

I 73-°5 

316.1 

1 X 77-9 

5-05 

. 198 

28 

57 -oi 

246.4 

1156.4 

14-37 

.0696 

86 

175-09 

316-9, 

1178. r 

5 

.2002 

2 9 

59-°4 

248.4 

H 57 -I 

13-9 

.071 9 

87 

I 77 -I 3 

317-8 

1178.4 

4.94 

.2024 

30 

61.08 

250.4 

1157.8 

13.46 

■074 3 

88 

179. 16 

318.6 

1178.6 

4.89 

.2044 

3i 

63.11 

252.2 

1158.4 

13-05 

.076 6 

89 

181.2 

3x9-4 

1178.Q 

4.84 

.2067 

32 

65-15 

254 -i 

1158.9 

12.67 

.078 9 

90 

183.23 

320. 2 

1x79.x 

4-79 

.2089 

33 

67.19 

255-9 

11 59-5 

12.31 

.081 2 

9 1 

185.27 

321 

xi 79-3 

4-74 

.2111 

34 

69.22 

257.6 

1160 

11.97 

•0835 

92 

187.31 

321.7 

1 X 79-5 

4.69 

•2133 

35 

71.26 

2 59-3 

1160.5 

11.65 

• 085 8 

93 

i8 9-34 

322.5 

1179.8 

4.64 

•2155 

3 & 

73-29 

260.9 

1161 

n -34 

.088 1 

94 

191.38 

323-3 

1180 

4.6 

.2176 

37 

75-33 

262.6 

1161.5 

11.04 

.0905 

95 

i 93 - 4 i 

324-1 

1180.3 

4-55 

.2198 

3 « 

77-37 

264.2 

1162 

10.76 

.092 9 

96 

195-45 

324.8 

1180.5 

4 - 5 i 

.2219 

39 

79-4 

265.8 

1162.5 

10.51 

.0952 

97 

X 97-49 

325.6 

1180.8 

4.46 

.2241 

40 

81.43 

267.3 

1162.9 

IO. 27 

.0974 

98 

199.52 

326.3 

1181 

4.42 

.2263 

4 i 

83-47 

268.7 

xi6 3 -4 

10.03 

.099 6 

99 

201.56 

327-1 

1181.2 

4 - 37 

.2285 

42 

85-5 

270.2 

1163.8 

9.81 

. 102 

IOO 

203 .59 

327-9 

1181.4 

4-33 

.2307 

43 

87-54 

271.6 

1164.2 

9-59 

. IO4 2 

IOI 

205.63 

328.5 

1181.6 

4.29 

.2329 

44 

89.58 

273 

1164.6 

9-39 

.106 5 

102 

207.66 

329.1 

1181.8 

4-25 

•2351 

45 

91.61 

274-4 

1165.1 

9.18 

.1089 

103 

2O9.7 

329.9 

1182 

4.21 

•2373 

46 

93-65 

275.8 

1165.5 

9 

.III I 

IO4 

211.74 

330.6 

1182.2 

4.18 

•2393 

47 

95-69 

277.1 

1165.9 

8.82 

•1133 

105 

213-77 

33 i -3 

1182.4 

4.14 

.2414 

48 

97.72 

278.4 

1166.3 

8.65 

.1156 

106 

215.81 

33 i -9 

1182.6 

4. IX 

•2435 

49 

99.76 

279.7 

1166.7 

8.48 

.1179 

IO7 

217.84 

332-6 

1182.8 

4.07 

.2456 

50 

101.8 

281 

1167.1 

8.31 

. 120 2 

108 

219. 88 

333-3 

1183 

4.04 

•2477 

5 i 

103.83 

282.3 

1167.5 

8.17 

. 122 4 

IO9 

221.92 

334 

1x83-3 

4 

•2499 

52 

105.87 

283.5 

1167.9 

8.04 

. 124 6 

no 

223-95 

334-6 

1x83-5 

3-97 

.2521 

53 

xo 7-9 

284.7 

1168.3 

7. 88 

.126 9 

III 

225.99 

335-3 

1x83.7 

3-93 

• 2543 

54 

109.94 

285.9 

1168.6 

7-74 

. 129 I 

112 

228.02 

336 

1183.9 

3-9 

2564 

55 

111.98 

287. 1 

1169 

7.61 

•1314 

113 

230.06 

336-7 

1184.1 

3.86 

.2586 

59 

II4.OI 

288.2 

1169.3 

7.48 

■1336 

114 

232.1 

337-4 

1184.3 

3-83 

.2607 

57 

116.05 

289.3 

1169.7 ) 

7 - 3 6 

.1364 

115 

2 34 -13 

338 

1184.5 

3-8 

.2628 



















































STEAM 


709 


Pei 

per 

Sa. 

Inch. 

f £ 

ISSUES = 

z 

. I Zi 

m i f=- 

Mer - 1 1 

cury. H 

- 5 
© ~ . 

l|* 

: •S 

id 

yi 

^ it - 

3J — 

2 

“* i. © 
°§ 

Pressure 

per | in 

So. ! Mer- 
Incn. I cury. 

. I 

® 

tm . — © 

-mm ^ W 

TS —* 3 

g 

~ © 0 s 

0 

0 • 

n 

: 0 M 

> 

[ 

if 1 

° 0 

Lbs. 

Ins. | 0 

0 

1 Cub. ft. 

Lb. 

Lbs. 

Ins. 

O 1 O 

Cub. ft. 

Lbs. 

116 

236-17 33 s - 6 

1184.7 

j 3-77 

.2649 

149 

! 303-35 

357.8 1190.5 

2.98 

•3357 

117 

238.2 339-3 

1184.9 

3-74 

.2652 

150 

305-39 

[358.3 1190.7 

2.96 

•3377 

118 

' 240.24 339.9 

1185.1 

: 3-7i 

|-2674 

155 

315-57 

361 1191.5 

2-8 7 

•3484 

119 

J 242.28 340.5 

1185.3 

i 3-68 

.2696 

160 

325-75 

363-4 1192-2 

2-79 

•359 

120 

2 44-3i 34i-1 

1185.4 

3-65 

!-2738 

165 

335-93 

366 1192.9 

2.71 

■3695 

121 

246-35 341-8 

1185.6 

3.62 

.2759 

170 

346-11 

368.2 1193.7 

2.63 

•3798 

122 

248.38 342.4 

1185.8 

3-59 

.278 

175 

356.29 

370.8.1194.4 

2.56 

•3899 

123 

250-42 343 

1186 

3-56 

.2801 

180 

366.47 

372.9! II95.1 

2-49 

.4009 

124 

252-45 343-6 

1186.2 

3-54 

.2822 

is 5 

376.65 

375-3 1195-8 

2-43 

.4117 

125 

254-49 344-2 

1186.4 

3-5i 

.2845 

190 

386.83 

377.5' 1196.5 

2-37 

.4222 

126 

256-53 ; 344-8 

1186.6 

3-49 

.2867 

*95 

397-oi 

379-7! 1197*2 

2.31 j 

•4327 

127 

258.56'345-4 

1186.8 

3-46 

.2889 

200 

407.19 

381.7 1197.8 

2.26 ; 

•4431 

128 

260.6 346 

1186.9 

3-44 

.2911 

210 

427-54: 

386 1199.1 

2.16 

•4634 

129 

262.64 340-6 

1 *87.1 

3*41 

2933 | 

220 

447-9 

389.9 | 1200.3 

2.06 f 

.4842 

130 

264.67 347.2 

1187.3 

3-38 

•2955 

230 

468.26 

393.8 ; 1201.5 

1.98 ! 

•5052 

131 

266.71 347.8 

1187.5 

3- 35 

2977 

240 

488.62 

397.5 • 1202.6 

1.9 

.5248 

132 

268.74 348.3 

1187.6 

3-33 

•2999 

250 

508.98 

401. I 1 I203.7 

i * - 5 

-5464 

133 

270.78 348.9 

1187.8 

3-3i | 

•3°2 | 

260 

529.34 

4O4.5 I2O4.8 

1.76 

.5669 

134 

272.81 349.5 

1188 

3-29 

.304 | 

270 

549-7 

407.9] 1205.8 

i-7 

.5868 

135 

274-85 350-1 

1188.2 

3-27 

.306 

280 1 

570.06 

411.2 i 1206.8 

1.64 

.6081 

136 

276- 89 | 350.6 

1188.3 

3-25 

-3°8 

29O 

590*42 

414.4 1207.8 

i-59 

.6273 

x 37 | 

278.92 351.2 

1188.5; 

3.22 

.3101 1 

3°0 j 

61O.78 

417.5 j 1208.7 

i-54 

.6486 

133 

280.96 351-8 | 

I188.7 

3-2 

.3121 

350 i 

712-57 

430.1 1212.6 

i-33 

.7498 

139 

282.99 352-4 

H88.9 j 

3 -i 8 

•3M2 1 

400 

8i4-37 

444-9'1217-1 ! 

1.18 

.8502 

140 

285-03 352.9 | 

1189 

3.16 

.3162 

450 | 

9l6.I7 

456.71220.7 

1.05 | 

•9499 

141 

287.07 j 353.5 ' 

I189.2 j 

3-i4 

.3184 

5 oo j 

IOl8 

467.511224 

-95 

1.049 

142 | 

289. t 354 

I189.4 j 

3.12 

.3206 

550; 

III9.8 

477*5 1227 

.87 j 

1.148 

143 j 

291.14 354-5’ 

1189.6 

3-i 

.3228 1 

600 • 

1221.6 ; 

487 11229.9 ‘ 

.8 

1.245 

144 

293-17 355 

H89.7 

3.08 

■325 

650 

1323-4 

495.6; 1232-5 

•74 

1.342 

145 

295.21 355.6 j 

I18Q.9 

3.06 

•3273 j 

700 

1425.8 

504-1 1235-1 

.69 

1-4395 

140 j 

297-251356-1 

II90 

3-°4 

•3294 , 

800 ; 

1628.7 

519-51239.8 

.61 

1.6322 

1471 

299.28 ! 356.7 j 

1190. 2 

3*02 

•3315 , 

9°° 

1832.3 

533-6 1 1244.2 1 

•55 

i8 235 

148 1 

30I.32I357-2! 

II90-3 

3- 1 

•3336 ! 

1000 | 

2035-9 1 

546-5 1 1248.1 

-5 

2.014 


Saturated. Steam. from 32° to 212°. (Claudel.) 


Tem¬ 

pera¬ 

ture. 

Pressure. 

Mercu- Per 

ry. 1 Sq. Inch. 

Weigrht 
of 100 
Cub. Feet. 

Volume 

Of | 

1 Lb. 

Tem¬ 

pera¬ 

ture. 

Pressure. 

Mercu- Per 

ry. Sq. Inch. 

Wieight 
of 100 
Cub. Feet. 

Volume 

of 

1 Lb. 

O 

Ins. 

Lbs. 

Lb. 

Cub. Feet. 

O 

Ins. 

Lbs. 

Lbs. 

Cub. Feet. 

32 

M 

00 

H 

.089 

.031 

3226 

125 

3-933 

1.932 

•554 

180.5 

35 

.204 

. 1 

•°34 

2941 

130 

4.509 

2-215 

• 6 3 

158-7 

4 ° 

.?48 

. 122 

.041 

2439 

135 

5-174 

2-542 

• 7 i 4 

140.1 

45 

•299 

•147 

.049 

2041 

140 

5-86 

2.879 

.806 

124.1 

5 ° 

.362 

,178 

-059 

1695 

145 

6.662 

3-273 

.909 

no 

55 

.426 

.214 

,07 

1429 

150 

7-548 

3.708 

1.022 

97-8 

60 

•517 

•254 

.082 

1220 

r 55 

8-535 

4-193 

i*i 45 

87.3 

65 

.619 

• 3°4 

•097 

1031 

l60 

9-63 

4-731 

i -333 

75 

7 ° 

•733 

•36 

.114 

877.2 

165 

10.843 

5-327 

1.432 

69-8 

75 

.869 

•427 

.134 

746-3 

170 

12.183 

5 - 9^5 

1.602 

62.4 

80 

1.024 

•503 

■156 

641 

175 

13.654 

6.708 

i -774 

56.4 

85 

1.205 

•592 

.182 

549-5 1 

180 

15.291 

7 - 5 II 

1.97 

50.8 

9 ° 

1.41 

•693 

,212 

471-7 | 

l85 

17.041 

8-375 

2.181 

45-9 

95 

1.647 

.809 

.245 

408.2 

190 

19.001 

9-335 

2.411 

4 i -5 

100 

I - 9 I 7 

•942 

-283 

353-4 

!95 

21.139 

10.385 

2.662 

37-6 

105 

2. 229 

1.095 

•325 

307.7 

200 

23.461 

11.526 

2-933 

34 -i 

no 

2.579 

1.267 

•373 

268.1 

205 

25-994 

12-77 

3.225 

3 i 

US 

2.976 

1.462 

.426 

234-7 | 

210 

28.753 

14.127 

3-543 

28.2 

120 

3-43 

1.685 

•488 

204.9 

3 

212 

0 

29.922 

14.7 

3.683 

27.2 






































7io 


STEAM. 


GASEOUS STEAM. 

When saturated steam is surcharged with heat, or superheated, it is termed 
gaseous or steam-gas. The distinguishing feature of this condition of steam 
is its uniformity of rate of expansion above 230°, with the rise of its tem¬ 
perature, alike to the expansion of permanent gases. 

To Compute Total Heat of Gaseous Steam. 

1074.6 -f- .475 t = H. t representing temperature , and H total heat in degrees. 
Hence, total heat at 212 0 , and at atmospheric pressure = 1175.3 0 . 

Specific gravity = .622. 

To Compute Velocity of Steam. 

Into a Vacuum. Rule. —To temperature of steam add constant 459, and 
multiply square root of sum by 60.2; quotient will give velocity in feet per 
second. 

Into Atmosphere. 3.6 ffh — Y. V representing velocity as above, and h height in 
feet of a column of steam of given pressure and uniform density, weight of which is 
equal to pressure in unit of base. 

Illustration. —Pressure of steam 100 lbs. per sq. inch, what is velocity of its 
flow into the air? 

Cube foot of water = 62.5 lbs., density of steam at 100 lbs. = 270 cube feet. Hence, 
62.5 : 100 :'. 270 : 432 = volume at 100 lbs. pressure , and 432 X 144 = 62 208 feet — 
height of a column of steam at a pressure of 100 lbs. per sq. inch. 

Then 3.6 62 208 = 898 feet. 

EXPANSION. 

To Compute IPoint of Catting off to Attain Limit of 

Expansion. 

b -\-f L -T- P —point of cutting off. b representing mean bade pressure for entire 
stroke , in lbs. per sq. inch , f friction of engine, P initial pressure of steam, all in lbs. 
per sq. inch, and L length of stroke, in feet. 

Illustration. —Assume stroke of piston 9 feet, pressure 30 lbs., mean back press¬ 
ure 3 lbs., and friction 2 lbs. 

3 + 2X9^- 30 = i -5 f eet - 


To Compute Actual Ratio of Expansion. 

L + c 

— — — R. c representing clearance or volume of space between valve seat and 

i “ 4 ” C 

mean surface of piston, at one or each end in feet of stroke, l length of stroke at point 
of cutting off, excluding clearance in feet, and R actual ratio of expansion. 

Illustration.— Assume length of stroke 2 feet, clearance at each end 1.2 ins., 
and point of cutting off 1 foot. 

2 -4- . I 

1.2 ins. — .1. Then — — = 1.9 ratio. 

1 + .1 


To Compute Pressure at any IPoint of IPeriod. of Ex¬ 
pansion. 

When Initial Pressure is given. P £-t-s=p. p representing pressure at period 
of given portion of stroke, both in lbs. per sq. inch, and s any greater portion of stroke 
than l. 

When Final Pressure is given. P'xL'-f« = p. P' representing final pressure, 
in lbs. per sq. inch, and 1/ length of stroke, including clearance, in feet. 

Illustration l— Assume length of stroke 6 feet, clearance at each end 1.2 ins., 
pressure of steam 60 lbs., point of cutting off one third; what is pressure at 4 feet? 

1.2 ins. = . 1 foot. 60 X 2 -j—. 1 -7" 4 -j- • 1 — 30.73 Ms. 

2.— What is pressure in above cylinder at 2.8 feet, when final pressure is 21 lbs. ? 


21 X 6-f-. 1 ■— 2.8 -f-. 1 = 44.17 lbs. 







STEAM. ^ I I 

To Compute IVLean or Total Average Pressure. 


P {V x + hyp. log. R — c) 


: p' or mean or average pressure. I' length of stroke at 


point of cutting off, including clearance. 

Illustration. —Assume elements of preceding cases: i + hyp. log. R = 2.065. 


60(2.1X2.065 — .1) 254.19 

6 = 6 


42.365 lbs. 


To Compute Einal Pressure. 

PXi'-rS = P'. 

Illustration. —Assume elements of preceding cases, steam cut olf at 2 feet. 


60 X 2 -f-. 1 -r- 6 -(-. 1 = 20. 65 lbs. 

To Compute Nfean Effective Pressure. 


P (l' 1 -f- hyp. log. R — c 


-b, or ( p' — b). 


Illustration. —Assume elements of preceding cases; b — 2 lbs. per sq. inch. 

= 2^? — 2 = 40.365 lbs. 


60 (2.1 X 2.065 —.1) 


To Compute Initial Pressure to Produce a Given A.v- 
erage Effective or Net Pressure. 


p' L 


V (1 +hyp. log. R) —c 

Illustration. —Assume elements of case 1. 

6 + -i_42-365X6 


= P. 


2 + -I 


: 2.9 ratio. 


(2.1 X 2.065)- 


^ = 60 lbs. 


4.2365 


To Compute Point of Crutting off for a Given Ratio of 

Expansion. 

L' R — c. Or, L 4- c R — c = l. 

6 - 4 -. 1 6 -f-. 1 

Illustration. —Assume elements of preceding cases: R= — — = 2.9, and — 1 —- 

. 2 + .1 2.9 

— .1 — 2 feet. 

To Compute Pressure in a Cylinder, at any Point of Ex¬ 
pansion, or at End of Stroke. 


P I'^rl+c — P, or P-t-R. 

Illustration. —Assume elements of preceding cases: 

60 x 2.1 _ „ .60 . „ 

-—— = 60 lbs ., and — = 20.69 lbs. 

2 + .1 2.9 

To Compute Initial Pressure for a Required Net Effec¬ 
tive Pressure for a Given Ratio of Expansion. 

P' h 


W -f a 6 L _ 

a (f 1 -f hyp. log. R — c) 


Or, 


V 1 -}- hyp. log. R — c 


— P. W representing net¬ 


work in foot-lbs. — a L p' — b, and a area of piston, in sq. ins. 

Illustration. —Assume elements of preceding cases: area of piston = 100 sq. 
ins., back pressure 2 lbs., and net effective pressure = 42.365 lbs. 


100 X 6 X 42.365 — 2 = 24 219 foot-lbs. 
24219 + 100X2X6 _ 25419 42-365X6 


100 X 2.1 X 2.065—- 1 100X4-2365 


: 60 lbs. 


254.19 


2.1X2.065 —.1 4-2365 


: 60 lbs. 





























712 


STEAM. 


^Points of Expansion. 

Relative points of expansion, including clearance 5 per cent., assuming 
stroke of piston to be divided as follows, and initial pressure = 1. 

Point .1 .75 .6875 .625 .5625 .5 .4375 .375 .333 .25 .2 .125 .1 

Ratio .1 1. 31 1.43 1.55 1.71 1.91 2.15 2.43 2.74 3.5 4.4 6. 7. 

Hyp. Log. of above Ratios. 

.0 1.27 1.36 1.44 1.54 1.65 x.77 1.9 2 2.25 2.43 2.79 2.95 


Hyperbolic LogaiMth.rris. 


No. 

f Log. 

I No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

x.05 

.0488 

2.65 

.9746 

4-25 

1.447 

5-8 

i-758 

7-4 

2.001 

I. I 

•0953 

2.66 

•9783 

4-3 

J -459 

5-85 

1.766 

7-45 

2.008 

i-i 5 

.1398 

2.7 

•9933 

4-33 

1.466 

5-9 

i -775 

7-5 

2.015 

1.2 

.1823 

2-75 

1.0116 

4-35 

1.47 

5-95 

1-783 

7-55 

2.022 

1.25 

.2231 

2. 8 

1.0296 

4.4 

1.482 

6 

1.792 

7.6 

2.028 

i -3 

.2624 

2.85 

10473 

4-45 

i -493 

6.05 

1.8 

7-65 

2-035 

i -33 

.2852 

2.9 

1.0647 

4-5 

1.504 

6.1 

1.808 

7.66 

2.036 

i -35 

.3001 

2-95 

1.0818 

4-55 

i- 5 i 5 

6.15 

1.816 

7-7 

2.041 

1.4 

• 33 6 S 

3 

1.0986 

4.6 

1.526 

6.2 

1.824 

7-75 

00 

0 

0$ 

i -45 

■ 37 l6 

3-05 

1-1151 

4-65 

1-537 

6.25 

1-833 

7.8 

2.054 

i -5 

•4055 

3 - 1 

1.1314 

4.66 

1-539 

6-3 

1.841 

7-85 

2.061 

i -55 

•4383 

3 -i 5 

!• I 474 

4-7 

1.548 

6-33 

1.845 

7-9 

2.067 

1.6 

•47 

3-2 

1.1632 

4-75 

i-558 

6-35 

1.848 

7-95 

2.073 

1.65 

.5008 

3-25 

1.1787 

4.8 

1.569 

6.4 

1.856 

8 

2.079 

1.66 

.5068 

3-3 

i ^939 

4-85 

i -579 

6-45 

1.864 

8.05 

2.086 

i -7 

.5306 

3-33 

1.203 

4.9 

1.589 

6-5 

1.872 

8.x 

2.092 

i -75 

•5596 

3-35 

I. 209 

4-95 

i-599 

6-55 

1.879 

8.15 

2.098 

1.8 

.5878 

3-4 

1.2238 

5 

1.609 

6.6 

1.887 

8.2 

2. IO4 

1.85 

.6152 

3-45 

1.2384 

5-05 

1.619 

6.65 

1.895 

8.25 

2 . II 

1.9 

.6419 

3-5 

1.2528 

5 -i 

1.629 

6.66 

1.896 

8-3 

2.116 

i -95 

.6678 

3-55 

1.2669 

5 -i 5 

1.639 

6.7 

1.902 

8-33 

2. 12 

2 

.6931 

3-6 

1.2809 

5-2 

1.649 

6-75 

1.909 

8-35 

2.122 

2.05 

.7178 

3- 6 5 

1.2947 

5-25 

1.658 

6.8 

1.917 

8.4 

2.128 

2. I 

.7419 

3.66 

1-2975 

5-3 

1.668 

6.85 

1.924 

8-45 

2.134 

2.15 

•7655 

3-7 

1-3083 

5-33 

1-673 

6.9 

I - 93 I 

8-5 

2. 14 

2.2 

.7885 

3-75 

1.3218 

5-35 

1.677 

6-95 

I -939 

8-55 

2.146 

2.25 

.8109 

3-8 

i -335 

5-4 

1.686 

n 

1.946 

8.6 

2.152 

2-3 

.8329 

3-85 

1.3481 

5-45 

1.696 

7-05 

i -953 

8.65 

2.158 

2-33 

.8458 

3-9 

1.361 

5-5 

1-705 

7-i 

1.96 

8.66 

2.159 

2-35 

•8544 

3-95 

1-3737 

5-55 

1.714 

7-i5 

1.967 

8.7 

2.163 

2.4 

•8755 

4 

1-3863 

5-6 

1-723 

7.2 

1.974 

8-75 

2.169 

2-45 

.8961 

4-05 

1-3987 

5-65 

1.732 

7-25 

1.981 

8.8 

2-175 

2.5 

.9x63 

4.1 

I.4II 

5.66 

1-733 

7-3 

1.988 

8.85 

2.18 

2-55 

•936 

4-i5 

1-4231 

5-7 

I - 74 I 

7-33 

1.992 

8.9 

2.186 

2.6 

•9555 

4.2 

I - 435 I 

5-75 

1.749 

7-35 

i -995 

8-95 

2.192 


To Compute IVEeaix Pressure of Steam upon, a Piston 
"by Hyperbolic Logarithms. 

Rule. —Divide length of stroke of a piston, added to clearance in cylinder 
at one end, by length of stroke at which steam is cut off, added to clearance 
at that end, and quotient will express ratio or relative expansion of steam or 
number. 

Find in table logarithm of number nearest to that of quotient, to which 
add 1. The sum is ratio of the gain. 

Multiply ratio thus obtained by pressure of steam (including the atmos¬ 
phere) as it enters the cylinder , divide product by relative expansion, and 
quotient will give mean pressure. 

Note.—H yp. log. of any number not in table may be found by multiplying a 
common log. by 2.302 585, usually by 2.3. 





































STEAM. 


713 


When Relative Expansion or Number falls between two Numbers in Table , 
proceed as follows: Take difference between logs, of the two numbers. 
Then, as difference between the numbers is to difference between these logs., 
so is excess of expansion over least number, which, added to least log., will 
give log. required. 

Illustration. —Expansion is 4.84, logs, for 4.8 and 4.85 are 1.569 and 1.579, and 
their difference .01. Hence, as 4.85 co 4.8 = .05 : 1.579 cc 1.569 =; .01 ;; 4.84 — 4.8 = 
.04 : .008, and 1.569 .008 = 1.577 = tog. required. 

Example. —Assume steam to enter a cylinder at a pressure of 50 lbs. per sq. inch, 
and to be cut off at .25 length of stroke, stroke of piston being 10 feet; what will be 
mean pressure ? Clearance assumed at 2 per cent, nz. 2 feet. 

10-J-.2 = 10.2 feet, stroke 10 = 44-. 2 = 2.38 feet. Then 10.2 — 2.38 = 4.29 rela¬ 
tive expansion. 

Hyp. log. 4.29 =z 1.456, which 1 = 2.456, and X 5 ° = 28.62 lbs. 

4.29 

Relative Effect of steam during expansion is obtained from preceding rule. 

Mechanical Effect of steam in a cylinder is product of mean pressure in 
lbs., and distance through which it has passed in feet. 

Effects of Expansion. {Essentially from D. K. Clarlc.) 

Back Pressure is force of the uncondensed steam in a cylinder, consequent 
upon impracticability of obtaining a perfect vacuum, and is opposed to the 
course of a piston. It varies from 2 to 5 lbs. per sq. inch. 

It must be deducted from average pressure. Thus: assume pressure 60 lbs., 
stroke of piston as in preceding case, and back pressure 2 lbs. 

At termination of._1st, 2d, 3d, 4th, 5th, and 6th foot of stroke. 

Pressure. 60 30 20 15 12 10 lbs. per inch. 

Back pressure. 2222 2 2 “ “ “ 

Effective pressure. 58 28 18 13 10 8 “ “ “ 

Total work done by expansion at termination of each foot or assumed 
division of stroke of piston is represented by hyp. log. of ratio of expansion, 
initial work =: 1. 


Thus, for a stroke of 10 feet and a pressure of 10 lbs.: 



At end of. 

1st, 

2d, 3d, 4th, 

5 th, 

6th, 

7 th, 8tli, 

9th, and 10th foot. 

Steam is expanded 1 
into vols., hyp. V=: 
log. of which... ) 


.69 1.1 1.39 

I.6l 

1.79 

1.95 2.08 

2.2 

2-3 

Initial duty. 

I 

I II 

I 

I 

I I 

I 

I 

Total dutv. 

I 

1.69 2.1 2.39 

2.61 

2.79 

2.95 3 -o 8 

3-2 

3-3 

Initial duty is rep- ) 
resented by 10. . ) 

IO 

16.9 21 23.9 

26.1 

27.9 

29.5 30.8 

32 

33 

Resistance for each ) _ 

foot of stroke... ) 

2 

468 

IO 

12 

14 16 

18 

20 

Total effective ) _ 
duty.J ~ 

8 

12.9 15 15.9 

l6. I 

I 5-9 

15.5 14.8 

14 

13 

Gain by expansion 

O 

61.25 87.5 98.75 

iox.25 98.75 93.75 85 

75 

62.5 


The same results would be produced if expansion was applied to a non-condens¬ 
ing engine, exhausting into the atmosphere. 

Again, assume total initial pressure in a non-condensing cylinder 75 lbs. per sq. 
inch, expanded 5 times, or down to 15 lbs., and then exhausted against a back press¬ 
ure of atmosphere and friction of 15 lbs. 


4th, and 5th foot of stroke. 
2.61 

195-75 foot-lbs. 

75 “ “ 

120.75 “ “ 

101.25 P er cent. 

From which it appears that the total duty performed by expanding steam 5 times 
its initial volume is full 2.5 times, or as 75 to 195.75. 

3 0 * 


At termination of.. 

1st, 

2 ( 3 , 

3 d, 

4th, 

Total duty. 

I 

1.69 

2. I 

2-39 

“ “ performed... 

75 

126.75 

157-5 

I 79-25 

“ backpressure.... 

i 5 

3 ° 

45 

60 

“ effective duty.... 

60 

96.75 

112.5 

119.25 

Gain by expansion. 

O 

61.25 

87-5 

98-75 























STEAM. 


7*4 


Relative Effect of Equal Volumes of Steam. 

Relative total effect or work of steam is directly as its mean or average pressure 
(A), and inversely as its final pressure (B). or volume of steam condensed. 

If former is divided by latter, quotient will give relative total effect or work (C) 
of a given volume of steam as admitted and cut off at different points of stroke of 
piston, with a clearance of 3.125 per cent. 

In following computations resistance of back pressure is omitted. If this press¬ 
ure is uniform with all the ratios of expansion, it is a uniform pressure, to be de¬ 
ducted from the total mean pressure in column (A). 



Pressure. 

(C) 


Pressure. 

(C) 

Cut off at 

(A) 

(B) 

Relative 

Cut off at 

(A) 

(B) 

Relative 


Meau. 

Final. 

Effect. 


Mean. 

Final. 

Effect. 

I 

I 

I 

I 

•375 

.761 

•394 

i -93 

•75 

.969 

.787 

1.28 

•33 

.702 

•335 

2.09 

.6875 

.946 

.697 

i -35 

• 2 5 

.628 

• 2 73 

2 -3 

.625 

.924 

.636 

i -45 

.2 

•559 

.224 

2.05 

•5625 

.889 

•576 

i -54 

.125 

•435 

•15 

2.9 

•5 

•857 

.501 

1.71 

. I 

.418 

•13 

3.21 


To Compute Total Effective Work iix One Stroke of Ris¬ 
ton, or as Given Toy an Indicator Diagram. 

dP(!'i| hyp. log. R — c) = w, and ab L — to', w representing total work, and 
w' back pressure. 

Note.— Pressure of atmosphere is to be included in computations of expansion; 
it is therefore to be deducted from result obtained in non-condensing engines. In 
condensing engines, the deduction due to imperfect vacuum must also be made, 
usually 2.5 lbs. per sq. inch. 

Illustration. —Assume cylinder of a condensing engine 26.x ins. in diameter, a 
stroke of 2 feet, pressure of steam 95 lbs. (80.3 -+-14.7) per sq. inch, cut off at .5 stroke, 
with an average back pressure of 2 lbs. per sq. inch, and a clearance of 5 per cent. 

Area of piston, deducting half area of rod — 530 sq. ins. 2X5-1- 100 = .1 clear¬ 
ance, and 2-|-.i-i-i-f.i = i.9 = ratio of expansion, and 1 -j- hyp. log. 1.9 .= 1.642. 

Then 530 X 95 X i-i X 1.642 — .1 — 530 X 2X 2 = 50 350 X 1706 — 2120=- 83 777 lbs. 

Illustration. — Assume cylinder of a non-condensing engine having an area of 
2000 sq. ins., a stroke of 8 feet, steam at a pressure of 50 lbs. (35.3 + 14-7), cut off at 
.25 of stroke, and clearance .25 foot. 

Ratio of expansion 3.66, back pressure 17 lbs., and 1 -J- hyp. log. 3.66 = 2.297. 

2000 X 50 (2.25 1 -J- hyp. log. 3.66 — .25) = 100000 X 2.25 X i-|- 1.297 — c — 49 1 825 
foot-lbs. 

2000 X 17 X 8 = 272 000 foot-lbs. or negative effect, and 491 825 — 272 000 — 219 825 
foot-lbs. 

Total Effect of One EIo. of Expanded Steam. 

If 1 lb. of water is converted into steam of atmospheric pressure = 14.7 lbs. per 
sq. inch, or 2116.8 lbs. per sq. foot, it occupies a volume equal to 26.36 cube feet; 
and the effect of this volume under one atmosphere = 2116.8 lbs. X 26.36 feet — 
55 799 foot-lbs. Equivalent quantity of heat expended is 1 unit per 772 foot-lbs., 
= 55 799 77 2 — 7 2 -3 units. This is effect of 1 lb. of steam of a pressure of one at¬ 

mosphere on a piston without expansion. 

Gross effect thus attained on a piston by 1 lb. of steam, generated at pressures 
varying from 15 to 100 lbs. per sq. inch, varies from 56000 to 62000 foot-lbs., equiv¬ 
alent to from 72 to 80 units of heat. 

Effect of 1 lb. of steam, without expansion, as thus exemplified, is reduced by 
clearance according to proportion it bears to volume of cylinder. If clearance is 5 
per cent, of stroke, then 105 parts of steam are consumed in the work of a stroke, 
which is represented by 100 parts, and effect of a given weight of steam without ex¬ 
pansion, admitted for full stroke, is reduced in ratio of 105 to 100. Having deter¬ 
mined, by this ratio, effect of work by 1 lb. of steam without expansion, as reduced 
by clearance, effect for various ratios of expansion may be deduced from that, in 
terms of relative operation of equal weights of steam. 



























STEAM. 


715 


Volume of 1 lb. of saturated steam of 100 lbs. per sq. inch is 4.33 cube feet, and 
pressure per sq. foot is 144X100 = 14400 lbs.; then total initial work = 14400X4-33 
= 62 352 foot-lbs. This amount is to be reduced for clearance assumed at 7 per cent. 

Then 62 352 X 100= 107 = 58273 foot-lbs., which, divided by 772 (Joule’s equiva¬ 
lent), = 75.5 units of heat. 

Total or constituent heat of steam of 100 lbs. pressure per sq. inch, computed from 
a temperature of 212 0 , is 1001.4 units; and from 102 0 (temperature of condenser 
under a pressure of 1 lb.) the constituent heat is mi.4 units. 

Equivalent, then, of net simple effect 75.5 units is 7.5 per cent, of total heat from 
212 0 , or 6.7 per cent, from 102 0 . 

When steam is cut off at 

1 .75 .5 .33 .25 .2 .125 and . 1 of stroke, 

comparative effects are as 

1 1.26 1.616 1.92 2.14 2.27 2.51 and 2.6. 

Total effects as given in table, page 718. 

Effect of 1 lb. of steam, without deduction for back pressure or other effects, varies 
from about 60000 foot-lbs., without expansion, to about double that, or 120000 foot- 
lbs., when expanded 3 times, cutting off at about 27 per cent, of stroke; and to 
about 150000 foot-lbs. when expanded about 6 times, and cut off at about 10 per 
cent, of stroke. 


Effect, of Clearance. 


Clearance varies with length of stroke compared with diameter of cylinder, 
with form of valve, as poppet, slide, etc. 

With a diameter of cylinder of 48 ins., and a stroke of 10 feet, and poppet 
valves, clearance is but 3 per cent., and with a diameter of 34 ins. and a 
stroke of 4.5 feet and slide valves, it is 7 per cent. 

Illustration op Effect. —Assume steam admitted to a cylinder for .25 of its 
stroke, with a clearance of 7 per cent. 

Mean pressure for 1 lb. = .637, and loss by clearance = 7 -4-100 = .07, which, added 
to .637, = .707, which is effect of a given volume of steam, if there was not any loss 
by clearance, or a gain of n per cent. 

When steam is cut off at.1 .75 .5 .33 .25 .125 and .1 stroke. 

Loss at 7 per cent, clearance. . = 7 7.2 8.1 9.6 n 15.3 17 percent. 

To Compute TSTet Volnme of Cylinder for Given Weight 
of Steam, XLatio of Expansion and One Stroke. 

Rule.— Multiply volume of 1 lb. of steam, by given weight in lbs., by 
ratio of expansion and by 100, and divide product by 100, added to per cent, 
of clearance. 

Example. —Pressure of steam 95 lbs., cut off at .5, weight .54 lbs., volume of 1 lb. 
steam 4.55, and weight = .2198 lbs., stroke of piston 2 feet, and clearance 7 per cent. 

Ratio of expansion 2 -f-. 14 - 4 -1 +. 14 = 1.88. 


4-55 X -54 X i-88 X 100 
100 -)- 7 


—— = 4.31 cube feet. 

107 


To Compute Volume of Cylinder for Given Effect with 
a Given Initial Pressure and ILatio of Expansion. 

Rule. — Divide given effect or work by total effect of 1 lb. of steam ox 
like pressure and ratio of expansion, and quotient will give weight of steam, 
from which compute volume of cylinder by preceding rule. 

Example. —Assume given work at 50766 foot-lbs., and pressure and expansion as 
preceding. 

Total work by 1 lb., 100 lbs. steam, cut off at .5, = by table 94 200 foot-lbs., and by 
table of multipliers for 95 lbs. = .998, which x 94 200 = 94 012 foot-lbs. 









STEAM. 


7l6 


Consumption of Expanded. Steam per H? of Effect per 

Tlonr. 

BP = 33 000, which x 60 — 1 980 000 foot - lbs. per hour, which -4- 1 lb. 
steam, the quotient = weight of steam or water required per IP per hour. 
Illustration.— Effect of 1 lb., 100 lbs. steam, without expansion, with 7 per cent. 

of clearance = 58273 foot-lbs., and — 9 f° °° ^ = 34 lbs. steam — weight of steam con- 

5 ° 2 73 

sumed for the effect per IP per hour. 

When steam is expanded, the weight of it per BP is less, as effect of 1 lb. of steam 
is greater, and it may be ascertained by dividing 1 980000 by the respective effect, 
or by dividing 34 lbs. by quotient of total mean pressure by final pressure, as given 
in table, page 718. 

When steam is cut off at 1 .75 .5 .375 .33 .25 and .2 of stroke. 

vo rp:rcr e . dp ?:}=34 ** » ^ ^ 

Hence, assuming 10 lbs. steam are generated by combustion of 1 lb. coal per IP 
of total effect per hour, 

The coal consumed per) _ 

ff> per hour.j— 3-4 2 -°9 


1.85 1.76 1.6 


1.49 lbs. 


SATURATED STEAM. 


V _ T >. 

y- R ’ 


s- D ; 


To Compute Energy and Efficiency of Saturated Steam. 

S 1 , X X D 33 000 

v = _ ; p-p Xa , or--or —t=P; — 

HD 

HD _ „„ IT 

R 


HD 

~IT 


-H': 


B or Fs = r; 


JD(l-OfL = HD; 


= C: 


= H; 


H' 


a 


P'; 


p — p' X ci R S z= X; 


XD=:H' 


h 

rs' 


— p' 


h — X = h'] 

= e ; nlap — p' =- x, and 


i j 

a p — a p' 


— H' 


15.5 I a S = h) 


or 


— E; 


1 980 000 
E 


or 1 980 000 ■ 


: A; 


33 000 


: cube feet. 


33 000 
1 980 000 


IBP; Rp—p'a: 


F C X 60 =/; 


„ — cube feet icater evaporated per hour per BP. 

pa—p a ~ 62.5 X * * 

V and v representing volumes of mass of steam entering cylinder and of it at 
termination of stroke of piston; S and s volumes of 1 lb. steam when admitted and 
when at termination of expansion ; C volume of cylinder per minute for each IBP ; 
R and r ratios of expansion and effective cut-off; ¥ feed water per cube foot of vol¬ 
ume of cylinder per stroke of piston , and f per IIP per hour , all in cube feet. D den¬ 
sity or weight of 1 cube foot of steam at temperature of operation , in lbs.; p mean 
pressure ; p' mean back pressure ; I initial pressure ; P mean effective pressure , or 
energy per cube foot of volume of cylinder; P' pressure per sq. inch or that equivalent 
to heat expended , and P" pressure equivalent to expenditure of available heat, or en¬ 
ergy, all in lbs. J Joule's equivalent = 772 foot-lbs.; L as per following table ; t and 
t' absolute temperatures of steam at initial pressure and of feed water in degrees; 
H D heat expended per cube foot of steam admitted ; H' heat expended per cube foot 
of volume of cylinder, or pressure equivalent to heat expended per sq.foot; H" heat 
rejected per cube foot of steam admitted; H'" heat rejected per cube foot of volume 
of cylinder ; A available heat per IBP per hour ; e energy per cube foot of volume of 
cylinder to point of cutting off, or of steam admitted; li and h' heat expended and 
rejected , and X energy exerted , all per lb. of steam and in foot-lbs. E efficiency ; x en¬ 
ergy exerted per minute and per cube foot of steam admitted; a area of piston in 
sq. ins. ; l length of stroke of piston in feet, and f feed water per IIP per hour , in 
cube feet. 

Illustrations.— Assume volume of cylinder and clearance (5 per cent. = .6 inch) 
1 cube foot, steam (86.3 + 14.7) 100 lbs. per sq. inch, cut off at .5, mean pressure by 
rule (page 711) 86 lbs., and back pressure 3 lbs. 

V = I. v — 2. S = 4.33. s = 8.31. p — 86. p' = 3. a := 144 ins. 

t and t' = 327.9 0 + 461.2 0 and ioo° + 461.2 0 . 1 = 2 feet. n—i. L = 157748. 













STEAM. 


717 


-1 = 2 ratio. 
33 000 


86 — 3 X 144 


4.33 - 4 - 8.31 =. 526 effective cut-off. 

1 7T. -231 

-= .231 lbs. - 

4-33 2 


2.76 cube feet. 


86 — 3 X 144 = 11952 lbs. 
or --- .1154 cube feet. 


2 X 4-33 


198389 


772 X • 231 (789. i° — 561.2 0 ) -f-157 748 = 198 389 foot-tbs. 

I 9& 389 0 „ Q q_ 7 /,„ 99195 


= 99 195 foot-lbs. 


; 858 827 foot-lbs. 


689 lbs. 


2 .231 ' 144 

86 — 3 X 144 X 2 X 4 - 33 = io 3 5°4 foot-lbs. 

198 389-7- .231 — 103 504 X 2.31 = 174 479 foot-lbs. 174 479 -7- 2 = 87 239 foot-lbs. 
15.5 X 100 X 144 X 4-33 = 966 456 foot-lbs. 966 456 — 103 504 = 862 952 foot-lbs. 


966 456 


2 X 4-33 


= hi 600 lbs. 


144 X 86 —144 X 3 


600 


. 107 E. 


1 980 000 


Or 1980 000 X 


966456 

103 504 


18 504 673 foot-lbs. 


. 107 
23904 


1 X 2 X 144 X 86 — 3 = 23 904 foot-lbs. 


1 980 c 


33000 


= 18 504673 foot-lbs. 
. 7 2 5 H*- 


2 X 86 — 3 X 144 = 23 904 foot-lbs. 
103 ^ =Iig 52 f ooUbs _ 


- =. 306 cube feet. 
62.5X103504 

.1154 X 2.76 X 60 = 19.11 cube feet. 
33000 


: 2.761 cube feet. 


2 X 4- 33 ^ 86 X 144 — 3 X 144 

In illustration of connection of expenditure of available heat (A) and consumption 
of fuel, assume coal to have a total heat of combustion of 10000000 * foot-lbs., cor¬ 
responding to an equivalent evaporative power under 1 atmosphere at 212 0 of 13.4 
lbs. water and efficiency of furnace .5; then available heat of combustion of 1 lb. 
coal = 5 000 000 foot-lbs. 

Hence, consumption of coal per IH? in an engine of like dimensions and opera¬ 
tion with that here given would be 19223000=5000000 = 3.8444 lbs. 


Properties of Steam of Maximum Density. (Rankine.) 

Per Cube Foot. 


Temp. 

L 

Temp. 

L 

Temp. 

L 

Temp. 

L 

Temp. 

L 

Temp. 

L 

O 


O 


O 


O 


O 


O 


3 2 

248 

95 

1999 

158 

9 687 

221 

33 180 

284 

88 740 

347 

197 700 

41 

348 

IO4 

2 57 i 

167 

11 760 

230 

38700 

293 

IOO 500 

356 

219 OOO 

50 

481 

113 

3 2 77 

176 

14 200 

239 

44 93 ° 

3 ° 2 

113400 

365 

242 OOO 

59 

655 

122 

4136 

185 

17 OIO 

248 

51 920 

3 ii 

127 500 

374 

266 600 

68 

881 

131 

SU 8 

194 

20 280 

257 

59720 

320 

143 OOO 

383 

293 IOO 

77 

II7I 

14O 

6430 

203 

24 020 

266 

68 420 

3 2 9 

159 800 

392 

321 400 

86 

1538 

149 

792 1 

212 

28 310 

275 

78050 

338 

178 000 

401 

351600 


L representing latent heat of evaporation per cube foot of vapor in foot-lbs. of en¬ 
ergy. To reduce this to units of heat divide by 772, or Joule's equivalent. 

SUPERHEATED STEAM. 

The results attained by imparting to steam a temperature moderately in 
excess of that due to the volume or density of saturated steam are: 

1. An increase of elasticity without a corresponding increase of water evaporated. 

2. Arresting or reducing passage of water, in suspension, to cylinder (foaming), as 
the heat contained in that water is wholly lost without affording any elastic effect. 

Both of these results, by increasing effect of the steam, economize fuel. 

Superheated steam should be treated as a gas. 

The product of its pressure, p in lbs. per sq. foot, and volume v of 1 lb. of it in cube 
feet, in the perfectly gaseous condition, is obtained by following formula: 

42 140 T-r -1 = p v =7 85.44 T. T temperature of steam- f-461.2 0 , and t 32° + 461.2°. 

Illustration.— Assume temperature of steam, 327.9 0 , superheated to 341. i°. 

Then 42 140 X 461.2 0 -j- 341. i° -7- 32 -f- 461.2 0 = 68 549 foot-lbs. 

Hence, as pressure of steam at 327.9°= 100 lbs. per sq. inch, and at 341.i° 120. 

120-7- 100 = 1.2 to 1 = a gain of one fifth. 


* Coal of average composition, 14133 X 772= 10910676. 












































STEAM. 


718 


To Compute Energy and. Efficiency- of Superheated 

Steam. 

In following illustrations elements are same as those in preceding cases for satu¬ 
rated steam, with addition of the steam being superheated, so that 
1 = 115 < = 338°4-461.2 0 = 799.2 0 , <' = 290 + 461.2° = 75i.2 0 , S = 3-8, s = 7-4. 


R _piS-R a2 >'S = X; 15.5 I a S = /t; aP =E; 


R S 


5 = p"; 


h — X 
RS : 


: H' 


33000 


a p — a p 


.cube feet ; 


I080 000 

—-z=z A. 

E 


Efficiency of saturated steam (p. 716). 107, and, as above,. 109; hence = 1.02 to x. 

107 

If, then, available heat of combustion of efficiency of furnace is assumed at 5000000 
foot-lbs., as above, consumption of coal per IIP 18 183486 = 5000000 = 3.637 lbs. 

Note.— For further illustrations Rankine's “Steam-engine,” London, 1861,p. 436. 


Wire-drawing. 


Wire-drawing of steam is difference between pressure in boiler and pressure in 
cylinder, and is occasioned as follows: 

Resistance or friction in steam-pipe to passage of steam to steam-chest and piston. 

Resistance of throttle-valve to passage of steam, when it is partly closed or of in¬ 
sufficient area in proportion to steam-pipe. 

Resistance from insufficient area of valves or ports. 

Mr. Clark, from his experimental investigation, declared, that resistance in a 
steam-pipe is inappreciable, when its sectional area is not less than . 1 area of piston, 
and its velocity not exceeding 600 feet per minute. 

When velocity of a piston is from 200 to 240 feet per minute, area of steam may 
be .04th of piston. 


Effect of Expansion 'with. Ecpua .1 Volumes, and Effect of 
One Eh. of IOO EEs. Pressure per Sq_. Inch. 
Clearance at each End of Cylinder, including Volume of Steam Openings , 7 per cent, 
of Stroke, and 100 per cent, of Admission = 1. 


Ratio 
of Ex¬ 
pansion. 

Initial 
Volume 
= 1. 

Point 

of 

Cut-off. 

Stroke 
= 1. 

Tot a 

Final. 

Initial 
Pressure 
= 1. 

L Pkessl 

Mean. 

Initial 
Pressure 
= 1. 

RES. 

Initial. 

Mean 

Pressure 

Weight 
of Steam 
of 100 Lbs. 

for one 
Stroke per 
Cube Foot. 

Actual 

By 1 Lb. 
of 100 Lbs. 
Stpam. 

Effect. 

Per 

Sq.Inch 
per Foot 
of Stroke 
by 100 Lbs. 
Steam. 

Volume 
of Steam 
expended 
per H? 
of Work 
per Hour. 

Heat 

con¬ 

verted. 






Lbs. 

Foot-lbs. 

Foot-lbs. 

Lbs. 

Units. 

I 

I 

I 

I 

I 

•247 

58273 

IOO 

34 

75-5 

I. I 

•9 

• 9°9 

.996 

1.004 

.225 

63 850 

99.6 

3 1 

82.7 

M 

M 

OO 

•83 

.847 

.986 

I.014 

.209 

67836 

98.6 

29.2 

87.9 

1.23 

.8 

.813 

.98 

1.02 

.201 

70 246 

98 

28.2 

9 1 

i -3 

•75 

.769 

.969 

1.032 

.19 

73 513 

96.9 

26.9 

95.2 

i -39 

•7 

.719 

•953 

1.049 

.178 

77 242 

95-3 

25.6 

IOO. I 

i -45 

.66 

.69 

.942 

1.062 

•17 

79 555 

94.2 

24.9 

102.9 

i -54 

•625 

.649 

• 9 2 5 

1.081 

. l6l 

83055 

9 2 -5 

23.8 

107.6 

1.6 

.6 

.625 

• 9 i 3 

1.095 

•155 

85125 

9 x -3 

23-3 

110.3 

1.88 

•5 

•532 

.86 

1.163 

•131 

94 200 

86 

21 

122 

2.28 

•4 

•439 

.787 

I.27I 

. 108 

104 466 

78.7 

x 9 

132-5 

2.4 

•375 

.417 

.766 

1-305 

.103 

107 050 

76.6 

18.5 

138.6 

2.65 

. -33 

•377 

.726 

1-377 

•093 

112 220 

72.6 

17.7 

* 45-4 

2.9 

•3 

•345 

.692 

i -445 

.085 

116855 

69.2 

16.9 

I 5 I -4 

3-35 

•25 

.298 

•637 

*•57 

.074 

124 066 

63 -7 

l6 

160.7 

4 

.2 

•25 

•567 

1.764 

.062 

132770 

56-7 

14.9 

171.9 

4-5 

. l6 

.222 

.526 

I. 901 

•055 

138130 

52.6 

x 4-34 

178.8 

5 

.14 

.2 

.488 

2.049 

.049 

142180 

48.8 

13.92 

184.2 

5-5 

.125 

.182 

•457 

2.188 

•045 

146325 

45-7 

J 3 - 53 

189.5 

5-9 

.11 

.169 

•432 

2 - 3 I 5 

.O42 

148 940 

43-2 

13.29 

192.9 

6 -3 

. I 

• 159 

•413 

2.421 

•039 

151 37 ° 

4 x -3 

13.08 

196.1 

6.6 

.09 

.152 

• 39 8 

2-513 

•037 

152955 

39-8 

12.98 

197.7 

7 

.083 

•143 

.381 

2.625 

•035 

155 200 

38.1 

I2 -75 

201. I 

7.8 

.066 

. 128 

•348 

2.874 

.032 

158414 

34-8 

12.5 

205.2 

8 

.0625 

.125 

•342 

2.924 

.031 

I 59 433 

34-2 

11.83 

206.5 



























STEAM. 


719 


HVInltipliers for Actual “Weight and Effect for other 
Pressures than IOO Lbs. 


Pressure 

Multipliers. 

Pressure 

Multipliers. 

Pressure 

Multipliers. 

per 

Sq.Inch. 

Weight. 

Actual 

Effect. 

per 

Sq.Inch. 

Weight. 

Actual 

Effect. 

per 

Sq.Inch. 

Weight. 

Actual 

Effect. 

Lbs. 



Lbs. 



Lbs. 



65 

.666 

•975 

90 

.901 

•995 

130 

1.28 

I-OI 5 

70 

.714 

.981 

95 

•952 

.998 

140 

i -37 

1.022 

75 

•763 

.986 

IOO 

I 

I 

15 ° 

1.46 

1.025 

80 

.806 

.988 

no 

I.O9 

I.OO9 

160 

i -55 

1-031 

85 

•855 

.991 

120 

1.17 

i.on 

170 

1.64 

1-033 


In this illustration, in connection with preceding table, no deductions are made 
for a reduction of temperature of steam while expanding, or for loss by back 
pressure. 

When steam is cut off at .0625, or one sixteenth, its expansion is 16 times, but as 
7 per cent, of stroke is to be added to it (.0625 + .07) = .1325 = 132.5 per cent., or 
nearly double of 16, or only a little over 7 times, as in 3d column of table on pre¬ 
ceding page. 

Column 7 is product of 58273 and ratio of total effect of equal weights of steam 
when expanded, or average total pressure divided by average final pressure. 


Thus, if steam is cut off at .5, with a clearance of 7 per cent., —- = 

’.532X100 = 53.2 

1.6165, and 58273 X 1.6165 = 94 200 foot-lbs. 

Column 9 gives volume of steam consumed per BP per hour. Thus, assume cyl¬ 
inder to have an area of 292 sq. ins., a stroke of 2 feet, and pressure of steam 100 lbs. 
without expansion. 

292 X 100 X 2 = 58400 foot-lbs. 1 and 292-(-7 per cent, of stroke for clearance = 
.14; hence, 292 X 2.14 A- 144 = 4.34 cube feet, and weight of a cube foot of such steam 
= .23 lbs., and 58400 : 4.34 X .23 :: 33000 : .564, which, x 60 minutes = 33.84, or 34 
as per table. 

The pressures are computed on premise that steam is maintained at a uniform 
pressure during its admission to cylinder, and that expansion is operated correctly 
to termination of stroke. 

Column xo is quotient of work in foot-lbs., divided by Joule’s equivalent 772. 

Thus, 94 200-7- 772 = 122. 

For percentage of constituent heat, converted from 102° and 212 0 , assume 
122 as in last case: 

Then 122 X 9 = 100 = 10.98 per cent, for 102 0 , and 122 X 10-f-100 = 12.2 per cent, 
for 212°. 

“Wire-drawing” will cause a reduction of pressure during admission, and clear¬ 
ance will vary from 3 to 8 per cent., according to design of valve, as poppet, long or 
short slide. 

In practice, wire-drawing of steam, and opening of exhaust before termination of 
stroke, involve deviations from a normal condition, for which deductions must be 
made, added to which there is the back pressure, from insufficient condensation in 
condensing engines, and from pressure of air in non-condensing engines, and com¬ 
pression of exhaust steam at termination of stroke. 


To Compute Grain in Weed Water at Higli Temperature. 

T — t -j- W — w = H. T and t representing total heat in steam and temperature of 
feed water , W and w temperature , of water blown off and fed — heat lost by blowing 
off , and H total heat required from fuel, all in degrees. 

Illustration.—A ssume steam at 248°, feed water'xoo° in one case and 150 0 in 

another, and density —, and total heat at 248° = 1157 0 ; what is gain? 

32 

1157 —100 + 248 — xoo = 1205 0 = total heat required from fuel. 

1157 — 150 + 248 — 150 = 1105°= “ “ “ 

H-H' 1205-1105 on . 

Then -= — - -- — . 083 = 8.3 per cent. 

H 1205 
























720 


STEAM. 


COMPOUND EXPANSION. 


Compound Expansion is effected in two or more cylinders, and is prac¬ 
tised in three forms. 


ist. When steam in one cylinder is exhausted into a second, pistons of the 
two moving in unison from opposite ends—that is, steam from top or for¬ 
ward-end of first cylinder being exhausted into bottom or after-end of the 
other, and contrariwise—this is known as the Woolf* engine. 

2d. Steam from the xst cylinder is exhausted into an intermediate vessel, 
or “ receiver,” the pistons being connected at right angles to each other. 

3d. Steam from receiver is exhausted into a 3d cylinder of like volume 
with 2d, pistons of all being connected at angles usually of 120 0 . 

The two latter types are those of the compound engine of the present time. 

Expansion from, Receiver. The receiver is filled with steam exhausted 
from ist cylinder, which is then admitted to 2d, or 2d and 3d. in which it is 
cut off and expanded to termination of stroke. 


Initial pressure in 2d, or 2d and 3d cylinders, is assumed to be equal to final press¬ 
ure in ist, and consequently the volume cut off in the one or the other cylinders 
must be equal in volume to that of 1st cylinder, for its full volume must be dis¬ 
charged therefrom. 

Inasmuch as 3d cylinder is but a division of the 2d, with addition of receiver, 
this engine, in following illustrations, will, for simplification, be treated as having 
but two cylinders. 

In illustration, assume ist and 2d cylinders to have volumes as 1 to 2, with like 
lengths of stroke, and that steam is cut off at .5 stroke, and equally expanded in 
both cylinders, the ratio of expansion in each cylinder being thus equal to their 
volumes. 


Volume received into 2d cylinder would be equal to that exhausted from ist, as¬ 
suming there would not be any diminution of pressure from loss of heat by inter¬ 
mediate radiation, etc. This is based upon assumption that expansion occurs only 
upon a moving piston; but in operation, expansion occurs both in receiver and in 
intermediate passages, as nozzles and clearances; the 2d cylinder, therefore, receives 
steam at a reduced pressure, increased volume, and reduction of ratio of expansion. 
To meet this, and. attain like effects, volume of 2d cylinder must be increased in 
proportion to increased volume of steam and its ratio of expansion. Consequently, 
there is no loss of effect aside from increased volume and weight of parts by inter¬ 
mediate expansion, provided primitive ratio of expansion is maintained by giving 
relative increased volume to ?d cylinder. 

Ir.uasTKATroN.— Assume cylinders having volumes as 1 and 3, initial steam of ist 
cylinder to be 60 lbs, per sq, inch, stroke of piston 6 feet, cut off at one third, and 
clearance 7 per cent. 


Final pressure, as per rule, page 711, =22.62 lbs., and pressure as exhausted into 
receiver, reduced one fourth, = 16.97 lbs,, assuming there is no intermediate fall of 
pressure. The steam, therefore, is expanded to 1.33 times volume of cylinder; a 
like volume, therefore, must be given to 2d cylinder, to admit of this at a like press¬ 
ure. If, therefore, the increased terminal volume of the steam in the ist cylinder 
was augmented, including a clearance of 7 per cent., the effect would be as follows: 

Volume admitted to 2d cylinder is equal to volume of ist added to its clearance, 
or to .33 volume of 2d cylinder added to its clearance; that is, to .33 of 107 per cent., 
or 35.66 percent., consisting of clearance, and 35.66 — 7 = 28.66 per cent, stroke of 
2d cylinder. The steam exhausted into 2d cylinder thus fills less than. 33 of its stroke 
by 4.67 (33.33—28.66). As steam is expanded from volume of ist cylinder, plus its 
clearance, to 2d cylinder, plus its clearance, ratio of expansion in 2d cylinder is equal 
to ratio of volume of both cylinders, which is 3, and 


100 (representing full stroke) 4 - 7 
28.66 -j- 7 


, and final pressure 


22.62 


; 7,54 lbs. per sq. inch. 


* I11 1825-28 James P. Allaire, of New York, adopted this design of engine in the steamboats Henry 
Eckford, Stin, Commerce, Swiftitire, Post Boy, and Pilot Boy. 





STEAM. 


721 


Assuming volume of receiver, or augmented terminal volume, for expansion in 2d 
cylinder, to have proportions of 1, 1.25, 1.33, and 1.5 times volume of 1st cylinder 
plus its clearance, the x’elations would be as tbllows: 


Augmented terminal volumes) 
in 2d cylinder.j 

j- i 

1.25 

i -33 

, 5 j 

times volume of 
1st cylinder. 

Equal to. 


i -337 

I - 4 2 7 

1.605 ] 

do. do. 

including clear- 

Final volumes in 2d cylinder) 
added to clearance. J 

| 3 - 2 i 

3.21 

3-21 

( 

3 - 2 i j 

ance. 

times volume of 
1st cylinder. 

Ratio of expansion in 2d cyl’r. 

• 3 

2.4 

2.25 

2 


Intermediate reductions of) 
pressure. j 

[ ° 

.2 

•25 

•33 -j 

f of terminal press- 
l ure in 1st cyl’r. 

Equal to. 


4-52 

5-65 

11.31 lbs. per sq. inch. 

Pressures in receiver and ini-1 
tial pressure in 2d cylinder.. ; 

| 22.62 

l8. I 

16.96 

11. 3 1 

do. do. 

Final pressure in 2d cylinder .. 

• 7-54 

7-54 

7-54 

7-54 

do. do. 


To Compute Expansion in a Compound Engine. 

RECEIVER ENGINE. 

Ratio of Expansion. In 1st cylinder , as per formula, page 710. In 2d cylinder. 


n 


r = ratio. Of Intermediate Expansion. 


n 


: ratio, n representing ratio 


n n — 1 

of intermediate reduction of pressure between 1st and 2 d cylinder , to final pressure in 
1 st cylinder , and r ratio of area of 1st cylinder to that of 2d. 


Illustration. —Assume n = 4, and r = 3. 

Then -- X 3 — 2.25 ratio , and 4 

4 4 — 1 


1.33 ratio. 


Total or Combined Ratio of Expansion, r R' —product of ratio of 1st and 2d cyl¬ 
inders by ratio of expansion in 1st cylinder. As when r — 3, and R' = 2.653, then 
2.653 X 3 = 7-959 total ratio. 


Hence, Combined Ratio of Expansion in both, cylinders, 
resenting ratio of expansion in 1 st cylinder , and R" combined ratio. 
Illustration.— Assume as preceding, and R / = 2.653. 


rR'=R". R'rep- 


Then 


4 — i 


X 3 X 2.653 = 5.969 combined ratio. 


To Compute Effect for One Strolxe and. a Given Ratio 
of Expansion in Eirst Oylinder. 


Without Intermediate Expansion. Rule. — Multiply actual ratio of ex¬ 
pansion in 1st cylinder bv ratio of both cylinders, and to hyp. log. of com¬ 
bined ratio add 1; multiply sum by period of admission to 1st cylinder plus 
clearance, and term product A. 

Divide ratio of both cylinders, less 1. by ratio of expansion in 1st cyl¬ 
inder ; to quotient add 1; multiply sum by clearance, and term product B. 

Subtract B from A, and term remainder C. Multiply area of 1st cylinder 
in sq. ins. by total initial pressure in lbs. per sq. inch, and by remainder C. 
Product is net effect in foot-lbs. for one stroke. 


With Intermedia.te Expansion. Add effect thereof to result obtained above, 
and by following formula: 

Or, l' 1 -j— hyp. log. R" — c ^1 -|- -j£>—^ a P — E. a representing area in sq. ins., 

P initial pressure in lbs. per sq. inch of 1st cylinder , V length of admission or point 
of cutting off plus clearance , c clearance in feet , and E effect in foot-lbs. 

3 P 














722 


STEAM. 


Illustration.— Assume areas of cylinders i and 3 sq. ins., length of stroke 6 feet, 
pressure of steam 60 lbs. per sq. inch, cut off at 2 feet, clearance 7 per cent., and 
area of intermediate space, as receiver, one third volume of 1st cylinder. 


R" = ratio of expansion in 2d cylinder - - - X 3 X 2.653 = 5-969 hyp. log. 

__ _ 4 

2.653 X 2.25 + 1 X 2.42 —3 — 1 - 4 - 2.653 + 1 X -42 X 1 X 60 = 1.7865 + 1 X 2.42 — 
2-r- 2.653 4 " i X - 42 X 60 = 6.743 — -737 X 60 = 360.36 foot-lbs. 


1st Cylinder. 


Effect on piston 60 lbs. X 1 inch X 2 feet. 

“ of clearance 60 lbs. X -42 foot. 

Total initial effect = 60 X 2 X - 42. 

Then 145.2 X i + byp. log. 2.653 or 1-976. 

Less effect of clearance. 

Net effect on piston above vacuum line. 

Less effect of back pressure 60 = 2.653 = 22.61, which, X 3 sq. 
ins. and 2 feet stroke. 

Net effect on piston. 


= 120 

foot-lbs. 

= 25.2 

U 

— 145 - 2 

foot-lbs. 

= 286.91 

foot-lbs. 

= 25.2 

U 

= 261.71 

foot-lbs. 

= 135.66 

(C 

= 126.05 

foot-lbs. 


2 d Cylinder. 

145.2 X 1 + hyp. log. 2.25 or 1.81. = 262.81 foot-lbs. 

Effect of clearance 22.61 X 3 X-42 . = 28.49 “ = 234.32 foot-lbs. 

360.37 foot-lbs. 

Intermediate reduction of pressure, as given at page 721, = .25 X 22.61 = 5.65 lbs. 
per sq. inch, which, x 3 sq. ins. and by 2 per foot of stroke’ = 33.9 foot-lbs. 

Hence 360.36 -f- 33.9 = 394.26 foot-lbs. 

Or, by sum of the three results, viz.: 

1st cylinder. 126.05 foot-lbs. 

Intermediate expansion. 33.9 ‘ “ 

2d cylinder. ..... 234.32 “ 

394.27 foot-lbs. 


WOOLF ENGINE. D. K. Clark. 

Ratio of Expansion.—In 1 st cylinder as per formula, page 710. In 2 cl cylinder , 

r=-+*- 4 -i-j-a; = ratio, r representing ratio of area of 1st-cylinder to that of 2 d, 
t 

l and l' lengths of stroke and of stroke added to clearance , in ins. or feet, and x ratio 
value of intermediate volume. 


Illustration. —Assume 1 = 6 feet, V = 7 per cent. = .42, r = 3, and x = .333. 

3 X 6 ^ + ' 333 . , 

Then - 1 = 2-353, ratio of expansion in 2d cylinder. 

Total Actual Ratio of Expansion. R' (y >- -J- xj = ratio. 

Illustration. —Assume preceding elements, R = 2.653. 

Then 2.653 ^3 x + - 333 ^ = 2.653 X 3.137 = 8.322, total actual ratio. 

Combined Actual Ratio of Expansion. R' (r y + xj 1 -j- x = ratio. 

Illustration. —Assume preceding elements. 

6 . 8.322 , _. 

3 X 7 -h • 333 A- 1 + • 333 = - = 6.242, combined actual ratio. 

6.42 1.333 































STEAM. 


723 


To Attain Combined Ratio of* Expansion and. Final 
Pressure in 2d Cylinder. 

Assuming four cases as taken for Receiver Engine with a clearance of 
7 per cent. The relations would be as follows: 


Volume of 1st cylinder. o 

Add to these 1.07, the volume of 1st) 
cylinder plus its clearance, and .... \ 

To same values of intermediate space 
add 3, the volume of 2d cylinder, I 
and the sums are the final volumes j 
by expansion in 2d cylinder.J 

Ratios of expansion in 2d cyl’r are quo- ) 


Intermediate falls of pressure are, in ) 
parts of final pressure in ist cylinder 1 


The initial pressures for expansion in ) 
2d cylinder are...... J 


Hence , final pressures in 2 d cyl’r are . 



•333 



| part of volume of ist cylin- 


•5 


( der plus its clearance, or, 

0 

•357 

•535 

1.605 

I.O7 

f total initial volumes for ex- 

1.07 

1.427 

2.14 

-j pansion in 2d cylinder or 
( times volume of ist cyl’r. 


3 

3-357 

3-535 

4.07 

f times volume of ist cyl- 
1 inder. 

K) 

00 

O 

4 * 

2.352 

2.202 

1.902 ratios of expansion. 





(■ of final pressure ; or, as- 


•25 

•333 


J suming initial pressure at 


• 5 

) 63 lbs., and final pressure 

l at 23.75 lbs., they are 



O 

5-94 

7.92 

11.87 lbs . per sq . inch . 

I 

•75 

.66 

•5 

f of final pressure in ist cyl- 
1 inder, or 

23-75 

17.81 

1583 

11.87 lbs . per sq . inch . 

• 8.47 

7-57 

7.19 

6.24 

lbs . per sq . inch . 


1 St. 
2d. 

3d- 

4th. 


Combined Ratios in these Four Cases. 

ist ratio Of expansion_ 1 to 2.653 Combined Ratio. 

2d do. do. .... 1102.804=22.653X2.804 = 7.44. 


ist 

2d 

ist 

2d 

ist 

2d 


do. 

do. 

do. 

do. 

do. 

do. 


do. 

do. 

do. 

do. 

do. 

do. 

do. 


1 to 2.653 

1 to 2.352 = 2.653 X 2.352 = 6.24. 
1 to 2.653 

1 to 2.202 = 2.653 X 2.202 = 5.84. 


1 to 2.653 

1 to 1.905 = 2.653 X 1-905 = 5.05. 


Initial effect of steam at 63 lbs. pressure, admitted to ist cylinder, for 2 feet, or oue 
third of stroke of piston, and with a clearance of 7 per cent, or .42 feet, is as follows: 

Effect on piston.63 x 2 feet = 126 foot-lbs. ( Total initial 

do. in clearance.. 63 X-42 foot = 26.46 = 63 X 2.42 = 152.46 foot-lbs. ( effect. 

This sum is initial effect, on which effect by expansion is computed, while it is 
26.46 foot-lbs. in excess of the initial effect on the piston. 

The total effect, then, is computed as follows: 

1 st case. 152.46 x (1 + hyp. log. 7.44) or 3.0069 = 458.27 Net Effect. 

Less effect of clearance. 26.46 431.81 foot-lbs. 

2d case. 152.46 X (1+ hyp. log. 6.24) or 2.831 =431.47 

Less effect of clearance. 26.46 405.01 “ 

3d case. 152.46 X (1 + hyp. log. 5.84) or 2.7647 = 421.35 

Less effect of clearance. 26.46 394 89 “ 

4th case. 152.46 x (i-fhyp. log. 5.05) or 2.6294 = 399.29 

Less effect of clearance. 26.46 372.83 “ 

The reductions of net effect in 2d, 3d, and 4th cases are 6.2, 8.6, and 13.7 per cent, 
of effect in ist case. 

To Compute Effect for One Stroke and a Given Com¬ 
bined Actual Ratio of Expansion. 

Rule.— To hyp. log. of combined actual ratio of expansion (behind both 
pistons) add i; multiply sum by period of admission of steam to ist cylin¬ 
der, added to clearance, and from product subtract clearance. 

Multiply area of ist cylinder in sq. ins. by initial pressure of steam in lbs. 
per sq. inch and by above remainder. Product is net effect in foot-lbs. for 
one stroke. 






















724 


STEAM. 


Example.— Assume elements of ist illustration page 723. 

Hyp. log. 6.24-f-1 = 2.831, which, X 2.42 = 6.85, and 6.85 — .42 and remainder 
X 60 — 385.8 foot-lbs. 

Or, V (1 + hyp. log. R') — C X a V = E. 

Comparative Effect of Steam in Receiver and. Woolf 

Engines. 

The effect of steam in a compound engine, without clearance and without any 
intermediate reduction of pressure, is the same whether operated in a receiver or 
Woolf engine. 

When, however, there is an intermediate space between the two cylinders, as a 
receiver, there is an intermediate reduction of the pressure of the steam, conse¬ 
quent upon the increase of its volume in the receiver; the reduction of pressure, 
therefore, being less rapid than with the Woolf engine, the effect is greater. 

In illustration, the following comparative elements of the effect of both engines 
is furnished. 

Receiver. (7 per cent, clearance.) Woolf. 



Ratio of Expansion. 

Net Effect. 

Ratio of Expansion. 

Net Effect. 

ist case.7.96. 

..422.3 foot-lbs. 

ist case... 

•••7-64. 

..431.71 foot-lbs. 

2d 

“ . 5-97 . 

..421.55 “ 

2d “ ... 

.. .6.24. 

..405.11 “ 

3 d 

“ . 5 - 3 i. 

..417.96 “ 

3 d “ 

• ••5-84. 

..394.99 “ 

4th 

“ . 3-98 . 

..402.78 “ 

4 th “ ... 

••• 5-05 . 

•• 372-93 “ 


From which it appears, that although the effect of a receiver engine is the great¬ 
est, its ratio of expansion is less than with the Woolf engine. 

Also, that by the addition of clearance to the pistons of each engine, the actual 
ratios of expansion are sensibly reduced, as compared with the ratios without 
clearance. 

INDICATOR. 


To Compute jMeau 



inch upon piston 


Pressure "by an Indicator. 

Rule.—D ivide atmosphere line, o o in fig¬ 
ure, into any convenient number of parts, as 
feet of stroke of piston, and erect perpendic¬ 
ulars at each point. Measure by scale of 
parts (alike to that of diagram) the actual 
mean pressure, as defined between the two 
lines at top and bottom of diagram, add the 
results, divide sum by number of points, and 
quotient will give mean pressure in lbs. per 


Example.— Pressures, as above given, are: 

35 + 35 + 35 + 34 + 3' 2 + 25+16-)- 10 + 8 + 6 = 236, which, - 4 -10, = 23.6 lbs. 

Note.— If it were practicable to run an engine without any load, and it some¬ 
times is, the mean pressure, as exhibited by an indicator, would be an exact meas¬ 
ure of the friction of the engine. 

Conclusions on Actual Efficiency of’ Steam. 

For development of highest efficiencies of steam, as used in an engine, means for 
protecting it from cooling and condensing in the cylinder must be employed. Super 
heating of it prior to its introduction into a cylinder is probably most efficient 
means that may be employed for this purpose. Application to cylinder of gases 
hotter than it is next best means; and next is the steam-jacket. 

In cases of locomotive and portable engines, consumption of steam per IBP per 
hour is lass than for that of single-cylinder condensing engines for like ratios of ex¬ 
pansion, which is due to effect of temperature of non-condensing cylinders, always 
exceeding 212 0 . 

It is deducible from these results that the compound engine is more efficient than 
the single-cylinder, and that, of the two kinds of compound engines, the receiver- 
engine is more efficient than the Woolf. 

Average consumption of bituminous coal per IIP per hour, for compound engines 
in long voyages, as shown by Mr. Bramwell, ranged from 1.7 to 2.8 lbs. (D. K. Clark.) 






















STEAM. 


725 


X'o Compute Yolume of "Water Evaporated per Lb. 

of Coal. 

v — v W 

p ^ =volume of water, in lbs. V and v representing volume of steam and 

relative volume of water, in cube feet, W weight of cube foot of water, and F weight of 
fuel consumed , both in lbs., and d density of water, in degrees of saturation. 

Illustration. — Take case at foot of page. V = 449887 cube feet, 11 = 838 cube 
feet, W = 64.3, E = 1, and F = 4061 lbs. 


449887 = 838 X 64.3 
4061 X x 


= 8.5 lbs. per hour. 


Gain in Fuel , and Initial Pressure of Steam required, when Acting Expan¬ 
sively , compared with Non-Expansion or Full Stroke. 


Point of 
Cutting off. 

Gain in 
Fuel. 

Cutting 

off. 

Point of 
Cutting off. 

Gain in 
Fuel. 

Cutting 

Point of 
Cutting off. 

Gain in 
Fuel. 

Cutting 

off 

Stroke. 

Per Cent. 

Lbs. 

Stroke. 

Per Cent. 

Lbs. 

Stroke. 

Per Cent. 

Lbs. 

•75 

22.4 

1.03 

•5 

4 i 

I. l8 

•25 

58.2 

1.67 

.625 

32 

I.O9 

•375 

49.6 

1.32 

.125 

67.6 

2.6 


Illustration.— What must be initial pressure of steam cut off at .5, to be equiv¬ 
alent to 100 lbs. per sq. inch at full stroke? 


100 at full stroke = 100, and 100 X 1.18 = 118 lbs. 


To Compute Grain in Enel. 


Rule. —Divide relative effect of steam by number of times the steam is 
expanded, and divide 1 by quotient; result is the initial pressure of steam 
required to be expanded to produce a like effect to steam at full stroke. 

Divide this pressure by number of times the steam is expanded, and sub¬ 
tract quotient from 1, remainder will give gain per cent, in fuel. 

Example.—W hen steam is cut off at ,5, what is gain in fuel, and what mechanical 
effect? 

Relative effect, including clearance of 5 per cent.,= 1.64; number of times of ex¬ 
pansion = 2. 

1.64 - 4 - 2 = .82, and 1 4 - .82 = 1.22 initial pressure. 

1.22 4 - 2 = .61, and 1 — .61 = .39 per cent. 

Mechanical effects of steam at full and half strokes are 2 —1.64 = .36 difference. 

Hence, 1.64 : .36 :: 50 (half volume of steam used) : 10.97 per cent, more fuel to 
produce same effect at half stroke, compared with steam at f ull stroke. 

To Compute Consumption of Euiel in. a Furnace. 

When Dimensions of Cylinder , Pressure of Steam , Point of Cut-off] Revo¬ 
lutions , and Evaporation per Lb. of Fuel per Minute are given. 

Rule. —Compute volume of cylinder to point of cutting off steam, in¬ 
cluding clearance. Multiply result by number of cylinders, by twice number 
of strokes of piston, and by 60 (minutes) ; divide product by density of steam 
at its pressure in cylinder, and quotient will give number of cube feet of 
water expended in steam. 

Multiply number of cube feet by 64.3 for salt water (62.425 for fresh), 
divide product by evaporation per lb. of fuel consumed, and quotient will 
give consumption in lbs. per hour. 

Example. —Cylinder of a marine engine is 93 ins. in diameter by 10 feet stroke 
of piston; pressure of steam in steam-chest is 15.3 lbs. per sq. inch, cut off at .5 
stroke; number of revolutions 14.5, and evaporation estimated at 8.5 lbs. of salt 
water per lb. of coal; what is consumption of coal per hour, when density of water 
is maintained at 2-32? (See Saturation, page 726.) 

Volume of steam at above pressure, compared with water (15.3-f-14.7), = 838. 
Area of 95 ins. + 2.5 per cent, for clearance 4 - 144 = 50.45 cube feet. Point of cut¬ 
ting off 5 feet-)- 2.5 per cent. = 5 feet 1.5 ins., and 50.45 X 5 feet 1.5 ins. X 14.5 X 2 
X 60 = 449 887 cube feet steam per hour. 

3 P* 






















STEAM. 


726 


Hence, 449887-4-838 = 536.86 cube feet water, which, X 64.3 = 34520 lbs., which, 
- 4 - 8.5 = 4061 lbs. coal per hour. 

Note. —Elements given are those of one engine of steamer Arctic , and consump¬ 
tion of clean fuel (selected) for a run of 12 days (one engine) was 3820 lbs. per hour. 

Utilization of Coal in a Marine Boiler. 

Experiment gives from .55 to .8 per cent, of the heat developed in the 
combustion of coal, as utilized in the generation of steam. Ordinarily it 
may be safely taken at .65. 

SxVLINE SATURATION IN BOILERS. 

Average sea-water contains per 100 parts: 

Chloride of sodium (com. salt) . .2.5; Chloride of magnesium .33- = 2.83 


Sulphuret of magnesium.53; Sulphuret of lime. 

Carbonate of lime and of magnesia. 


.01- = .54 

...... .02 

Saline matter. 3.39 

Water. 96.61 


Hence, sea-water contains .0339th part of its weight of solid matter in solution, 
and is saturated when it contains 36.37 parts. 

Mean quantity of salts, or solid matter, in solution, is 3.39 per cent., three fourths 
of which is common salt. 

Removal of Incrnstation of Scale or Sediment. 

Potatoes , in proportion of .033 weight of water. Molasses , in proportion of 1.6 
lbs. per BP of boiler. Oak , suspended in the water, and Mahogany or Oak sawdust , 
and Tanner" 1 s and Slippery Elm, bark, renewed frequently, according to volume of it, 
and the evaporation of the water. Muriate of Ammonia and Carbonate of Soda, in 
quantity to be determined by observation. 

BLOWING OFF. 

To Compute Loss of Heat by Blowing Off of Saturated 
Water from a Steam-Boiler. 


S-T E + t 
t 


water evaporated in degrees , and 


proportion of heat lost , S — T X E = heat required from fuel for 
t 


i = loss of heat per cent. S representing 


S — T E-f t 

sum of sensible and latent heals of water evaporated , T temperature of feed water , 
t difference in temperature of water blown off and that supplied to boiler , E volume 
of water evaporated, proportionate to that blown off, the latter being a constant quan¬ 
tity, and represented by 1. 

Values of E at following degrees of saturation, and volumes to be blown off: 


32. 

Value 

E. 

Volume 

to 

Blow off. 

32 - 

Value 

E. 

Volume 

to 

Blow off. 

32. 

Value 

E. 

Volume 

to 

Blow off. 

32. 

Value 

E. 

Volume 

to 

Blow off. 

1.25 

•25 

4 

1.65 

•65 

1-54 

2 

I 

I 

2-75 

I -75 

•57 

i -35 

•35 

3 

i -75 

•75 

i -33 

2.25 

1.25 

.8 

3 

2 

• 5 

i -5 

•5 

2 

1.85 

•85 

1.18 

2-5 

i -5 

.66 

2.25 

2.25 

•45 


Thlts, when water in a boiler is maintained at a density of — , 1 volume of it is 

32 

evaporated, and an equal volume, or 1, is to be blown off. 

Hence 1 —(- 1 —1 = 1 = ratio of volume evaporated to volume blown off. 

Illustration l— Point of blowing off is 2 (32), pressure of steam is 15.3 lbs. mer¬ 
curial gauge, and density of feed water 1 (32); what is proportion of heat lost? 

S = 1157.8°. T = ioo°. t = 15.3 + 14.7 = 250.4° — ioo° = 150.4° E = 1. 


Then 


1157.8 —100 X x-f 150.4 


8.03 proportion of heat lost. 


150.4 



































STEAM.-STEAM-ENGINE. 


727 


2 -—Assume point of blowing off 1.75 (32); what would be loss of heat per cent, in 
preceding case? 

^ _ 150-4 

111 — 75 - -—- —5 -—-:- = 15.9 per cent, lost by blowing off. 

1157.8 — 100X .75 + 150.4 * y M 

3.—Assume elements of preceding case. What is total heat required from fuel 
for water evaporated ? 

1157.8 —100 X -75 = 793 - 35 °- 

To Compute Volume of Water Blown Off to tliat 

Evaporated. 

When Degree of Saturation is Given. Rule.—D ivide 1 by proportionate 
volume of water evaporated to that blown off, or value of E as above, for 
degree of saturation given, and quotient will give number of volumes blown 
off to that evaporated. 

Illustration.— Degree of saturation in a marine boiler is —-- • what is volume 

3 2 

of water blown off? 


E = i. 25. Then i- i-1.25 = .8 blown off. 


Baltic.... 
Black Sea 
Red Sea.. 


Proportional Volumes of Saline Matter in Sea-water. 


1 in 152 British Channel... .1 in 28 

1 “ 46 Mediterranean.1 “ 25 

1 “ 131 Atlantic, Equator.. 1 il 25 


Atlantic, South. .1 in 24 
“ North. .1 “ 22 
Dead Sea.1 “ 2. 59 


When saline matter at temperature of its boiling-point is in proportion of 10 per 
cent., lime will be deposited, and at 29.5 per cent. salt. 

Temperature of water adds much to extent of saline deposits 


STEAM-ENGINE. 

The range of proportions here given is to meet the requirements of 
variations in speed, pressure, length of stroke, draught of water, etc., 
in the varied purposes of Marine, River, and Land practice. 


CONDENSING. 


For a. Range of Pressures of from 30 to SO lbs. (Mercu¬ 
rial Grange) per Sq. Inch, Cut Off at Half Stroke. 

Piston-rod. Cylinder or Air-pump (Wrought Iron), .1 to .14 of its diam.; 
(Steel), .8 diam.; and (Copper or Brass), .11 and .125. 

Condenser (Jet). Volume, .35 to .6 of cylinder. (Surface.) Brass tubes, 
16 to 19 B W G, .625 to .75 in diameter by from 5 to 10 feet in length, and 
.75 to 1.25 in pitch, condensing surfaces, .55 to .65 area of evaporating sur¬ 
face of boiler with a natural draught; .7 to .8 with a blower, jet, or like 
draught. Or, for a temperature of water of 6o°, 1.5 to 3 sq. feet per IIP. 

W'ith a very effective and sufficient circulating pump, areas may be reduced to 
.5 and .6. 


Effect of vertical tube surface, compared to horizontal, is as 10 to 7. 

Air-pump (Single acting and direct connection ), Volume from .15 to .2 
V4-d 

steam cylinder. Or, ——— 2.75. For Double acting put 4 for 2.75. V and v 

representing volumes of condensing and condensed water per cube foot, and n strolces 
of piston per minute. 


Foot and Delivery Valves. Area, .25 to .5 area of air-pump. 

Delivery Valve (Out-board). With a Reservoir. Area from .5 to .8 Foot 
valve. 

Note. —Velocity of water through these valves should not exceed 12 feet per 
second. 











STEAM-ENGINE. 


728 


CL S YL CL S YL 

Steam and Exhaust Valves. — (Poppet ).—-— = area for steam, - for 

v 1 ^ ' 24 000 J 20 000 ^ 

CL S Yt CL S Yl 

exhaust: (Slide), - for steam , ancl — : — for exhaust, a representing 

' 30000 J 22750 ^ 

area of steam cylinder in sq. ins., s stroke of piston in ins., and n number of revolu¬ 
tions per minute. 

Injection Pipes. —One each Bottom and Side to each condenser; area of 
each equal to supply 70 times volume of water evaporated when engine is 
working at a maximum; and in Marine and River engines the addition of 
a Bilge , which is properly a branch of bottom pipe. 

Note i.— Proportions given will admit of a sufficient volume of water when en¬ 
gine ig in operation in the Gulf Stream, where the water at times is at temperature 
of 84°, and volume required to give water of condensation a temperature of ioo° is 
70 times that of volume evaporated. 

2. Velocity of flow of water through cock or valve 20 feet per second in river or at 
shallow draught, and 30 feet in sea or deep draught service. 

Feed Pump.* — (Single acting, Marine), Volume, .006 to .01 steam cylinder. 
(River and Land), or when fresh water alone or a surface condenser is used, 
.004 to .007. 

NON-CONDENSING. 

For a Range of Pressures of from SO to ISO IDs. GVTercn.- 
rial Grange) per Sq. Incli, Cat Off at Tlalf Stroke. 

Piston-rod.—(Wrought Iron), Diam., .125 to .2 steam cylinder. (Steel), 
.8 diam. 

Steam and Exhaust Valves. —Area is determined by rules given for them 
in a condensing engine, using for divisors 30 000 and 22 750. 

A decrease in volume of cylinder is not attended with a proportionate decrease 
of their area, it being greater with less volume. 

Feed-pump .*■ —(Single acting. Marine), Volume, .008 to .016 steam cylin¬ 
der. (River and Land), or where fresh water alone is used, .005 to .oil. 


G-eneral Rules. 

Engines. 

• • D j) • T) j) 

Cylinder. Thickness .— (Vertical), = t; (Horizontal), —L — t; (In¬ 
clined), divide by 2000 in a ratio inversely as sine of angle of inclination. 

D representing diameter of cylinder, p extreme pressure in lbs. per sq. inch that it 
may be subjected to, and t thickness in ins. 

Shafts, Gudgeons, Journals, etc. To resist Torsion. See rules, pp. 790, 796. 
Coupling Bolts. — / -K-, — d. n representing number of bolts, D diameter 

2 V YL CL 

of shaft, d' distance of centre of bolts from centre of shaft, and cl diameter of bolts, 
all in ins. 

Cross-head, Wrought-iron. (Cylinder), ~~ = S, and y/y = d, or ~ =. h. 

a representing area of cylinder in sq. ins., I length of cross-head between centres of its 
journals in feet, and S product of square of depth d, and breadth , b, of section, 'both 

in ins. (Air-pump), — = S, and as above for d and b. 

If section of either of them is cylindrical, for S put ^Sx 1.7. 

Diam. of boss twice, and of end journals same as that of piston-rod. Sec¬ 
tion at ends .5 that of oentre. 


* See Formulas, page 736. 








STEAM-ENGINE. 


729 


Steam-pipe ,— Its area should exceed that of steam-valve, proportionate to 
its length and exposure to the air. 

Connecting-rod. — Length, 2.25 times stroke of piston when it is at all 
practicable to afford the space; when, however, it is imperative to reduce 
this proportion, it may be twice the stroke. 

Comparative friction of long and short connecting-rods is, for length of stroke ot 
piston, 12 per cent, additional; twice, 3 per cent.; and for thrice, 1.33. 

Neck. — Diam. 1 to 1.1 that of piston-rod. Centre of body ( Horizontal ), 
1.25 ins.; (Vertical), .06 inch per foot of length of rod. 

With two connecting-rods or links, area of necks .65 to .75 area of attached rod. 

When a second set of rods is used, as in some air-pump connections, area of 
necks, in a ratio, inversely as their lengths to that of first set. 


Straps of Connecting-rods, Links, etc. — (Strap), area at its least section 
.65 neck of attached rod ; (Gib and Keg), .3 diam. of neck, width, 1.25 times, 
(Slot) 1.35 times (Draft) of keys .6 to .8 inch per foot. Distance of Slot 
from end of rod .5 diam. of pin. 


Pins (Cranks, Beams , etc.). 



P representing pressure or thrust 


of rod or beam, l length of journal in ins., and C for Wrought iron — 640, Cast, 560. 
Puddled steel, 600, and Cast, 1200. 


Length, 1.3 to 1.5 times their diam., and pressure should not exceed 750 
lbs. per sq. inch for propeller engine, and 1000 for side-wheel. 


Cranks (Wr ought-iron ).— Hub, compared with neck of shaft, 1.75 diam., 
and 1 depth. Eye, compared with pin, 2 diam., and 1.5 depth. Web, at pe¬ 
riphery of hub, width, .7 width, and in depth .5 depth of lmb ; and at periph¬ 
ery of eye, width, .8 width, and in depth, .6 depth of eye. 

(Cast-iron.) Diameters of Hub and Eye respectively, twice diam. of neck 
of shaft, and 2.25 times that of crank pin. 

Radii for fillets of sides of web .5 width of web at end for which fillet is designed; 
for fillets at back of web, .5 that at sides of their respective ends. 


Beams, Open or Trussed. —Length from centres 1.8 to 2 stroke of piston, 
and depth .5 length. If strapped, Strap at its least dimensions .9 area of 
piston-rod, its depth equal to .5 its breadth. End centre journals each 1, and 
main centre journals 2.5 times area of piston or driving-rod. 

This proportion for strap is when depth of beam is .5 length, as above; conse¬ 
quently, when its depth is less, area of strap must be increased; and when depth of 
strap is greater or less than .5 width, its area is determined by product of its bd 2 , 
being same as if its depth was .5 its width. _ 


P X l - 7 - 2 

(Cast-iron). Area of Section of Centre. ^ ood = A - prepresenting 
extreme pressure upon piston in lbs ., d depth in ins., and l length in feet. 

Depth at centre .5 to .75 diam. of cylinder, and, when of uniform thick¬ 
ness, a thickness of not less than .1 of depth. 


Vibration of End Centres. — l -7-, 2 — V(l -i- 2) 2 — (s 2) 2 = vibration at each 

end; s representing stroke of piston , in feet. 

Plumber Blocks (Shaft).—Binder d S J-^C = depth. d representing diam. 

of bolts when two to binder, l distance between bolts, b breadth of binder, all in ins .. 
and C for wrought iron 1, steel .85, and cast iron .2. 

Holding-down Bolts. P-r 3 C = area at base of thread of each bolt. C for mild 
steel for small and large bolts 6000 and 7000, for wrought iron 4500 and 6000, if but 
two are used. 

Binder (Brass). — ,^Jfj — depth. 




730 


STEAM-ENGINE. 


Cocks .—Angles of sides of plug from *7° to 8° from plane of it. 

Pumps .—Velocity of water in pump openings should not exceed 500 feet 
per minute. 

Fly-wheels and Governors .—See Rules, pages 451 and 452. 

Water-wheels. 

Water-wheels ( Arms ).—Number from .75 to .8 diam. of tvheel in feet. 
(Blades) Wood .—For a distance of from 5 to 5.5 feet between arms, thick¬ 
ness from .09 to .1 inch for each foot of diam. of wheel. 

Area of blades, compared with area of immersed amidship section of a 
vessel, depends upon dip of wheels, their distance apart, model and rig of 
vessel. , 

In River service , area of a single line of blade surface varies from .3 to .4 
that of immersed section; in Bay or Sound service , it varies from .15 to .2; 
and in Sea service , it varies from .07 to .1. 

Note.—A wrought-iron blade .625 inch thick bent at a stress withstood by an 
oak blade 3.5 ins. thick. 


Radial and Feathering. 

Radial .—Loss of effect is sum of loss by oblique action of wheel blades 
upon the water, their slip, and thrust and drag of arms and blades as they 
enter and leave the water. 

Loss by oblique action is computed by taking mean of square of sines of 
angles of blades when fully immersed in the water. 

Loss by oblique action of blades of wheel of steamer Arctic, when her wheels 
were immersed 7 feet 9 ins. and 5 feet 9 ins., w T as 25.5 and 18.5 per cent., which 
was the loss of useful eftect of the portion of total power developed by engines, 
which was applied to wheels. 

Feathering .—Loss of effect is confined to thrust and drag of arms and 
blades as they enter and leave the water. 


Comparative Effects .—In two wheels of a like diameter (26 feet, and 6 feet immer¬ 
sion), like number and depth of blades, etc., the losses are as follows : 

Radial. 26.6 per cent. | Feathering. 15.4 per cent. 

Loss of effect by thrust and drag in a feathering wheel, having these elements 
and included in the above given loss, is computed at 2 per cent. 

Relative loss of effect of the two wheels is, approximately, for ordinary immer¬ 
sions, 20 and 15 per cent, from circumference of wheel. 

2 <J3_ d'3 

Centre of Pressure , ———— — d = c. d and d' representing depths of blades 

3 a a 


below surface of water, and c centre of pressure, all in like dimensions, from bottom 
edge. 

In the cases here given, centres of pressure are as follows: 

Radial.6.4 ins. | Feathering.8.5 ins. 


IPropellers. 

Propellers (Screw). — Pitch should vary with area of circle described bv 
screw to area of midship section of vessel. 

AREA, TWO-BLADED. 


Area of disk of propeller to mid -1 
ship section being 1 to.j 

6 

5 

' 

4-5 

4 

3-5 

3 

2-5 

2 

Ratio of pitch to the diameter of \ 
propeller is 1 to.j 

.8 

1.02 

1.11 

1.2 

1.27 

I - 3 I 

1.4 

1.47 


For Four-bladed screws, multiply ratio of pitch to diam. as given above, 
by 1-35. Length , .166 diam. 






















STEAM-ENGINE. 


731 


Slip .—Slip of a screw propeller is directly as its pitch, and economical 
effect of a screw is inversely as its pitch; greater the pitch less the effect. 

An expanding pitch has less slip than a uniform pitch, and, consequently, 
is more effective. 


IS* ^7: 


To Compute Thrust of a Lropeller. 
: T. S representing speed of vessel in knots per hour. 


SLIDE VALVES. 

A ll Dimensions in Inches. 

To Compute Lap required on Steam End, to Cnt-ofT at 
any- given 3 ?art of Stroke of Liston. 

Rule. —From length of stroke subtract length of stroke that is to be made 
before steam is cut off; divide remainder by stroke, and extract square 
root of quotient. 

Multiply this root by half throw of valve, from product subtract half lead, 
and remainder will give lap required. 

Example.— Having stroke of piston 60 ins., stroke of valve 16 ins., lap upon ex¬ 
haust side .5 in. =one thirty-second of valve stroke, lap upon steam side 3.25 ins., 
lead 2 ins., steam to be cut off at five sixths stroke; what is the lap? 

60—of 60 = 10. = . 408. .408 X — = 3-264, and 3.264 — — = 2.264 ins. 

To Ascertain Lap required on Steam End, to Cut-off 
at various Portions of Stroke. 

I Distance of piston from end of its stroke when steam is cut off, 
Valve in parts of length of its stroke. 


without Lead. 

1 

5 

1 

7 

1 

5 

1 

1 

1 

1 


2 

12 

3 

2 ¥ 

4 

2 T 

6 

8 

1^ 


Lap in parts of) 
stroke.j 

•354 

• 3 2 3 

.286 

.27 

•25 

.228 

.204 

.177 

.144 

.102 


Illustration.— Take elements of preceding case. 

Under i is .204, and .204 X 16 = 3.264 ins. lap. 

When Valve is to have Lead .—Subtract half proposed lead from lap as¬ 
certained by table, and remainder will give proper lap to give to valve. 

If, then, as last case, valve was to have 2 ins. lead, then 3.264 — 2 -f- 22.264 ins. 


To Compute at what Lart of Stroke any given Lap on 
Steam Side will Cut off. 

Rule. —To lap on steam side, as determined above, add lead; divide sum 
by half length of throw of valve. From a table of natural sines (pages 390- 
402) find the arc, sine of which is equal to quotient; to this arc add 90°, 
and from their sum subtract arc, cosine of which is equal to lap on steam 
side, divided by half throw of valve. Find cosine of remaining arc, add x 
to it, and multiply sum by half stroke, and product will give length of that 
part of stroke that will be made by piston before steam is cut off*. 

Example.— Take elements of preceding case. 

Cos. (sin. — 4 ~t - 2 + 90 0 — cos. - 2 . 264 ) + 1 X — = cos. (32 0 13' + 90 0 — 73 0 34') 
V 16-^2 y 16 — 2/ 2 

= 48° 39', and cos. 48° 39'+1 X — = 1.66 X 30 = 49.8 ins. 

To Ascertain Breadth of Ports. 

Half throw of valve should be at least equal to lap on steam side, added to breadth 
of port. If this breadth does not give required area of port, throw of valve must be 
increased until required area is attained. 




















732 


STEAM-ENGINE. 


Portion, of 1 Stroke at which Exhausting Port is Closed. 


Lap on 
Exhaust 
Side of 
Valve in 
Parts of 
its Throw. 


A 
.125 
.062 5 
.031 25 
o 


and Opened. 


Portion of Stroke at which Steam 
is cut off. 


178 

13 

113 

092 


7 15 

H T H 


. 161 
. 118 
. 101 
.082 


143 

1 

085 

067 


.126 

085 

.069 

055 


109 

071 

053 

,041 


•093 

.058 

•043 

•033 


1 

Ta 


.074 

.043 

•033 

.022 


Lap on 
Exhaust 
Side of 
Valve in 
Parts of 
its Throw. 


B 

.125 
.062 5 
.031 25 
o 


Portion of Stroke at which Steam 
is cut off. 


033 

,06 

•073 

.092 


7 

■gr 


.026 

.0^2 

.066 

.082 


.019 

.04 

.051 

.067 


5 

ad 


012 

03 

042 

055 


.008 

.022 

•°33 

.044 


1 

¥ 


,004 

015 

,023 

033 


1 

Ta 


.001 

.008 

.013 

.022 


Units in columns of table A express distance of piston, in parts of its stroke, from 
end of stroke when exhaust port in advance of it is closed; and those in columns 
of table B express distance of piston, in parts of its stroke, from end of its stroke 
when exhaust port behind it is opened. 

Illustration. —A slide valve is to be cut off at one sixth from end of stroke, lap 
on exhaust side is one thirty-second of stroke of valve (16 ins.), and stroke of piston 
is 60 ins. At what point of stroke of piston will exhaust port in advance of it be 
closed and the one behind it open. 

Under one sixth in table A, opposite to one thirty-second, is .053, which X 60, 
length of stroke = 3.18 ins. ; and under one sixth in table B, opposite to one thirty- 
second, is .033, which X 60 = 1.98 ins. 

If lap on exhaust side of this valve was increased, effect would be to cause port in 
advance of valve to be closed sooner and port behind it opened later. And if lap on 
exhaust side was removed entirely, the port in advance of piston would be shut, 
and the one behind it open, at same time. 

Lap on steam side should always be greater than that on exhaust side, and differ¬ 
ence greater the higher the velocity of piston. 

In fast-running engines, alike to locomotives, it is necessary to open exhaust valve 
before end of stroke of piston, in order to give more time for escape of the steam. 


To Compute Stroke of a Slide Valve. 

Rule.—T o twice lap add twice width of a steam port in ins., and sum 
will give stroke required. 


Expansion by lap, with a slide valve operated by an eccentric alone, cannot be 
extended beyond one third of stroke of a piston without interfering with efficient 
operation of valve; with a link motion, however, this distortion of the valve is 
somewhat compensated. When lap is increased, throw of eccentric should also be 
increased. 

When low expansion is required, a cut-off valve should be resorted to in addition 
to- main valve. 


To Compute Distance of a Liston from End of its 
Stroke, when Lead produces its Effect. 

Rule. — Divide lead by width of steam port, both in ins., and term the 
quotient sine; multiply its corresponding versed sine by half stroke, and 
product will give distance of piston from end of its stroke, wdien steam is ad¬ 
mitted for return stroke and exhaustion ceases. 

Example.— Stroke of piston is 48 ins., width of port 2.5 ins., and lead .5 inch; 
what will be distance of piston from end of stroke when exhaustion commences? 

• 5 -l - 2 -5 = • 2 — sine, ver. sin. of. 2 = .0202, and .0202 x — = .4848 ins. 


To Compute Lead, when Distance of a Liston from 
the End of Stroke is given. 

Rule.—D ivide distance in ins. by half stroke in ins., and term quotient 
versed sine; multiply corresponding sine by width of steam port, and prod¬ 
uct will give lead. 

Example. —Assume elements of preceding case. 


.4848 -4- — = .0202 = ver. sin., and sine of ver. sin. .0202 = .2, and .2 x 2.5 = .5 inch. 


2 
































STEAM-ENGINE. 


733 


To Compute Distance of a Piston from Kind, of its Stroke, 
wlieix Steam is admitted for its Return Stroke. 

Rule.—D ivide width of steam port, and also that width, less the lead, by 
.5 stroke of slide, and term quotients versed sines first and second. Ascer¬ 
tain their corresponding arcs, and multiply versed sine of difference between 
first and second by .5 stroke, and product will give distance. 

Example.— Assume elements of preceding case, lap = .5 inch, and stroke of 
slide 6 ins. 

y 2 ' 5 ■ and —— =. 8333, and .667 and ver. sin. 8o° 24' 7 o° 33' x — =.3328 inch. 

6-7-2 6-7-2 2 

To Compute Lap and Lead of Locomotive "Valves. 

To cut off at .33, .25, and .125 of stroke of piston, lap == 289, .25, and ,i 77 t , outside 
lead =. o 7 t, and inside lead = . 3 t. t representing stroke of valve, all in ins. 

HORSE-POWER. 

Horse-power is designated as Nominal, Indicated , and Actual. 

Nominal , is adopted and referred to by Manufacturers of steam-engines, 
in order to express capacity of an engine, elements thereof being confined 
to dimensions of steam cylinder, and a conventional pressure of steam and 
speed of piston. 

Indicated , designates full capacity in the cylinder, as developed in opera¬ 
tion, and without any deductions for friction. 

A dual, refers to its actual power as developed by its operation, involving 
elements of mean pressure upon piston, its velocity, and a just deduction for 
friction of operation of the engine. 

To Compute Horse-power of an Engine. 

J ) 2 d # D 2 'y 

Nominal.— Non-condensing, - q - q , and Condensing , —— = HP D repre¬ 
senting diameter of cylinder in ins., and v velocity of piston in feet per minute. 

Non-condensing is based upon uniform steam-pressure of 60 lbs. per sq. 
inch (steam-gauge), cut off at .5 stroke, deducting one sixth for friction and 
losses, with a mean velocity of piston, ranging from 250 to 450 feet per 
minute. 

Condensing is based upon uniform steam-pressure of 30 lbs. per sq. inch 
(steam-gauge), cut off at .5 stroke, deducting one fifth* for friction and 
losses, with a mean velocity of piston of 300 feet per minute for an engine 
of short stroke, and of 400 feet for one of long stroke. 

A PI* — (fX — 1 — 1 a. 7) 2 s v 

Actual.— Non-condensing. - - - ‘33000 - = B?- A representing 

area of cylinder in sq. ins., P mean effective pressure upon cylinder piston, inclusive 
of atmosphere, f friction of engine in all its parts, added to friction of load, both in 
lbs. per sq. inch, s stroke of piston in feet, and r number of revolutions per minute. 

Sum of these resistances is from 12.5 to 20 per cent., according to pressure of 
steam, being least with highest pressure. 

* This value may be safely estimated at 2.5 lbs. per sq. inch for friction of engine in all its parts, and 
friction of load may be taken at 7.5 per cent, of remaining pressure. 

t This value is best obtained by an Indicator; when one is not used, refer to rule and table, pp. 710-12. 
In estimating value of P, add 14.7 lbs., for atmospheric pressure, to that indicated by steam gauge or 
safety-valve. Clearance of piston at each end of cylinder is included in this estimate. 

1 This value may be safely estimated in engines of magnitude at 1.5 to 2 lbs. per sq. inch, for friction 
of engine in all its parts, and friction of load may be taken at 5 to 7.5 per cent, of remaining pressure. 

Sum of these resistances in ordinary marine engines is from 10 to 20 per cent., according to pressure 
of steam, exclusive of power required to deliver water of condensation at level of discharge or load-line 
of a vessel. For pressure representing friction for different designs and capacities of engines as esti¬ 
mated by English authority, see pp. 473-5 and 662. 

3 Q 







734 


STEAM-ENGINE. 


Illustration. —Diameter of cylinder of a non-condensing engine is io ins., stroke 
of piston 4 feet, revolutions 45 per minute, and mean pressure of steam (steam 
gauge) do lbs. per sq. inch. 

A = 78.54 sq. ins. P 60+14.7 = 74.7 lbs. /=2.5 +(60+14.7 — 2.5) X .075 = 7.92 lbs. 

Then 78-54 X (60+14.7 —7.92+ 14. 7 )X 2 X 4X45 = 6 jp 

33 000 

Note i. —Power of a non-condensing engine is sensibly affected by character of its 
exhaust, as to whether it is into a heater, or through a contracted pipe, to afford a 
blast to combustion. 

2.—If an indicator is not used to determine pressure of steam in a cylinder, a 
safe estimate of it, when acting expansively, is .9 of full pressure, and when at full 
stroke from .75 to .8. 

Condensing. - - - — fi 2 s 1 _ jp 

33000 

Power required to work the air-pump of an engine varies from .7 to .9 lbs. per sq. 
inch upon cylinder piston. 

Illustration. —Diameter of cylinder of a marine steam-engine is 60 ins., stroke 
of piston 10 feet, revolutions 15 per minute, pressure of steam 50 lbs. per sq. inch, 
cut off at .25 stroke, and clearance 2 per cent. 

A = 2827.4 S <Z• i- ns - P (P er Ex -> P a g e 713) = 28.62 lbs. f— 1.5 + 28.62 — 1.5 X .05 
= 2.467 lbs. 


Then 


2827.4 X 28.66 — 2.856 X 2 X 10 X 15 


: 662.23 H*. 


33000 

From which is to be deducted iu marine engines power necessary to discharge 
water of condensation at level of load-line, which is determined by pressure due to 
elevation of water, area of air-pump piston, and velocity of its discharge in feet per 
second. 


A P 2 s r __ , 33 coo IP 

Indicated. - - = IP, and — -- — A. 

33 000 1 2 s r 

British. Admiralty Role.— Nominal. 


•7 A v D 2 v 

— - or -— = IP. 

33 000 6000 


French.—( Force de Cheval.) 1.695 D' 2 L r = IP. I) and L in meters. 

Illustration. — Assume a diameter of cylinder of .254 meters, with a stroke of 
piston of .3 meters and 250 revolutions per minute. 

1.695 X .254“ X .3 X 250 = 8.18 H?. 

A Force de Cheval— 4500 kilometers per minute = 32 549 foot-lbs. = .987 57 IP. 
One IP = 1.0139 Force de Chevaux. 

Compound Indicated. A L r 1 hyp. log. R"— l^j .000053 — IP. 

L representing length of stroke in feet , R" combined ratio of both cylinders , and b 
back pressure. 

Illustration. — Assume area of cylinder 3 sq. ins., stroke 6 feet, one stroke of 
piston, and steam 60 lbs. per sq. inch, cut off at .25. 

A n= 3 sq. ins. , L = 6 feet , n = 1 stroke , P = 60 lbs. , R" — 5.969, b = 3 lbs. 
per sq. inch, and r = . 5, ’ and 1 -f- hyp. log. R' / — 1 -f- 1.7865. 

Then 3 X 6 X .5 X {~^ X 1 + 1-7865 — 3^ X .000053 = 9 X 10.052 X 2.7865—3 

X .000053 — -on 93 IP> which, X 2 for 1 revolution, =.023 86 BP per revolution. 

To Compute Volume of Water required to he Evapo¬ 
rated in an Engine. 

Rule.— Multiply volume of steam expended in cylinder and steam-chests 
by twice number of revolutions, and multiply product by density of steam 
at given pressure. 


+ f For reference see 2d and 3d foot-note on previous page. 















STEAM-ENGINE. 


735 


Example. —What volume of water will an engine require to be evaporated per 
revolution, diam. of cylinder being 70 ins., stroke of piston 10 feet, and pressure of 
steam 34 lbs. per sq. inch, including atmosphere, cut off at .5 of stroke? 

Area of cylinder m 3848.5 ins. 10 X 12 -1- 2 = 60 ins. , 60 x 3848.5 = 230 910 cube ins. 

Add, for clearance at one end, volume of nozzle, steam-chest, etc., 17 317 cube ins. 

Then 2309104-17317-5-1728X2=^.287.3 cube feet, which, x • 001 336, density of 
steam at 34 lbs. pressure (see Note 2), =3838 cube feet. 

Note i.— This refers to expenditure of steam alone; in practice, however, a large 
quantity of water “foaming,” differing in different cases, is carried into cylinder in 
combination with the steam; to which is to be added loss by leaks, gauges, etc. 

2.—Volume of steam is readily computed by aid of table, pp. 708-9. Thus, den¬ 
sity or weight of one cube foot of steam at above pressure 1= .0835 lbs. Hence, as 
62.5 lbs. : x cube foot ;; .0833 lbs. : .001 336 cube foot. 

To Compute Volume of Circulating Water required lay 

an .Engine. 

1114 ~t~' 3 7 — - = V. T representing temperature of steam upon entering the con¬ 


denser , t, t', and t" temperatures of feed water , of water of condensation discharged , 
and of circulating water , all in degrees. 

Illustration. —Assume exhaust steam at 8 lbs. per sq. inch, temperatures of dis¬ 
charge xoo°, feed water 120 0 , and sea-water 75 0 . 


Temperature at 8 lbs. pressure = 183°. 


11144-.3 x 183 — 
100 — 75 


120 

— = 4i-95 times. 


To Compute "Volume of Flow tlrrouglr an Injection Fipe. 

Rule.— Multiply square root of product, of 64.33 and depth of centre of 
opening into condenser, from surface of external water, added to height of a 
column of water due to vacuum in condenser, all in feet, by area of opening 
in sq. ins.; and .6 product, divided by 2.4 (144 -4- 60) will give volume in 
cube feet per minute. 


Example.— Diameter of an injection pipe is 5.375 ins., height of external water 
above condenser 6.13 feet, and vacuum 24.45 ins.; what is volume of flow per min.? 

» . r • ^ 24.4s ins. .. 

Area of 5.375 ins. = 22.69 ins-, c = .6. vacuum -- - = 12 lbs.: 12X2.24 

2.04 

feet (sea-water) = 26.88 feet, and 26.88 -(- 6.13 =r 33.1 feet. 

Then V6 4 .33 _ X _ 3 _ 3. _ . _ X^6 9 X.6 = = ^ mbefid 

2.4 2.4 


To Compute Area of an Injection Fipe.. 

Rule.— Ascertain volume of water required by rule, page 706, in cube ins. 
per second, multiply it by number of volumes of water required for con¬ 
densation, by rule, page 707, divide it by velocity due to flow in feet per 
second, and again by 12, and quotient will give area in sq. ins. 

Example. —An engine having a cylinder 70 ins. diam., stroke of piston 10 feet, 
revolutions per minute 15, and steam 19.3 lbs., mercurial gauge cut off at .5; what 
should be area of its injection pipe at its maximum operation? 

Volume of cylinder 267.25 cube feet,-cut off at .5 = 133.625 ins. 

Density of steam at 34 lbs. (19.3 4 - 14-7) = .001 336. Velocity of flow of injected 
water (computed from vacuum and elevation of condensing water) 33 feet per second. 

Then 133.625 X 15 X 2 X 1728—- 60 = 115452 cube ins. steam per second , and 
115 452 X .001 336 = 154.24 cube ins. water per second. 

Maximum volume of water required to condense steam is about 70 times volume 
of that evaporated, which only occurs in the Gulf of Mexico; ordinary requirement 
is about 40 times. 

154.24-)- n.59 (= 7-5 P er cent, for leakage of valves, etc.) = 165.83, which, X 70 
as above, == n 608.1 cube ins., and n 608. i -j-33 X 12 = 29.31 sq. ins. 








STEAM-ENGINE. 


736 


Coefficient of velocity for flow under like conditions = .6; hence, 29.31- 3 -.6 = 
48.85 sq. ins. 

Note.—T his is required capacity for one pipe. It is proper and customary that 
there should he two pipes, to meet contingency of operation of one being arrested. 


To Compute Area of a Feed. Pump. ( Sea-water.) 

Rule. —Divide volume of water required in cube ins. by number of single 
strokes of piston, both per minute, and divide quotient by stroke of pump, in 
ins.; multiply this quotient by 6 (for waste, leaks, “running up,” etc.), and 
product will give area of pump in sq. ins. 

Example. —Assume volume to be 5 cube feet and revolutions of engine 15 per 
minute, with a stroke of pump of 3.5 feet. 
c X 172S , -*- 

--— 576, which -7- 3.5 x 12 = 13.72, and 13.72 X 6 = 82.32 sq. ins. 

Note.—I n fresh water, this proportion may be reduced one half. 


STEAM-INJECTOR. William Sellers & Co. 
Self-adjusting. 

Volume of Water FiscHarged. per Hour. 


No. 

Pressure of Steam in Lbs. 

No. 

Pressure of Steam in Lbs. 

60 

80 

100 

120 

60 

80 

100 

120 


Cub. feet. 

Cub. feet. 

Cub. feet. 

Cub. feet. 


Cub. feet. 

Cub. feet. 

Cub. feet. 

Cub. feet. 

3 

N 

H 

GO 

Cl 

31.66 

35-2 

38.75 

7 

162.65 

182.1 

201.55 

221 

4 

52.16 

58.44 

64.72 

7 i 

8 

213.2 

238.8 

264.4 

29O 

5 

82.18 

92.02 

101.86 

hi. 7 

9 

269.97 

302.28 

334-59 

366.9 

6 

II9.O9 

I 33.33 

147-57 

161.82 

IO 

333.64 

373-57 

4 I 3.49 

453-41 


Highest temperature admissible of feed water i35 c 


To Compute Size of Injector required. 

One nominal IP per hour will require one cube foot of water per hour. 
When the lbs. of coal burned per hour can be ascertained, divide them by 
7.5, and quotient will give the volume of water in cube feet per hour. 

When the area of grate-surface is known, multiply it by 1.6 for HP. 

In case of plain cylindrical boilers, divide the number of sq. feet of heat¬ 
ing-surface by 10 for the HP. In case of flue boilers, divide by 12, and with 
multi-tubular boilers, by 15, for the nominal HP. 

Minimum capacity of Injectors, about 50 per cent, of Tabular capacity. 


To Compute Volume of Injection W r a.ter required. per 

IIP per Hour. 

Operation.— Assume temperature of water 8o°, and of condensation ioo°. Vol¬ 
ume of cylinder per IIP as per formula, page 716, and illustration, page 717, = 2.76 
feet per minute. 

Then, as per rulo page 707, --- 1= 52.3 cube ms. per cube foot of steam. 


2.76 X 52-3 X 62.5 
1728 


= 5.22 lbs., which, X 60, — 313.2 lbs. 


To Compute ISTet Volume of Feed. Water required per 

IIP per Hour. 

Operation.— Assume elements of formula, page 716, and illustration, page 717. 
Then .1154 X 2.76 X 60 = 19. u lbs. 

Feed Pipes, — y/v — diameter for small, and — y/ v, for large pumps. 

20 32 

d representing diameter of plunger in ins., and v its velocity in feet per minute. 
























STEAM-ENGINE. 737 

Results of Operations of Steam-engines. ( D. K. Clark.) 


Condensing Engine. 


SINGLE. 

Corliss, Saltaire. 

Pumping, Crossness. 

“ East London. 

Sulzer, Corliss valves.. . 

Superheated, Hirn. 

COMPOUND. 

Receiver. 

Marine, jacketed 

Receiver_... 

stationary. 

Woolf, stationary 

jacketed. 

ist cylinder. 

both... 

ist cylinder. 

both. 

NON-CONDENSING. 

Marshall, Sons, & Co. 

Davey, Taxman, & Co. 

Locomotive “Great Britain”. 

U U U 


J. Elder & Co. 

J. & E. Wood. 

Ponkin. 

American, Woolf. 

“ “ jacketed 


Actual 
Ratio of 
Expan¬ 
sion. 

Steam 
per 
IIP as 
cut-off. 

Coal 

per 

IIP. 

Initial 

Pressure 

at 

cut-off. 

Steam 
per I HP 
per hour. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

5-2 

I 4 - 5 I 

2-5 

34-5 

17.4 

6.07 

14.27 

2.2 

46 

18.7 

3.62 

12.92 

— 

23.25 

20.72 

IO 

— 

3-3 

50 

19.6 

4.132 

— 

— 

60 

18.62 

1.85 

1.852 

14-45 

14.85 

I.6l 

56 

— 

4.01 

1-857 

10.94 

13-34 

2.14 

85-5 

— 

2.486 

13.18 

— 

50-5 

— 

3.221 

13-87 

— 

-‘ 

22.51 

2.31 

actual 



15-37 

J3 

23.21 


90 

3-77 

9 - 1 9 

20.71 

— 

90 

14.1 

4.8 

16.87 


76 

25-9 

5 

14-93 

— 

73 

29.6 

i -45 

31-36 

— 

102 

31-36 

2.94 

21.24 

— 

87 

21.24 


Practical Efficiency of Steam-engines. 


Initial Volume = i. 


Cylinders. 


CONDENSING. 

Single cylinder, jacketed... 

Single cylinder. 

“ “ superheated 

Compound, jacketed, Re-) 
ceiver..j 

* From boiler. 


Most Efficient 
Ratio of Ex¬ 
pansion. 

Jtj 

* 

S s 

X Q. 

55 

Cylinders. 

Most Efficient 
Ratio of Ex¬ 
pansion. 


Lbs. 

Compound, jacketed, Woolf 

IO 

0 

19-5 

24 

18.5 

Compound, Woolf.. 

7 

*T 

4 

NON-CONDENSING. 

Single cylinder,! jacketed.. 

4 


1 9 

Single cylinder,!. 

3 


+ 70 lbs. pressure. J 90 lbs. pressure. 


0 > w 

o-5 
* .a 

is 

qj —h 

In 


Lbs. 

20.5 


23 


24 

21 


Standard Operation of a Portable Engine. 


Grate. 5.5 sq. feet. 

Heating surface.220 “ “ 

Coal per EP per hour- 6.25 lbs. 

“ “ sq. foot of grate. 9 “ 

“ “ hour... 50 “ 


Water evaporated from 1 

and at 212 0 .) ‘ ‘ ‘' 

Water evaporated per EP... 
“ “ “ sq. foot 

of grate. 


Ratio of surface.40 to 1. 


450 lbs. 
6.25 “ 
81.8 


MIXTURE OF AIR AND STEAM. 

Water contains a portion of air or other uncondensable gaseous matter, and when 
it is converted into steam, this air is mixed with it, and when steam is condensed 
it is left in a gaseous state. If means were not taken to remove this air or gaseous 
matter from condenser of a steam-engine, it would fill it and cylinder, and obstruct 
their operation; but, notwithstanding the ordinary means of removing it (by air- 
pump), a certain quantity of it always remains in condenser. 

20 volumes of water absorb i volume of air. 

3 Q* * 






























































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c 3 


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►H 

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ft. 


si 

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c*> 

v w 
*S =3 
S 3 £? 
J2, 

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PRICE. 

STEAM-ENGINE. 

u-)0mu">000ir>0000u - >iA0, . i i . i i 

Cl 0 ) ^Osrou-)M fO CMO H CN N rn On 

m d CO CO Tt- ir)VO VO vo t^oo t>-co CO 1 1 1 1 ' 1 ' 

i 

0 

b£ 

Jd d 

5 3 

*0 • 03 

►< co —> 

8 Q 

S d 

• 

m ^ 

3 h (N ci co co rt* ^ to in io mvo \ovovo n n c^oo no n 

M H 

for Short Le 
\ length incre 

Suction- 

pipe. 

« iom 

3 h ci m iovo vo vo vo n n n n.co good oco « n 

H MM 

eter of Pipes 
e increased as 

Exhaust- 

pipe. 

m 

oii%.m mm m mm m mmm 

£ ' H d d ci ci co ci cocococi cococococococo covo vo 

Pi am 
To b 

Steam- 

pipe. 

®m mm m mm mm mm 

3 , HHHNNCiONcicONcicicONcicOCOCO'tTh 

Diam. of Plunger 
in any single 
Cylinder Pump 
for like Volume 
and Speed. 

mm mm 

d n- mmmm m 

oo vo co m mco co d d ci d 

^ tj- mvo n*»oo on on ci ci cm ci 

MMHMMHHHMMHMHHCl 

Volume delivered 
per Minute, at 
stated Number of 
Strokes. 

0 0 OmmmmO 0 0 0 m m m m 0 0 0 0 0 0 m 

COCO ^t* 0 O VCOCOONCNONONH m Hi h CO 00 OO CO Cl (N MD 

• H M Cl COrOVV^-tNNNNONONONCOONCOO 

g M M Cl 

^ooooooooocoooooooooooo 

0 

rh m o 0 0 mmmmmmmmmmmo 0 0 mo mm 

h m n On ci vo vo rt-rt-Tj-rt-mmmmovOv ov'O h co rt- 

h h m ci ci ci ci cocococov v Tt-vo m t>. m 

H 

Single Strokes or 
Displacements 
per Minute of one 
Plunger. 

No. 

75 to 150 

75 t0 I2 5 

75 to IOO 

75 to 100 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to IOO 

50 to 90 

50 to 90 

50 to 90 

Displace¬ 

ment 

per Stroke 
of one 
Plunger. 

• 

a m on cn ci vo vo mmmmN*t^.t>»t^.OvOv owo 
jd h cnvo On ci vo vo Tt-^Tt-Tj-mmm moo co 00 vo h co -«t 

*3 * * * ’HHHCicicio 5 cocococo 44 4 -vo m m 

0 M 

Length 

of 

Stroke. 

“ ^vo OOOOOOOOOOOOOOOO mm 

Diameter 

of 

Water- 

Plungers. 

m m mmmm 

w n m cn mmmm ci ci ci ci 

3 ci ^ 4 - mvo Cx t^oo oooocdddddcicici^ocim 

Diameter 

of 

Steam- 

cylinders. 

.mm m mmm 

G 4 VO N Cn 0 Cl ^ Cl ^vo 06 Cl ^vo 00 Tj-VO cd CO N 0 0 

HH MHMMHMHMKHHHHHIMMdCl 


Many of the above sizes, or those of any desired capacity, can be compounded by the addition of a non-condensing cylinder, resulting in a sav 
ing of 33 per cent, of fuel for like service, by any non-condensing form over that*required. 

Exterior packed plungers, for pumping water or oil against extreme pressure. 































STEAM-ENGINE.-BOILER. 


739 


BOILER. 

Its efficiency is determined by proportional quantity of heat of com¬ 
bustion of fuel used, which it applies to the conversion of water into 
steam, or it may be determined by weight of water evaporated per lb. 
of fuel. 

In following results and computations, water is held to be evaporated from stand¬ 
ard temperature of 212 0 . 

Proportion of surplus air, in operation of a furnace, in excess of that which is 
chemically required for combustion of the fuel, is diminished as rate of combustion 
is increased; and this diminution is one of the causes why the temperature in a 
furnace is increased with rapidity of combustion. 

When combustion is rapid, some air should be introduced in a furnace 
above the grates, in order the better to consume the gases evolved. 

Natural Draught. 

Grate (Coal) should have a surface area of 1 sq. foot for a combustion of 

15 lbs. of coal per hour, length not to exceed 1.5 times width of furnace, and 
set at an inclination toward bridge-wall of 1 to 1.5 ins. in every foot of length. 

When, however, rate of combustion is not high, in consequence of low ve¬ 
locity of draught of furnace, or fuel being insufficient, this proportion of area 
must be increased to one sq. foot for every 12 lbs. of fuel. 

Width of bars the least practicable, spaces between them being from .5 to 
.75 of an inch, according to fuel used. Anthracite requiring less space than 
bituminous. Short grates are most economical in combustion, but generate 
steam less rapidly than long. 

Level of grate under a plain cylindrical boiler gives best effect with a fire 
5 ins. deep, when grate is but 7.5 ins. from lowest point. 

Depth, Cast-iron, .6 square root of length in ins. 

(Wood), their area should be 1.25 to 1.4 that for coal. 

Automatic (Vicar’s). — Its operation effects increased rapidity in firing 
and more effective evaporation. 

Ash-pit. —Transverse area of it, for a combustion of 15 lbs. of coal per 
hour, 2 to .25 area of grate surface for bituminous coal, and .25 to .3 for 
anthracite. Or 15 to 20 ins. in depth for a width of furnace of 42 ins. 

Furnace or Combustion Chamber. — (Coal) Volume of it from 2.75 to 3 cube 
feet per sq. foot of grate surface. ( Wood) 4.6 to 5 cube feet. 

The higher the rate of combustion the greater the volume, bituminous 
coal requiring more than anthracite. Velocity of current of air entering 
an ash-pit may be estimated at 12 feet per second. 

Volume of air and smoke for each cube foot of water converted into steam is, 
from coal, 1780 to 1950 cube feet, and for wood, 3900. 

Rate of Combustion. — In lbs. of coal per sq. foot of grate per hour. 
Cornish Boilers, slowest, 4; ordinary, 10. Stationary, 12 to 16. Marine , 

16 to 24. Quickest: complete combustion of dry coal, 20 to 23; of caking 
coal. 24 to 27 ; Blast or Fan and Locomotive, 40 to 120. 

Bridge-wall (Calorimeter). —Cross-section of an area of 1.2 to 1.6 sq. ins. 
for each lb. of bituminous coal consumed per hour, or from 18 to 24 sq. ins. 
for each sq. foot of grate, for a combustion of 15 lbs. of coal per hour. 

Temperature of a furnace is assumed to range from 1500° to 2000°, and 
volume of air required for combustion of 1 lb. of bituminous coal, together 
with products of combustion, is 154.81 cube feet, which, when exposed to 
above temperatures, makes volume of heated air at bridge-wall from 600 to 
750 cube feet for each lb. of coal consumed upon grate. 


740 


STEAM-ENGINE.-BOILER. 


Hence, at a velocity of draught of about 12 feet per second, area at bridge- 
wall, required to admit of this volume being passed off in an hour, is 2 to 2.5 
sq. ins., and proportionately for increased velocity, but in practice it may be 
1.2 to 1.6 ins. 

When 20 lbs. of coal per hour are consumed upon a sq. foot of grate, 20 X 1.2 or 
1.6 = 24 or 32 sq. ins. are required, and in a like proportion for other quantities. 

Or, When area of flues is determined upon, and area over bridge-wall is 
required, it should be taken at from .7 to .8 area of lower flues for a natural 
draught, and from .5 to .6 for a blast. 

When one half of tubes were closed in a fire-tubular marine boiler, the evapora¬ 
tion per lb. of coal was reduced but 1.5 per cent. 

Firing. —Coal of a depth up to 12 ins. is more effective than at a less 
depth. Admission of air above the grate increases evaporative effect, but 
diminishes the rapidity of it. 

Air admitted at bridge-wall effects a better result than when admitted at 
door, and when in small volumes, and in streams or currents, it arrests or pre¬ 
vents smoke. It may be admitted by an area of 4 sq. ins. per sq. foot of grate. 

Combustion is the most complete with firings or charges at intervals of 
from 15 to 20 minutes. 

With a fuel economizer (Green’s) an increased evaporative effect of 9 per 
cent, has been obtained. 

When external flues of a Lancashire boiler were closed, evaporative power was 
slightly increased, but evaporative efficiency was decreased; and when 25 per cent, 
of like surface in setting of a plain cylindrical boiler was cut off, evaporation was 
reduced but 1.5 per cent. When temperature at base of chimney was 630°, with a 
fire i2 ins. in depth, it was decreased to 556° with one 9 ins. in depth, and to 539 0 
with one 6 ins. 

High wind increases evaporative effect of a furnace. 

Stationary or Land. —Set at an inclination downward of .5 inch in 10 feet. 

Smoke Preventing .—A test of C. Wye Williams's design of preventing smoke, at 
Newcastle, 1857, as reported by Messrs. Longridge, Armstrong, and Richardson, 
gave an increased evaporative effect with the “practical prevention of smoke.” 
Hence it was concluded, “ That by an easy method of firing, combined with a due 
admission of air in front of furnace, and a proper arrangement of grate, emission 
of smoke may be effectually prevented in ordinary marine multi-tubular boilers, 
with suitable coals. 2d. That prevention of smoke increases economic value of fuel 
and evaporative power of boiler. 3d. That coals from the Hartley district have an 
evaporative power fully equal to that of the best Welsh steam-coals.” 


Heating Surfaces. 

Marine (Sea-water). — Grate and heating surfaces should be increased 
about .07 over that for fresh water. 

Relative Value of Heating Surfaces. 

Horizontal surface above the flame = 1 I Horizontal beneath the flame.=. 1 

Vertical.= .5 | Tubes and flues.=.56 

Minimum Volumes of Fuel Consumed per Sq. Foot of 
Grate per Hour, for given Surface-ratios. (D. K. Clark.) 

Description of Surface-ratios of Heating Surface to Grate. 


Boiler. 

10 

is 

20 

30 

40 

50 

60 

75 

90 

100 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Stationary. 

•7 

i -7 

3 

6.8 

12. I 

18.9 

26 

'— 

— 

— 

Marine. 

•7 

1.6 

2.8 

6-3 

II .2 

17-5 

24 

_ 

_ 

_ 

Portable. 

.2 

•4 

.8 

1.8 

3-2 

5 

— 

_ 

_ 

Locomotive (coal). 

.3 

•7 

i -3 

2.9 

5-2 

8.1 

11.7 

18.3 

26.3 

3 2 -5 

“ (coke). 

•4 

1 

1.8 

4 

7 

II 

l6 

25 

36 

44 


At extreme consumption of fuel (120 lbs.) coke will withstand disturbing effect 
of a blast better than coal. 




























STEAM-ENGINE.-BOILER. 


74i 


A scale of sediment one sixteenth of an inch thick will effect a loss of 14.7 per 
cent, of fuel. 

One sq. foot of fire surface is held to be as effective as three of heating. 

Relation, of Grate, Heating Surface, and. Fuel. 

When Grate and Heating Surface are constant, greater the weight of fuel 
consumed per hour, greater the volume of water evaporated; but the volume 
is in a decreased proportion to fuel consumed. 

In treating of relations of grate, surface, and fuel. D. K. Clark, in his valuable 
treatise, submits, that in 1852 he investigated the question of evaporative perform¬ 
ance of locomotive-boilers, using coke; and he deduced from them, that, assuming 
a constant efficiency of fuel, or proportion of water evaporated to fuel, evaporative 
effect, or volume of water which a boiler evaporates per hour, decreases directly as 
grate-area is increased; that is to say, larger the grate, less the evaporation of water, 
at same rate of efficiency of fuel, even with same heating surface. 

2d. That evaporative effect increases directly as square of heating surface, with 
same area of grate and efficiency of fuel. 

3d. Necessary heating surface increases directly as square root of effect—viz., for 
four times effect, with same efficiency, twice heating surface only is required. 

4th. Necessary heating surface increases directly as square root of grate, with same 
efficiency; that is, for instance, if grate is enlarged to four times its first area, twice 
heating surface would be required, and would be sufficient, to evaporate same vol¬ 
ume of water per hour with same efficiency of fuel. 

Result of 40 experiments with a stationary boiler (fresh water), with an 
evaporation of 9 lbs. water per lb. of fuel consumed, the coefficient .002 22 
was deduced. 

/ h \ 2 

Hence, (— 1 .00222 =W. W representing volume of water in cube feet, and g 

and h areas of grate and heating surfaces in sq. feet. 

Illustration; —Assume a heating surface of 90 feet, and a grate of 3; what will 
be the evaporation ? 


.2 


Then 90-4- 3 X -002 22 = 1.998 cube feet. 


Note.—A Galloway stationary boiler, with a ratio of grate area of 34.3 and a con¬ 
sumption of 21.8 lbs. coal per hour, evaporated 2.9 cube feet of water per sq. foot of 
grate. Hence the coefficient in this case would be .002 466. 

To Compnte Areas of Gfrate and. Heating Surfaces, 
Volume of Water, and. Weigh, t of ITuel. 

For cb^Temperature of 281°, or Pressure of 50 lbs. per Sq. Inch. 

To Compute "Weight of Fuel. 

When Water per Sq. Foot of Grate per Hour and Surface Ratio are Given. 


—— -— F, and x R 2 = (E — C) F. 

0 

Illustration. —Assume elements as preceding. 



To Compute Ratio of Heating Surface to Area of Grate, 
and. to Effect, a Given Evaporation. 



When Water and Fuel per Sq. Foot of Grate are Given. 


W representing water evaporated per sq. foot of grate , arid F fuel consumed, both 
in lbs. per hour. C and x specific constants for each type of boiler, and R (A -F g) 
ratio of heating surface to grate. 

Illustration.— Assume W = 200, C = 10, F = 15, and x = .02. 



'200 — 10X1S 200 — .02 X so 2 , 

— 50; -—- — p 15; and 



(13.33 — 10) X 15 


— S°- 


.02 


.02 













742 


STEAM-ENGINE.-JBOILEK. 


When Efficiency of Fuel and Fuel consumed per Sq. Foot of Grate per 

w 

Hour are given. — = E or efficiency of fuel or weight of water evaporated per lb. 


offuel. 


f 


F 

(E — C) F 


:R. 


To Compute Fuel tliat may "be consumed per Sq. Foot 
of Grate per Hour, corresponding to a Given Efli- 


ciency. 

When Efficiency of Fuel , that is. Weight of Water evaporated per Lb. of 
Fuel, and the Surface Ratio, are given. 


x R 2 + C F 
F ’ 


C 



E, and 


x R 2 
K — 0 



Illustration.—A ssume elements as preceding. 


.02 x 5° 2 -f~ X 15 


. 00 V 


Combustion of Coal per sq. foot of grate .—Natural Draught , from 20 to 25 lbs. can 
be consumed per hour.— Steam-jet , 30 lbs., and Exhaust-blast 65 to 80 lbs. 

From Results of Experiments upon Marine Boilers, see Manual of D. K. Clark, 
page 808; lie deduced the following formula, as applicable to all surface ratios in 
such boilers. 


Newcastle .021 56 R 2 4-9.71 F, and for Wigan .ox R 2 -(- 10.75 F = W in lbs. 

And the general formulas he deduced from all the various experiments are as 
follows. 


Portable. 

Stationary... 


From and at 212°. 

.008 R 2 + 8.6 F = AY. Marine.016 R 2 -f ^>.25 F = W. 

.0222 R 2 -f- 9.56 F =r W. Locomotive, coal, .ooqR 2 -]- 9.7 F = W. 
Locomotive, coke. 0178 R 2 -f-7.94 F — W. 


As the maximum evaporative power of fuel is a fixed quantity, the preceding 
formulas are not fully applicable in minimum rates of its consumption and evapo¬ 
rative quality. 

With coal and coke the limits of evaporative efficiency may be taken respectively 
at 12.5 and 12 lbs. water from and at 212 0 . 

Illustration i.— Assume a marine fire-tubular boiler with a surface ratio of heat¬ 
ing surface to grate of 30 and a consumption of coal of 15 lbs. per sq. foot of grate 
per hour, what will be its evaporation per sq. foot of grate? 


.016 X 3o 2 -f- 10.25 X 15 = 168.15 tbs. 

2.—Assume a like boiler, using fresh water, to have a ratio of heating surface to 
grate of 30 and an evaporation of 165 lbs. water per sq. foot of grate per hour, what 
would be consumption of coal per sq. foot of grate per hour? 


165 — .016 X 30 2 
10.25 


14.69 lbs. 


Tube Surface (Iron) per lb. of coal 1.58, per sq. foot of grate 32, and per IIP 4.27 
sq. feet. 

Locomotive Boiler has from 60 to 90 sq. feet per foot of grate, and consumes 65 
lbs. coal per sq. foot per hour. 


Evaporative Capacity of Tubes of Varying Eengtla. 
Total Length of Tubes 12 Feet 3 ins. (M. Paul Hevrer , 1874.) 


Surface and Water. 

Furnace and 

3 ins. in Length 
of Tubes. 

* 

3.02 

Feet. 

TUI 

3.02 

Feet. 

E S. 

3-02 

Feet. 

3.°2 

Feet. 

Surface in sq. feet. 

76-43 

24-5 

179 

8.72 

179 

4.42 

X 79 

2.52 

179 

1.68 

Water evaporated per sq. 1 
foot per hour in lbs.j “ '' 
































STEAM-ENGINE.-BOILER 


743 


Besnlts of' Operation, of Boilers under Varying Propor¬ 
tions of Grate, Area, and Length of Heating Surface, 
Draught of Furnace, and Bate of Combustion. 


Description. 

Area of 
Grate. 

Heating 

Surface. 

Grate to 
Heating 
Surface. 

Coal per 
Sq. Foot 
of Grate 
per Hour. 

Evapor 
Water fi 
per sq. ft. 
of grate. 

ation of 
*om 212° 
per lb. 
of Coal. 

Fuel. 

Fire-tubular. 

Sq. Feet. 

Sq. Feet. 

Ratio. 

Lbs. 

Lbs. 

Lbs. 


Agricultural and Hoisting 

4-7 

158 

34 

13 

119 

9-33 

Welsh. 

u u u 

3-2 

220 

69 

12.8 

I 5 i 

n.83 

4 4 

Locomotive.) 

(26.25 

963-5 

36-7 

30.86 

327 

10.6 

4; 

English.| 

(16 

818 

5 i 

38 

375 

i °-47 

44 

it 

10.5 

788 

75 

45 

419 

11.04 

44 

44 

10.6 

1056 

100 

157 

1401 

10.41 

4 4 

Marine 1 . 

22 

748 

34 

24-3 

265 

10.7 

44 

u 1 

18 

749 

41.6 

23.6 

264 

11.2 

4 4 

44 2 

10.3 

9*5 

50 

41.25 

468 

n.36 

44 

U 2* 

10.3 

508 

49-3 

27.63 

309.8 

n -54 

Lanc’r 

“ 3 . 

10.8 

I 5 1 - 2 

14 

27.76 

205 

7-39 

Anth’e 

Stationary 4 . 

31-5 

945 

30 

28.87 

293-7 

10.17 

Welsh. 

4 C 2 

31*5 

767 

24.4 

14 

i 4 i *4 

10.1 

4 4 


i New Castle. 2 and 4 Wigan. 3 Experimented at New York. 

* Effect of reducing the tube-surfaces was tried by stopping one half the number of tubes in alter¬ 
nate diagonal rows, so that the tube surface was reduced 206.5 sq. feet. The results with fires 12 ins. 
deep were as follows : 

Tubes open. Tubes half closed. 

Coal per sq. foot of grate per hour. 25 lbs. 24 lbs. 

Water from 212 0 per lb. of coal. 12.41 u 12.23 u 

Smoke per hour, very light.,. 2.8 minutes. 8 minutes. 

Evaporative Effects of Boilers for Different Bates of 
Combustion, and Surface Batios. (Z>. K. Clark.) 

Water from and at 212 0 per Hour . 

Surface Batio 30 . 


Fuel per 
Sq. Foot 
of Grate 
per Hour. 

Statio 

Wa 

per 

Sq. foot. 

S’ARY. 

ter 

per lb. 
of Coal. 

Mar 

Wa 

per 

Sq. foot. 

INE. 

ter 

per lb. 
of Coal. 

Port; 

Wa 

per 

Sq. foot. 

IDLE. 

ter 

per lb. 
of Coal. 

Cot 

Wa 

per 

Sq. foot. 

Locom 

il. 

ter 

per lb. 
of Coal. 

rOTIVE. 

Co 

W« 

per 

Sq. foot. 

4 . 

ke. 
iter 
per lb. 
of Coal. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

10 

116 

11.6 

117 

11.7 

93 

9-3 

105 

10.5 

95 

9-5 

15 

163 

10.9 

168 

11.2 

136 

9 

154 

10.3 

135 

9 

20 

211 

10.6 

219 

10.9 

179 

9 

202 

10.1 

175 

8.7 

3 ° 

307 

10.2 

322 

10.7 

265 

8.8 

299 

10 

254 

8-5 


Surface 


15 

187 

12-5 

187.5 

12.5 

20 

247 

12.3 

248 

12.5 

30 

342 

11.4 

348 

11.6 

40 

438 

10.9 

450 

ir -3 

5 ° 

534 

10.7 

552 

11 


Surface 

1 




Water. 

30 

Locomotive, 

44 

coal.. 

Per sq. foot. 

Lbs. 

342 

4 4 

u lb. coal. 

11.4 

4 4 

coke . 

u sq. foot. 

338 

44 

44 

“ lb. coal. 

“•3 


Batio SO. 


149 

9.9 

OO 

VO 

H 

II .2 

164 

192 

9.6 

217 

IO.9 

203 

278 

9-3 

3 H 

IO.4 

283 

364 

9 - 1 

411 

IO.3 

362 

450 

9 

508 

IO. I 

442 


Batio To. 


Fuel per Sq. Foot of Grate per Hour in Lbs. 


40 

50 

60 

75 

90 

100 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

439 

536 

633 

778 

927 

1020 

11 

10. 7 

10. 7 

10. 4 

10.3 

10.2 

418 

497 

576 

695 

815 

894 

10.4 

9.9 

9.6 

9-3 

9 

8.9 


When a heater is used, and temperature of feed-water is raised above that ob¬ 
tained in a condensing engine, the proportions of surfaces may be correspondingly 
reduced. 














































































744 


STEAM-ENGINE.—BOILER. 


Results of Operation of various Resigns of Boiler, un¬ 
der varying Proportions of Orate, Calorimeter, Area 
and Length, of Beating Surface, Draught, Firing, and 
Rate of Combustion. 


Stationary. 

Area 

of 

Grate. 

Heat¬ 

ing 

Surface. 

Grate 

to 

Heating 

Surface. 

Circuit of 
Heating 
Surface. 

Temperature 

of Chimney. 

Coal per 
Sq. Foot 
of Grate 
per Hour. 

Water I 

from 212 0 
per lb. of 
Coal. 

Ivaporated 

per Sq. Foot 
of Grate 
per Hour. 


Sq. Ft. 

Sq. Ft. 

Ratio. 

Feet. 

O 

Lbs. 

Lbs. 

Lbs. 

Lancashire double 1 









internal and ex- > 

20.5 

612 

29.8 

79 

5 ii 

15-35 

8.32 

125 4 

ternal Sued 1 ... ) 









U U 2 

21 

767 

3 6 -5 

80 

505 

21.5 

10.88 

204 4 

Galloway vertical ) 









water-tubular 2 , j 

21 

719 

22.8 

79 

505 

22.7 

10.77 

212 4 

t & it 2 

31-5 

719 

34-3 

SO 

630 

18.3 

IO. 17 

162 4 

Fairbairn 1 . 

20.5 

1017 

49-5 

— 

387 

15-27 

8.67 

133 

French 1 . 

20. I 

607 

30-3 


5 io 

16.42 

8.12 

133 

Cylindrical flued 3 ... 

14.2 

377-5 

26.8 

56 

292 

7-43 

9.08 

59 5 

Marine. 

At Pressure of Atmosphere. 



Horizontal fire-tub. 2 

10.3 

508 

49-3 

— 

— 

27-5 

11.92 

328 7 

U U 2 

10.3 

508 

49-3 

— 

— 

41.25 

11.36 

469 8 

U U 2 

10.3 

302 

3 ° 

— 

— 

24 

12.23 

268 9 


19-3 

749 

39 

— 

— 

21 

10 

182 10 

u a 

28.5 

749 

26.3 

— 

— 

21.15 

8.94 

164 10 

u u 

28.5 

749 

26.3 

— 

— 

*9 

11.13 

335 11 

(C. Wye Williams) j 

i 5-5 

749 

48.3 

— 

600 

37-4 

10.63 

398 

(( u | 

22 

749 

34 

— 

600 

17.27 

n.7 

202 12 

U << j 

u u \ 

42 

749 

17.6 

— 

— 

16 

9- 6 5 

154 13 

.1 u 3 

10.8 

150 

* 3-9 

8-5 

— 

10.99 

8-95 

H 

OO 

00 

“ O 3 

A 3 2 

i 47 

34 

8-5 

— 

27.58 

7.24 

40 r 4 


1 Trial in France. 2 At Wigan, 1866-68, height of chimneys 100 feet. 3 Navy- 
yard, Washington, U. S., chimney 61 feet. 4 At pressure of atmosphere, fires 12 ins. 
deep, at 40 lbs. pressure, evaporation was reduced 12 per cent. 5 Bituminous coals. 
6 Anthracite, at pressure of 6.5 lbs. above atmosphere. ^ Fires 14 ins. deep, air ad¬ 
mitted through furnace - doors. 8 Ditto do., jet blast. 9 Half tubes closed up. 
10 Air through grate only. 11 Air through grate and door, no smoke. 12 One open¬ 
ing in door, temp. 625°, with two 633°, with four 638°, and with six 6oo°. *3 Long 

grates, air spaces fully open, no smoke. HOne furnace, anthracite coal, 5 ins. deep. 


Draught. 

Draught of Furnace .— Volume of gas varies directly as its absolute tem¬ 
perature, and draught is best when absolute temperature of gas in chimney 
is to that of external air as 25 to 12. 


T 4 - 461.2 0 V 

— :— f - — - — V". V V', and V" representing absolute temperatures at T 

32 0 461.2 0 V 

or temperature given, and at 32 °, in degrees and volume of furnace gas at tempera¬ 
ture T in cube feet. 


Illustration. —Assume temperature of furnace or T =z 1500 0 , and 12 lbs. air per 
lb. of fuel. 


1500 0 -f- 461.2 C 


3.98, and as 150 cube feet is volume of gas per lb. of fuel at 12 


32° -j- 461.2° 
lbs. supply of air, 150 X 3-98 = 597 cube feet. 


—— — C. W representing weight of fuel consumed in furnace per second in lbs., 
a V 

v volume of air at 32 0 supplied per lb. of fuel in cube feet, t absolute temperature of 
gas discharged ]by chimney in degrees, a area of chimney in sq.feet, and C velocity of 
current in chimney in feet per second 






























STEAM-ENGINE.-BOILER. 


745 


Illustration. —Assume W = .16, v = 150, t 
.16X150X1000 24000 


1000 °, and a — 5. 
- 9-73 feet - 


V' 


5 x 493 - 2 ° 2466 

— -084 to .087 = D. D representing weight of a cube foot of gas discharged by 
chimney, in lbs. Illustration. 


4 Q 3 . 2 w 

- - -- X .086 = .0424 lb. 

1000 


C* 
2 g 


(■+ g +£)= h - G representing a coefficient of resistance and friction of 

air through grate and fuel* f coefficient of friction of gas through flues and over 
sooty surfaces ,t l length of flues and chimney . m hydraulic mean depth,% and H height 
of chimney, alt in feet. 

Illustration i. —Assume C = 9.73, l — 60, and m — .72, all in feet. 


9-73 

^ 4-33 


(' + 


.012 X 6o\ _94-67 


•72 


X 14 = 20.6 feet. 


2.—Assume preceding elements. 


64-33 
9-73X5X493-2° 
150 X 1000 


C a V' 
v t 


= W. 


.16 lb. 


When II is given. 



Illustration.— Assume preceding elements. V20.6X 64.33 A- 14 = 9.73 feet. 

. 192 X pressure in lbs. per sq. foot = head in ins. of water. 

Temperature at base of smoke-pipe or chimney, or termination of flues or 
tubes, is estimated at 500° ; and base of chimney, or its calorimeter , should 
have an area of 1.3 to 1.6 sq. ins. for every lb. of coal consumed per hour. 
With tubes of small diameter, compared to their length, this proportion may 
be reduced to 1 and 1.2 ins. 

Admission of air behind a bridge-wall increases temperature of the gases, 
but it must be at a point where their temperature is not below 8oo°. 


Loss of Pressure by 3 Flow of Air in Pipes. 
Length 3280 Feet, or 1000 Meters. 


Velocity at 
Pij 

Feet 

per Second. 

Entrance of 
le. 

Meter 
per Second. 

4 

Diameter of Pipe in Ins . 

6 | 8 | 10 | 12 j 

Loss of Pressure in Lbs. per Sq. Inch. 

*4 

3.28 

I 

.114 

.076 

•057 

•057 

.038 

.038 

6.56 

2 

•5 

•343 

•25 

.21 

. 172 

• 153 

9.84 

3 

1.183 

.8 

•592 

•477 

•394 

•343 

13.12 

4 

2.06 

1-374 

1.03 

.84 

.687 

.6 

16.4 

5 

3-2 

2.16 

1.61 

I. 29 

I. I 

•923 

19.68 

6 

4.446 

2.964 

2.223 

1-778 

1.482 

1.28 


At Mount Cenis Tunnel, the loss of pressure from 84 lbs. per sq. inch, in a pipe 
7.625 ' ns - * n diameter and 1 mile 15 yards in length, was but 3.5 per cent. 


Artificial Draught. 

In production of draught in an ordinary marine boiler, from 20 to 33 per 
cent, of total heat of combustion of fuel is expended. 

Blast. _By experiments of D. K. Clark and others it was deduced that the vacuum 

in back connection is about .7 of blast pressure, and in the furnace from .33 to .5 
of that in back connection; that rate of evaporation varies nearly as square root of 
vacuum in back connection; that best proportions of chimney and passages thereto 
are those which enable a given draught to be produced with greatest diameter of 
blast-pipe; for the manifest reason, that the greater that diameter, the less the back¬ 
pressure due to resistance of orifice, and that these proportions are best at all rates 
of expansion and speeds. 

* Which, ill furnaces consuming from 20 to 24 lbs. coal per sq. foot of grate per hour, is assigned by 
Peclet at 12. t Estimated by same authority at .012. 

J For a square or circular flue is .25 its diameter. 

3 R 
























74 6 STEAM-ENGINE.-DRAUGHT.-SAFETY YALYES. 


Velocity of Draught. Locomotive. 36.5 v/H (T — t) — V. H representing 
height of chimney or pipe in feet, T and t temperatures of air at base and top of chim¬ 
ney, and V velocity in feet per second. 

Sectional area of tubes within ferrules.2 grate. 

“ “ of smoke-pipe.066 “ 

Area of blast-pipe (below base of smoke-pipe). • .0x5 “ 

Volume of back connection.3 feet x area of grate. 

Height of smoke-pipe 4 times its diameter. 

Steam-jet .—Rings set above base of smoke-pipe, and should equally divide 
the area; jets .06 to .1 inch in diameter, 3 ins. apart at centres. 

A Steam-jet, involving 50 per cent, increased combustion of coal, produced 
45 per cent, more evaporation at nearly same evaporation per lb. of coal. 

Fan Blowers .—See page 447. 

Comparative Result o f Experiments with a Steam-jet in a Marine Boiler , 
with Bituminous Coal. (Nicoll and Lynn, Eng.) 

Without Jet. With Jet. 

Area of grate.sq. feet. 10.3 10.3 

Coal per sq. foot of grate per hour_lbs... 27.5 41-25 

Water “ “ “ “. 293.1 4 I 9-37 

“ from 212 0 per lb. of coal “. 11.9 11-36 

Duration of smoke in an hour,) minuteB .. I . I _ 

very light.j 

Comparative Effect of Draught and. Blasts. 

By late experiments in England, with boilers of two steamers, to deter¬ 
mine relative effects of the different methods of combustion, the results were: 
Natural draught 1, Jet 1.25, and Blast 1.6. 

Flow of .Adr. ( Hawksley.) 

7 . , _. , th d Tr V 2 Z , /hd* n 

In Cylindrical Pipes. = ^ Qod = ^ 311^/—=Q, 

Vd 2 h , \^d 2 l 

-, and - - = IP. 

135 21200 000 

In Conduits of Various Sections. 796 /~ — v, v C 1 . — h 

^ V 1 1 633 000 a 

/a 3 li Vah Qh V 3 Cf . . 

706 / -= Q, —7- = — 7 j and -= IP. In which 1 inch water is 

' J \] C l 106 106 67 000 000 

taken as equivalent to a pressure of 5.2 lbs. per sq. inch for any passage. 

V representing velocity in feet per second , h head of water in ins., d diameter of 
pipe, l length, and C perim eter, all in feet, a area of section in sq.feet, Q (V a) volume 
of air discharged per second in cube feet, and IP horse-power. 

Safety Valves. 

Up to a pressure of 100 lbs. per sq. inch, area in sq. ins. equal product of 
weight of water evaporated in lbs. per hour by .006. 

Act of Congress (U. S .).— For boilers having flat or stayed surfaces, 30 ins. for 
every 500 sq. feet of effective heating surface; for cylindrical boilers, or cylindrical 
flued, 24 sq. ins. 

y Q. 

- — 

452 

diameter. G representing area of grate in sq. ins. 

Locked Safety-valves .—Effective heating surface, less than 700 sq. feet, valve 2 ins. 
in diameter; less than 1500, 3 ins. in diameter; less than 2000, 4 ins. in diameter; 
less than 2500, 5 ins. in diameter; and above 2500, 6 ins. in diameter. 

Or, (.05 G -f- .005 S) yJ-jr — area <f each of two valves. G representing sq. inch, 
per sq.foot of grate, and S sq. inch, per sq.foot of heating surface. 



















STEAM-ENGINE.-FLUES AND TUBES. 


747 


Illustration.— Assume G = 50 sq. feet, S = 1600 sq. feet, and P — 80 lbs. (m. g.). 

Then, (.05 X 50 + .005 X 1600) X V100- 4 - 80 = 2.5 -f-8 X 1.118 1= 11.73 sq. ins. 

IPipes. 

Area. .25 G + .oi G representing area of grate and S area of heat¬ 

ing surface, both in sq. feet, and P pressure per mercurial gauge in lbs. 

ct 73 ct 73 

(Copper), Thickness. Steam, . 125 -j-— ; Feed ,. 125 + ; Blow (Bottom 

and Surface), . 125 —- - ; Supply, .1 4 --^- ; Discharge, .1 + — : Feed, Suction, 
9OOO 300 2CO 

and Bilge discharge, .09 -f- ---, and Steam Blow-off, .05 + —-- d representing 

internal diam. of pipe, and p internal pressure per sq. inch in lbs. 

Flanges. — Of brass, thickness 4 times that of pipe; breadth , 2.25 times 
diam. of bolt; bolts, diam. equal to and pitch 5 times thickness of flange. 

For lower pressure or stress, pitch of bolts 6 times. 

3 JTu.es and. Tubes. 

Flues and Tubes. —Cross section, for 15 lbs. of coal consumed per hour, 
an area of from .18 to .2 area of grate, area being measurably inverse to 
diameter, and directly increased with length. Thus, in Horizontal Tubular 
Boilers , area .18 to .2 area per sq. foot of grate, and in Vertical Tubular .22 
to .25, area decreasing with their length, but not in proportion to reduction 
of temperature of the heated air, area at their termination being from .7 
to .8 that of calorimeter or area immediately at bridge-wall. 

Large flues absorb more heat than small, as both volume and intensity of heat is 
greater with equal surfaces. 

Tubes. — Surface 1 sq. foot, if brass, and 1.33, if iron, for each lb. of coal 
consumed per hour; or 20 of brass and 27 of iron for each sq. foot of grate, 
and 2.6 sq. feet of brass and 3.7 of iron per UP. 

Set in vertical rows, and spaces between them increased in width with 
number of the rows. 

Temperature of base of Chimney or Smoke-pipe, or termination of the 
flues or tubes, is estimated at 500° ; and base of chimney, or its calorimeter , 
with natural draught, should have an area of 1.33 sq. ins. for every lb. of 
coal consumed per hour. With tubes of small diameter, compared to their 
length, this proportion may be reduced to 1 and 1.2 ins. 

When combustion in a furnace is very complete, the flues and tubes may 
be shorter than when it is incomplete. 

Evaporation. 

1 sq. foot of grate surface, at a combustion of 15 lbs. coal per hour, will 
evaporate 2.3 cube feet of salt water per hour. 

A sq. foot of heating surface, at a like combustion of fuel, will evaporate 
from 5 to 6.2 lbs. of salt water per hour; and at a combustion of 40 lbs. coal 
per hour (as upon Western rivers of U. S.), from 10 to 11 lbs. fresh water, 
exclusive of that lost by being blown out from boilers. 

13.8 to 17.2 sq. feet of surface will evaporate 1 cube foot of salt water per 
hour, at a combustion of 15 lbs. coal per hour per sq. foot of grate. 

Relative evaporating powers of Iron, Brass, and Copper are as 1, 1.32, and 1.56. 

Note.—B oilers of Steamer Arctic, of N. Y., vertical tubular, having a surface of 
33.5 to 1 of grate, consuming 13 lbs. of coal per sq. foot of grate per hour, evapo¬ 
rated 8.56 lbs. of salt water per lb. of coal, including that lost by blowing out of 
saturated w r ater. 




74 § STEAM-ENGINE.-SMOKE-PIPES AND CHIMNEYS. 


Water Surface. 

At low evaporations, 3 sq. feet are required for each sq. foot of grate sur¬ 
face, and at high evaporation 4 to 5 sq. feet. 


Steam Room. 

From 15 to iS times volume that there are cube feet of steam expended 
for each single stroke of piston for 25 revolutions per minute, increasing 
directly with their number. Or, .8 cube feet per HP for a side-wheel engine, 
and .65 for an ordinary and .55 for a fast-running screw-propeller. 

Space is required proportionate to volume of steam per stroke of piston. 
Thus, with like boilers, the space may be inversely as the pressures. 

Steam-drums and steam-chimneys, by their height, add to the effect of 
their volume, by furnishing space for water that is drawn up mechanically 
by the current of steam, to gravitate before reaching the steam-pipe. 


Grate. — Area in sq. feet per lb. of coal per hour for following boilers. 
Width , 1.5 diameter of furnace: 


Cornish and Lancashire, slow 

combustion.2 sq. foot. 

Marine, tubular.&5 to .066 u “ 


Portable, moderate forced .. .03 sq. foot. 
Locomotive and like, strong 
blast.01 “ “ 


Thickness of Tubes per B W G. 

External diameter in ins. 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 

Thickness for pressure of 50 lbs., number.. 12 12 11 u n 10 10 10 9 

“ “ “ 100 “ “ ..11 10 99 98 88 7 


Smoke-pipes and. Chimneys. 

Area at their base should exceed that of extremity of flue or flues, to 
which they are connected. 

In Marine service smoke-pipe should be from .16 to .2 area of grate. In 
Locomotive, it should be .1 to .083. 

Intensity of their draught is as square root of their height. Hence, rela¬ 
tive volumes of their draught is determined by formula: 

fh . 1 a — volume in sq. feet, h representing height of pipe or chimney in feet, and 
a its area in sq. feet. 

When wood is consumed their area should be 1.6 times that of coal. 


Chimneys {Masonry ).—Diameter at their base should not be less than from 
.1 to .11 of their height. 

Batter or inclination of their external surface .35 inch to a foot, which is 
about equal to 1 brick (.5 brick each side) in 25 feet. 

Diameter of base should be determined by internal diameter at top, and 
necessary batter due to height. 

Thickness of walls should be determined by internal diameter at top; 
thus, for a diameter of 4 feet and less, thickness may be 1 brick, but for a 
diameter in excess of that 1.5 bricks. 


A rea. 


<5C 


= a. C representing weight of coal consumed per hour in lbs., and 
a area of ditto at top , in sq. ins. 


(Brick masonry .)—25 tons weight per sq. foot of brickwork in height is 
safe if laid in hydraulic mortar. 

Less the height of a smoke-pipe or chimney, the higher the temperature of 
its gases is required. 







STEAM-ENGINE.-PUMPS.-PLATES AND BOLTS. 749 


Velocities of Current of Heated Air in a Chimney 100 Feet in Height. 

In Feet per Second. . 


Air at Base of Chimney. 

45 °° 


External Air. 

150° 

250° 

350 ° 


Feet. 

Feet. 

Feet. 

IO° 

24 

30 

33 

3 2 ° 

22 

28 

3 i 

50 0 

20 

27 

3 ° 


Feet. 

35 

34 

33 


External Air. 


6 o° 

70 0 

8 o° 


Air at Base of Chimney. 


150° 

250° 

350 ° 

450 ° 

Feet. 

Feet. 

Feet. 

Feet. 


26 

29 

33 

l8 

25 

29 

32 

W 

24 

28 

32 


When Height of Chimney is less than 100 feet. —Multiply velocity as ob¬ 
tained for temperature by .1 square root of height of chimney in feet. 

Draught consequent upon a steam-jet in a smoke-pipe or chimney is 
nearly equal to that of a moderate blast. 


The most effective draught is when absolute temperature of heated air or 
gas is to that of external air as 25 to 12, or nearly equal to temperature of 
melting lead. 

In chimneys of gas retorts, ovens, and like furnaces, the draught is more 
intense for a like height of chimney than in ordinary furnaces, in con¬ 
sequence of the great mass of brick masonry, which, becoming heated, adds 
to intensity of draught. 

Chimneys. Lawrence Manufacturing Co., Mass. Octagonal. 

Height above ground 211 feet. Diameters 15, and 10 feet 1.5 ins. Wall at base 
23.5, and at top 11.5 ins. Shell at base is ins., at top 3.75 ins. 

Foundation 22 feet deep. 


England. —Square. 

U 

Round. 


Height .190 feet. Diameter at base .20 feet. 

“ .300 “ “ “ .29 “ 

££ .312 11 u “ .30 “ 

“ “ .300 “ “ “ .20 “ 

Diameter at base usually .1 of height above ground. 

Vacuum at base of chimney ranges from .375 to .43 ins. of water. 


Circulating Pumps. 

Single-acting. — .6 volume of single-acting air-pump and .32 of double¬ 
acting. 

Double-acting. — .53 volume of double-acting air-pump. 

Volume of Pump compared to Steam Cylinder or Cylinders. 


Engine. Pump. Volume. 

Expansive, 1.5 to 5 times.Single-acting.08 to .045. 

Compound. do. 045 to .035. 

Expansive, 1.5 to 5 times.Double-acting.045 to .025. 

Compound. do. 025 to .02. 

Valves. —Area such as to restrict the mean velocity of the flow to 450 feet 
per minute. 

PLATES AND BOLTS. 


Wrought-iron. —Tensile strength ranges from 45500 to 70000 lbs. 
per sq. inch for plates, and 60000 to 65000 lbs. for bolts, being increased 
when subjected to a moderate temperature. 

English plates range from 45000 to 56000 lbs., and bolts from 55000 to 
59 000 lbs. 

D. K. Clark gives best quality of Yorkshire 56000 lbs., of Staffordshire 44800 lbs. 

Test of Plates. (U. S.) — All plates to bo stamped at diagonal corners at 
about four ins. from edge, and also in or near to their centre, with name of manu¬ 
facturer, his location, and tensile stress they will bear. 

Plates subjected to a tensile stress under 45000 lbs. per sq. inch, should contract 
in area of section 12 per cent., 45000 and under 50000, 15, and 50000 and over, 25, 
at point of rupture. 

































750 


STEAM-ENGINE.-PLATES. 


Brands. (C No. i) Charcoal No. i.— Plates, will sustain a stress of 40000 lbs. per 
sq. inch; hard and unsuited for flanging or bending. 

(C No. 1 R H) Reheated , hard and durable, suited for furnaces, unsuited for con¬ 
tinued bending. 

(C H No. 1 S) Shell , will sustain a stress of 50000 to 54000 lbs. in direction of fibre, 
and 34000 to 44000 across it: hard and unsuited for flanging or even bending with 
a short radius. 

(C H No. 1 F) Flange, will sustain a stress of 50000 to 54000 lbs., soft and suited 
for flanging. 

(C H No. 1 F B) Furnace and (C H No. iFFB) Flange Furnace. The first is 
hard, but capable of being flanged, the other is hard, and suited for flanging. 

The especial brands are Sligo, Eureka, Pine, etc. 

The best English plates known are the Yorkshire, as Low Moor, Bowling, Farnley , 
Monk Bridge , Cooper (& Co., etc. (See Steam-boilers, W. H. Shock, U. S. N., 1880.) 

Steel.— Tensile strength ranges from 75000 to 96000 lbs. Mr. Ivirkaldy 
gives 85 966 lbs. as a mean. 

When used in construction of boiler-plates should be mild in quality, containing 
but about .25 to .33 per cent, of carbon; for when it contains a greater proportion, 
although of greater tensile strength, it is unsuited for boilers, from its hardness and 
consequent shortness in its resistance to bending. 

Crucible steel may be used, but that obtained by the Bessemer or Siemens-Martin 
process is best adapted for boiler-plates. Its strength becomes impaired by the 
processes of punching and shearing, rendering it proper thereafter to submit it to 
annealing. 

Steel rivets, when of a very mild character and uniformly heated to a bright red, 
are superior to iron in their resistance to concussion and stress. 

Copper.— Tensile strength is 33000 lbs., being reduced -when subjected 
to a temperature exceeding 120 0 . At 212 0 being 32 000, and at 550° 25 000 lbs. 

AVrcmglit-irori Shell Plates. 

Pressure and. Thiclviiess. (U. S. Law.) 

Based upon a Standard of One Sixth of Tensile Strength of Plates. Iron or Steel. 

Results with a Tensile Strength of 50000 Lbs. 

Diameters in Ins. 


ness. 

36 

38 

40 

42 

44 

46 

48 

54 

60 

66 

72 

78 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•25 

Il6 

no 

104 

99 

95 

9 1 

87 

77 

69 

63 

58 

53 

•3125 

145 

137 

130 

124 

Il8 

113 

IO9 

90 

87 

79 

72 

67 

•375 

174 

165 

156 

149 

142 

136 

130 

Il6 

104 

95 

87 

80 

•5 

232 

220 

208 

198 

190 

182 

174 

154 

138 

126 

Il6 

106 


84 

90 

96 

102 

108 

114 

120 

126 

132 

135 

140 

144 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

74 

69 

65 

6l 

58 

55 

52 

49 

47 

46 

44 

43 

•4375 

86 

80 

76 

7 i 

68 

64 

6l 

57 

55 

53 

5 i 

50 

•5 

99 

92 

87 

81 

77 

73 

69 

65 

63 

6l 

59 

57 

.5625 

III 

103 

98 

9 i 

87 

82 

78 

73 

7 i 

69 

67 

64 

•75 

148 

I 3 8 

130 

121 

115 

IO9 

103 

97 

94 

9 1 

88 

85 

.875 

172 

160 

152 

142 

136 

128 

122 

114 

no 

106 

102 

IOO 


198 

184 

W 4 

162 

i 54 

146 

138 

130 

126 

122 

Il8 

114 


To which 20 per cent, is to be added for double riveting and drilled holes. 


Iron plates .375 inch in thickness will bear, with stay bolts at 4, 5, and 6 ins. 
apart from centres, respectively 170, 150, and 120 lbs. per sq. inch. 

Iron plates, as tested by Mr. Phillips at Plymouth Dockyard, .4375 inch in thick¬ 
ness, with screw stay bolts 1.375 ins. in diameter riveted over heads, 15.75 and 15.25 
ins. from centi - e:=: 240 sq. ins. of surface for each bolt; bulged between bolts and 
drew from bolts at a pressure of 105 lbs. per sq. inch of plate. 

Iron plates .5 inch in thickness, under like conditions with preceding case, bulged 
and drew from bolts at a pressure of 140 lbs. per sq. inch of plate. Hence, it ap¬ 
pears, resistances of plates are as squares of their thickness. 

When nuts were applied to ends of bolt through .4375 inch plate, its resistance in¬ 
creased to 165 lbs. per sq. inch of plate. 












































STEAM-ENGINE.-PLATES. 


75i 


To Compute Pressure for a, (Given. Thickness and 
Diameter, or Thickness Tor a Given Pressure and 
Diameter. 

For Pressure. Rule.— Multiply thickness of plate in ins. by one sixth 
of tensile strength of metal, and divide product by radius or half diameter 
of shell in ins. 

When rivet-holes are drilled, and longitudinal courses are double riveted, 
add one fifth to result as above attained. 

Example. — Assume boiler 8 feet in diam., and plates .5 inch thick; what work¬ 
ing pressure will it sustain, tensile strength of plates equal to a stress of 60000 lbs.? 

w 2 - : - r-Tr .8X12 5000 , 

. 5 X 60 000 — one sixth —- zzz - - == 104.16 lbs. 

2 48 


For Thickness. Rule. —Multiply pressure by radius of shell, and divide 
product by one sixth of tensile strength of metal. 

Example.— Assume pressure, radius, and tensile strength as preceding. 


104.16 X 96 A- 2 


; — 5 inch. 

10 000 


: t. t representing thickness of metal in six- 


60000 A- one sixth 

For Evaporation of Salt Water. —Add one sixth to thickness of plates and sec¬ 
tional area of stay bolts. 

For Freight Steamboats on Mississippi River and its Tributaries. 

Standard of 150 lbs. pressure for a boiler 42 ins. in diameter and plates 
.25 inch thick. 

For Pressure. Rule.—M ultiply thickness of plate by 12600, and divide 
result by radius of boiler in ins. 

Example. — A ssume a boiler 42 ins. in diameter, and plates .25 inch in thickness; 
what working pressure will it sustain ? 

.25 x 12 600 A- 42 A- 2 = 150 lbs. 

All boilers by U. S. Law to be tested to a pressure of 50 per cent, above that of 
their working pressure. 

To Compote Thickness of Tlates for a Griven Pressure. 

teentlis of an inch , p pitch of stays or distance apart at centres in ins. , P working 
pressure in lbs. per sq. inch , and F a constant, as follows : 

Cylindrical Shells. (Board of Trade , England.) Coefficient or 
Factor of Safety. — When shells are of best material and workmanship, 
rivet-holes drilled when plates are in place, abut strapped, plates at least 
.625 inch in thickness and double riveted, with rivets computed at a re¬ 
sistance not to exceed 75 per cent, over the single shear,* the coefficient is 
taken at 5. 

For all practicable deficiencies in drilling, punching, and riveting in trans¬ 
verse courses, if existing, this coefficient is increased up to 6.75, and in lon¬ 
gitudinal courses to 8.75, and when courses are not properly broken, an 
addition is made to above of .4. 

Tensile strengths of plates are taken, with fibre 47000 lbs. per sq. inch, 
across it 40000 lbs., and when used for superheaters at from 22400 to 
30000 lbs. 

47 oooB<2 , DPC . _ .. .. . , „ 

——— - = P and -— — — t. P representing pressure that shell will sus- 

D C 47 000 B 2 

tain per sq. inch in lbs., B least per cent, of strength of rivet or plate at lap, D diam. 
of shell and t thickness of plate, both in ins., and C coefficient of safety. 


Shearing or detrusive resistance of wrought iron is from 70 to 80 per cent, of its tensile strength. 











752 


STEAM-ENGINE.—STAY BOLTS. 


Diameter of rivets should not be less than thickness of plates. 

Note. —Solid plate = p X t. 

Illustration. —Assume T —. 50000 lbs. tensile strength of plate, B — 75 per cent , 
D = 120 tins., C ±= 5, and t =s 15. What pressure will shell sustain, and what should 
be thickness of plates for such pressure and diameter? 

50000 X-75 X .5 X 2 120X62.5X5 • • j. 

3 -—- 62.5 lbs ., and --—— = . 5 inch. 

120X5 50000 X -75 X 2 

"Wrouglit-iron Stay Bolts, 
j For a Tensile Strength of Metal of 50 000 Lbs. per Sq. Inch. 

Screwed Stay Bolts and Riveted. — Plates up to .4375 inch in thickness 
F = 95, above that 105. 

Screwed Stay Bolts and Nuts. —Plates up to .4375 inch in thickness F = 
117, above that 128. 

Resistance of a flat surface decreases in a higher ratio than space between 
stays. Hence, F must be decreased in proportion to increase of pitch above 
that of ordinary boiler-plates. 

Illustration. — Assume thickness of metal 5 sixteenths inch thick, stay holts 
screwed and riveted over its threads, and working pressure steam 80 lbs. per sq. inch. 


■ 95 - 


Then 


f 


5 2 X 9 5 
80 


= 5.45 ins. pitch. 


Screwed. 


To Compute Diameter of* Stay Bolts. 

Socket. tVJ. 

70 95 


— d. d representing diameter in ins. 


Illustration.— Assume pitch or distance apart of stay bolts 6 ins., and working 
pressure 100 lbs. per sq. inch; what should be diameters of bolts, both screw and 
socket ? 

(fXj/100 _ i nc Ji screwed , and ^ * ^' /r °° = . 7 -f- inch soclcet. 

72 85 

To Compute Distance Apart of Stay Bolts and Press¬ 
ure tlaey will Sustain. 

d 72 


V** 

.833 x 72 


5200 d 2 , 7350 d 2 _ 

p, -— —- = P, and —= P. 

p 2 ’ p 2 


Illustration.— Assume elements 
of preceding case. 


_ 6 ins. - ' = 100 lbs. screwed , and -— _ IOO n )S _ socket. 

-yioo 6 2 6 2 

Note.— Where stays are secured by keys, their ends should be 1.25 times diameter 
of stay, depth of slot 1.6 diameter of stay, and width .3. 

Proportions of Eyes of Staj^s, Rods, etc. 


Dimensions. 


No. 1. a and a — i inch. 

6 = .9 “ 

c - .75 “ 

No. 2. a and a — i “ 
b = .6 “ 

*= -75 “ 

No. 3. a and a — i “ 
6 = .75 “ 

c = .875 “ 


No. 1. No. 2. 

Forged and Welded. 








No. 3. 

Drilled from Bar. 



When drilled from upset bar, dimensions same as for No. 1. 

.66 neck of rod. 

Oblique Stays. —Stress upon an oblique stay is equal to stress which a 
perpendicular stay supporting a like surface would sustain, divided by co¬ 
sine of angle which it forms with perpendicular to surface to be supported. 































STEAM-ENGINE.-PLATES AND BOLTS.-RIVETING. 753 


Illustration.—A ssume pressure no lbs. per sq. inch, area of supported surface 
36 sq. ins., and angle of stay 45 0 ; what would be pressure or stress upon stay ? 

Cosine 45 0 =. 707 11. Then 110 X 36 -f-. 707 11 = 5600 lbs. 

From Experiments of Mr. Fairbairn. 

.75 in. iron bolts screwed into .375 in. iron plates and riveted drew at 28760 lbs. 

.75 in. “ “ “ .375 in. copper “ “ “ 24160 “ 

.75 in. “ “ “ .375 in. “ aud not riveted “ 18260 

Bolts were upset at their ends, to bring diameter at base of threads equal 
to diameter of bolt. Hence, stay-bolts when screwed and riveted are .33 
stronger than when screwed alone. 

Tensile or retaining strength of an iron screw-bolt .75 inch in diameter at 
base of thread , screwed and riveted into an iron plate .375 inch in thickness 
= 12654 lbs. per sq. inch of bolt, also measured at base of thread , or about 
.25 tensile strength of plates. 

Blades and. Bolts. (Molesworth.) 

^ = C, — — P, and = t. d representing diameter in ins. 

21 a 2 C 

Single riveted. Double riveted. 

Best Yorkshire plates. C = 6200 and 7800 

“ Staffordshire plates. “ — 5000 “ 6200 

Ordinary plates. “ = 3000 “ 3700 

Working stress not to exceed .2 tensile strength of joint or riveted plate, 
and .125 is sometimes used. 

Hence, taking C for mean of best plates as above when single riveted at 5600. 

Then for a pressure of no lbs., a thickness of .25 inch, and a diameter of 42 ins., 
as given for a standard U. S. boiler. 
cZ 1 x o ^ 42 

-= C =-— = 0240. Taking C as above for best single-riveted plate at 

2 f 2 X .25 

no X 42 , . • , 

6200, -— = . 472 4 - ins. m thickness. 

’ 2 X 6200 

RIVETING. 


Diameter, Length, and Bitch of Rivets, and Lap ofPlate. 
Deduced essentially from Experiments of Mr. Fairbairn. 


Plate. 

Rivet. 

Diameter. j Leng 

th.* 

Pitch. f 

L 

Single Riveted. 

AP. 

Double Riveted. 

In 16th. 

Ins. 

Mult’r. 

Ins. 

Mult’r. 

Ins. 

Mult’r. 

Ins. 

Mult’r. 

Ins. 

Mult’r. 

3 

•38 

2 

.88 

4-5 

1.25 

6 

1.25 

6 

2.1 

IO 

4 

•5 

2 

I - 1 3 

4-5 

i-5 

6 

i-5 

6 

2-5 

IO 

5 

•63 

2 

*• 38 

4-5 

1.63 

5 

1.88 

6 

3-i5 

IO 

6 

•75 

2 

1.63 

4-5 

l -75 

5 

2 

5-5 

3-33 

9.2 

8 

.81 

i-5 

2.25 

4-5 

2 

4 

2.25 

4-5 

3-75 

7-5 

IO 

•94 

i-5 

2-75 

4-5 

2-5 

4 

2-75 

4-5 

4-58 

7-5 

12 

113 

i-5 

3-25 

4-5 

3 

4 

3-25 

4-5 

5'4 2 

7-5 

14 

1.25 

I. 42 

4 

4-5 

3-5 

4 

3-5 

4 

5-8 

6-75 

16 

i-37 

i-37 

4-5 

4-5 

4 

4 

3-75 

3-75 

6.25 

6.25 


* From inside of head. + From centre to centre. 

Shearing strength of a rivet of best iron = 40 000 c l~. 

Multipliers are for computing Diameter, Length, and Pitch for Distance between 
centres of rivets; also for Laps for Single and Double Joints, by multiplying thick¬ 
ness of plate by Multiplier for element required. 

In Riveted Joints exposed to a tensile stress, area of rivets should be equal 
to areas of section of plates through line of rivets, running a little in excess 
up to .5625 inch, and somewhat less beyond that diameter of rivet, the area 
being determined by the relative shearing and tensile resistances of rivet 
and plate. 


























754 


STEAM-ENGINE.-EIVETING. 


Strength of Riveted Joints per Sg. Inch of Single Plate. 

Single Lapped. — Machine riveted. Pitch = 3 diameters, 25 000 lbs. — 
Hand riveted, 24 000 lbs. 

Rivets “ Staggered,” and equidistant from centres, 30500 lbs. 

Abut Joints. —Hand riveted. Rivets equidistant from centres, single cover 
or strip, 30 000 lbs. 

Rivets “Square,” single strips, 42000 lbs.; double strips, 55000 lbs. 

Relative Mean Strength of Riveted Joints compared to that of Plates. 
Allowances being made for Imperfections of Rivets, etc. 

Plates , 100; Triple, .72 to .75; Double or “Square,” .68 to .72; Double 
with double abut straps, .7 to .75 ; Staggered, .65 ; Single, .56 to .6. 

Cylindrical Shells and Rivets. (Lloyd's Rules, Eng.) 


t JC 


P D 


= «, 


p — d 


: x, and 


n a 


z. t representing thickness of plate 


D ’ C J P P t 

and D diameter of shell, p pitch and d diameter of rivets, all in ins. ; J per cent, of 
strength of joint, the least to be taken ; C a constant as per table ; P working press¬ 
ure in lbs. per sq. inch ; n number and a area of rivets ; x per cent, of strength of 
plate at joint compared with solid plate, and z per cent, of strength of rivets com¬ 
pared with solid plate. 

When plates are drilled, take .9 of z, and when rivets are in double shear, 
put 1.75 a for a. 

Constants. 


Joint. 

Ir 

.5 inch 
and 
under. 

on Plate 
. 75 inch 
and 
under. 

s. 

Above 
.75 inch. 

.375 inch 
and 
under. 

Steel I 
•5625 

inch and 
under. 

’lates. 

.75 inch 
and 
under. 

Above 
.75inch. 

T (punched holes. 

155 

170 

170 

l80 

165 

180 

170 

190 

190 

200 

200 

215 

230 

240 

Lap (drilled do. 


Double abut j punched holes 
strap (drilled do. 

l80 

190 

215 

230 

250 

260 


When plates, as in steam-chimneys, superheaters, etc., are exposed to direct ac¬ 
tion of the flame, these constants are to be reduced .33. 

Illustrations.— Assume pitch 4 ins., diam. of rivet 1.375 ins., and thickness of 
plate 1 inch, both single and double riveted. Area 1.375 = 1.48 sq. ins. 

4 — 1-375 


1X1.48 

4X1 


.656 per cent, strength of joint compared to solid plate. 

1.75 X 1.48 


.37 per cent, strength of rivet to solid plate, and 


cent, strength of rivet to solid plate when double riveted. 
C t 2 


4X1 


.647 per 


IPlates. 


p“ 


/C t 2 /P p 2 

= P, /—- — p, and / -- - — t. t representing thickness of 


plate in sixteenths of an inch. 

Illustration.— Assume pressure 80 lbs., pitch of rivets 3 ins., and 0=190. 


Then 


80 X 3* 
90 


= 2.83: 


3 sixteenths. 


C = 90 for plates up to .4375 inch, stay-bolts screwed and riveted heads. 

100 “ above .4375 inch, “ “ 

no “ up to .4375 inch, “ “ and nut heads. 

120 “ above .4375 inch, “ “ “ 

140 “ — — “ “ double nut heads. 

When stay-bolts are not exposed to corrosion, these constants may be reduced .2. 

Stay-loolts.— Iron, are not to be subjected to a greater stress than 
6000 lbs. per sq. inch of section ; Steel, 8000 lbs., both areas computed from 
weakest part of rod, and when of steel they are not to be welded. 


























STEAM-ENGINE.-RIVETING AND GIRDERS. 


755 


Essentially by Formulas of Nelson Foley, London , 1881. 
Single Lap Riveting. 

d I 27 b' 

:p, p tb =z a, and — t 


— -- — b /or plate. — — b' for rivets, 

P pt 


d. p representing pitch and d diameter 


i — b ’ i — b 

of rivets in ins., a sectional area of rivets in sq. ins., and b and b' least per cent, of 
strength of course at lap of plates, or of rivet's section to solid plate, i. e. plate before 
being punched = B, or plate and rivets in ultimate tension. 

Illustration.— Assume p = 3 ins ., d = 1 inch, a — .7854 inch, and < =. 5 inch. 


3 : .66 strength of lap, - — .523 rivet to solid plate, and 


3 ' " 3 .X .5 1 —.66 

Ritclies as Determined by Diameter of Rivets. 


= 3 ins. 


Distance 
between 
Edges 
of Holes. 

Pitch 

= Diam. of 
Rivet X 

Distance 
between 
Edges 
of Holes. 

Pitch 

= Diam. of 
Rivet X 

Distance 
between 
Edges 
of Holes. 

Pitch 

= Diam. of 
Rivet X 

Distance 
between 
Edges 
of Holes. 

Pitch 

= Diam. of 
Rivet X 

Per Cent. 


Per Cent. 


Per Cent. 


Per Cent. 


50 

2 

58 

2.38 

65 

2.86 

72 

3-57 

52 

2.08 

60 

2-5 

68 

3 -i 3 

75 

4 

55 

2.22 

62 

2.63 

70 

3'33 

78 

4-55 


Operation. —If distance between edges of holes, or p — d, =65 per cent, of solid 
plate, and diam. of rivet 1 inch, then 2.86X1 = 2.86 ins. pitch. 

When Plate and Rivets are of equal, strength in ultimate tension, V — b, = B. 

I. 27 13 

Hence, --— t = d. In illustration of B, assume p = 3, d = 1.1, and t — .5. 


Then 3 — x.i = 1.9, and — = .633 = b, or per cent, of strength of punched to 


solid plate. Area 1.1 = .95, and 


•95 


3 X .5 


: .633 = b', or per cent, of section of rivet to 


solid plate. Hence, B — .633. 

Illustration. —Assume as shown, B = .633- 
1.27 x .633 

Then — -— - — - X - 5 = 1-095 or 1.1 ms. diam. 

1 —-633 

When shearing strength of metal of rivets varies from that of tensile strength of 
1 27 b' T 

plate. — / t — d. T and S representing ultimate strength of plate and rivets, 

1 — b S 

which may be taken at 5 and 4 for iron, and 7 and 6 for steel. 

Illustration. —Assume as in preceding case and for iron. 

1.27 X -633 X 5 


Then 


1 — -633 X 4 


X .5 = 1.37 ins. diam. 


Girders. 


C d? t 


„ P (L p) D L_ 4 

’ C d 2 


P (L— p) D L 


= d. L 


(L — p) D L ’ Cd 2 ’ V C t 

representing length of girder, d its depth, t its thickness at centre or sum of its.thick¬ 
nesses, D their distance apart from centre to centre, and p pitch of stays, all in ins., 
and C a constant as per following : 

One stay to each girder , C = 6000. If two or three = 9000. Four stays to each 
girder —10 200. 

Illustration. —Assume triple stayed or bolted girders, 6 ins. apart, 24 ins. in 
length, 3 ins. in depth, 1 inch thick, and stayed at intervals of 6 ins.; what work¬ 
ing pressure will it sustain? 

qooo X 6 2 X x 324000 

— - 77^— - = £ — 1 -—- = 125 lbs. 

(24 — 6) X 6 X 24 2592 


C = 9000. Then 
































STEAM-ENGINE.-RIVETING. 



. 89 600 t 2 

Flues or ^Arclied. or Circular Furnaces.* — :—=r — = r, 

u 


t. ^- = D, auc| ^ = L. D representing external diameter 

i L x JL) 


/P L D 

\/ 89 600 

of flue or furnace, and t thickness of plate, both in ins., L length of flue or furnace 
between its ends or between its rings, in feet, and P working pressure in lbs. per sq. 
inch. 

Illustration.— Assume diameter of flue 16 ins., length 6 feet, and working press¬ 
ure of steam 80 lbs. per sq. inch. 


Then 


V 


80 X 6 X 16 


-^.0857 — .29 inch. Furnace .—P not to exceed 


8000 t 


89 600 D 

Illustration.— Assume diameter of a circular furnace or width of a semicircular 
one 48 ins., working pressure of steam 80 lbs., and length 6 feet. 


/80 X 6 X 48 . . , 

Then V~ 89 600 ~ ^' 257 = ' 5 °7 mch - 


Diameter of Rivets as Determined by 'TTiickness of 

X > late. 


B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

Per Cent. 

T = S. 

.9 per cent, 
of Section 

Per Cent. 

T = S. 

.9 per cent, 
of Section 

Per Cent. 

T = S. 

.9 per cent, 
ot Section 



of Rivet. 



of Rivet. 



of Rivet. 

52 

1.38 

i -53 

55 

1.56 

i -73 

58 

1.76 

i -95 

S 3 

1.44 

i -59 

56 

1.62 

1.8 

60 

i. 9 r 

2. 12 

54 

i -5 

1.66 

57 

1.69 

1.87 

62 

2.08 

2.31 


Operation.— If thickness of plate = .5 inch and plate and rivet have equal resist¬ 
ance, or B —62 per cent., then .5 x 2.08 = 1.04 ins. diameter. 


Note i. —When full resistance of rivet section for drilled holes is not computed as 


by Lloyd’s Rules and 90 per cent, is taken, then above formulas are extended by 
2.—Rivets should be of less diameter than thickness of the plate. 


100 
90 ' 


Double Lap Riveting. All the preceding formulas for single lap 


riveting apply to this, with substitution of 2 a for a ancl .64 for 1.27 


Illustration.— Assume as preceding, p = 3 ins., t 
3 X -5 X -58.9 — . 44 i8 area of d. 

1 —-75 


X-5 = -75 d. 


.5 inch, and — .589 
3 —-75 


.75 b, and 


.4418 x 2 


3 X -5 


=. 589 b'. 


Diameter of Rivets as Determined Toy Thickness of 

I 3 ! ate. 


B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 



.9 per cent. 

Per Cent. 


.9 per cent, 
of Section 
of Rivet. 



.9 per cent. 

Per Cent. 

T = S. 

ot Section 
of Rivet. 

T — S. 

Per Cent. 

T = S. 

of Section 
of Rivet. 

68 

i -35 

1-5 

7 i 

1.56 

I> 73 

74 

M 

00 

M 

2 

69 

1.42 

i -57 

72 

1.64 

1.82 

75 

1.91 

2. 12 

70 

1.48 

1.65 

73 

1.72 

1.91 

76 




Operation. —Assume t = .5 inch and B = 70 per cent., tensile strength compared 
to shearing being as 7 to 6. What should be diameter of the rivets? 


7 » _ 

.5 x 1.48 X -g = .863 inch. When rivets are in double shear, put 1.9 a for a. 


* The U. S. law for a furnace crown or flue for a like diameter requires a thickness of .875 inch_viz., 

.3125 inch for each 16 ins. of diameter. English iron, being harder than American, is better constructed 
to resist compression, and consequently a less thickness of metal is required for like stress. 












































STEAM-ENGINE.-RIVETING. 


757 


Triple Lap Riveting. Preceding formulas for single lap riveting 
apply to this, with substitution of 3 a for a and .42 for 1.27. 


Illustration.—A ssume as preceding, p — 3 ins., t .5 inch, and b' = .883. 

3X.5X.883 .42 X-883 . 3 — .75 t 

- = .4417 area of d, — - -f X-5 = -74 m. diam., -- — — .75b, 


and 


• 44 l8 X 3 


1 —-75 


.883 V. 


3 X -5 

Diameter of Rivets as Determined by Thickness of 


Blate. 


B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

B 

or Strength 
at Joint. 

Diam. = Thickness 
of Plate X 

Per Cent. 

T = S. 

.9 per cent 
of Section 

Per Cent. 

T = S. 

.9per cent, 
of Section 

* 

Per Cent. 

T = S. 

.9 per cent, 
of Section 

70 

•99 

of Rivet. 

1.1 

73 

I-I 5 

of Rivet. 
1.27 

76 

i -34 

of Rivet. 
1.49 

7 1 

I.O4 

115 

74 

I.21 

i -34 

77 

1.42 

I. 5 8 

72 

I.O9 

1.21 

75 

I.27 

1.41 

78 




Operation.—A s shown by preceding tables. 


Greneral Formulas and Illustrations. 


Rivets in Single Shear. 
Rivets in Double Shear. 
Rivets in Triple Shear. 


1.27 B T 


1 (1 — B) S 
1.27 B T 
1.75 (1 — B) S 
1.27 B T ^ 
2.5 (1 — B) S 


t = d. and 


a S 


p t T 




t 


. , a 1,75 S 

d, and - = b . 

’ p t T 

, , a 2.5 S 

d, and —— — =zb. 

’ p t T 


Zigzag Riveting. Strength of plate between holes diagonally is 
equal to that horizontally between holes, when diagonal pitch = .6 and hor¬ 
izontal = diameter of rivet + .4. 

Thus, .6 p -J- .4 p — diagonal pitch. 


Duty of Steam-engines. 

The conventional duty of an engine is the number of lbs. raised by it 1 
foot in height by a bushel of bituminous coal (1x2 lbs.). 

Cornish Engine. —Average duty, 70 000 000 lbs.; the highest duty ranging 
from 47000000 to 101 900000 lbs. 

A condensing marine engine, working with steam at .75 lbs. (mercurial 
gauge), cut off at .5 stroke, will require from 1.75 to 2 lbs. bituminous coal 
per IP per hour. 

Relative Cost of Steam-engines for Equal Effects. 

In Lbs. of Coal per IP per Hour. .. 


A theoretically perfect engine.......66 

A Cornish condensing engine.2.38 

A marine condensing engine.1.75 to 3 


Evaporative Power of Boilers. 

The Evaporative power of a boiler, in lbs. of water per lb. of fuel consumed, 
is ascertained approximately by formula 

1.833 (- —-— A e = lbs. S representing total heating surface in sq. feet, F fuel 

\2 S T I / 

consumed in lbs. per hour, and e theoretical evaporative power of the fuel. 

Illustration. — Assume evaporative power of the fuel at 15, consumption per 
hour 800 lbs., and heating surface 1600. 

Tb0 “ ‘' 833 ( .fooxAfeo ) X 15 = ,0 '« 8 m - 

3 S 































753 


STEAM-ENGINE.—WEIGHTS. 


/ l6oO \ 

Efficiency of idler. ,.833 ( l6oo x „ + 8 J = - 733 - 

The evaporative power of different fuels, from and at 212 0 , is, for coals, from 14.5 
to 16.8 lbs., the average of Newcastle being 15.3, for patent fuels 15.66, Lignite 13.5, 
Coke 13.3, Peat 10.3, and Woods, when dry, 8.1. See A. E. Seaton, London , 1883. 

ISTotes 011 Horse-power. 

A Lancashire boiler with a heating surface of 610 sq. feet and a grate-area of 25 
will evaporate in ordinary operation 50 cube feet of water per hour; 3.12 sq. feet of 
horizontal section per cube foot of water, and .5 sq. foot of grate-area per cube foot. 

Nominal. Flue Boilers .—Usually computed at 5.5 to 6 sq. feet of horizontal 
section, 15 sq. feet of heating surface, and 1 sq. foot of grate-area. 

The IIP of such boilers will range from 3 to 4 times that of the nominal. 

Multitubular Boilers .—.75 sq. foot of grate-area and 2.5 of heating surface. 


“Weights of Steam-engines. 
SicLe-wLieels .—American Marine ( Condensing). 


Engine. 

Frame. 

Water¬ 

wheels. 

Cylinders. 

No. | Volume. 

Weight per 
Cube Foot. 

Service 

Vertical beam. 

Wood.* 

Wood 

I 

Cube Feet. 

63 

216 

430 

Lbs. 

1100 

1040! 

River 

ii 

Wood.* 

Wood. 


Coast. 

« 

Wood. * 

Wood. 


Coast. 

Coast. 

a 

Wood.* 

Wood. 


1480+ 

10S9I' 

850 

55 °§ 

IIOO 

a 

Wood. * 

Iron. 



Sea. 

Sea. 

Oscillating. 

Iron. 

I ron. 

2 

/ 

ii 

Iron. 

Iron. 


1502 

535 

Sea. 

Sea. 

Inclined. 

Iron. 

Iron. 

2 


* Without frame. f With frame nog. J Including boilers. § Single frame. 


Screw ^Propellers .—American Marine ( Condensing ). 


Engine. 


Vertical direct, Jet Condcns’g. 
“ “ Surface Cond’g 

“ “ Jet 

It 1 C It 1 C 

11 ci cc ii 

Horizontal back-action. 

“ direct.... 

Vertical compound.. 


f. 

J & 


direct 




, = & 

ci cc o 

.[ u 

“ Non-Condensing. 

u cc cc 


Cyli 

No. 

nders. 

Volume. 

Engine. 

Weights. 

Boilers. 

Per C. Ft. 
Cylinder. 

Ser¬ 

vice, 


CubeFeet. 

Lbs. 

Lbs. 

Lbs. 


I 

4 

22 O4O 

12 IOO 

8 535 

Sea. 

I 

12.5 

59000 

32 OOO 

7 280 

Sea. 

I 

12.5 

48130 

35000 

6650 

Sea. 

I 

33 

120 45O 

98 000 

6 620 

Coast, 

4 

506 

x 523 060 

q8s 600 

4 958 

Sea. 

2 

68 

289 680 

200 800 

7 212 

Sea. 

2 

67 

201 OOO 

200 593 

6 009 

Sea. 

2 

4.8 

24 705 

26372 

10641 

Coast, 

2 

24-3 

94196 

88 050 

7500 

Sea. 

2 

425 

I 022 4OO 

840 000 

4380 

Sea. 

I 

3-6 

30534 

27301 

16 066 

Coast, 

I 

35 

172 028 

100065 

7 774 

Sea. 

I 

x .86 

14 41O 

22481 

i 9 8 34 

River. 

I 

2.77 

*4 759 

22 417 

13421 

Coast. 


English Marine ( Condensing ). 


Cylinders. 


Description. 

No. 

Volume. 

Trunk. 

2 

Cube Ft. 
230 

Horizontal direct. 

2 

382 

Vertical direct. 

2 

393 

Oscillating. ■.. 

2 

440 

Vertical compound. 

2 

24 

it U 

6 

707 

Horizontal compound... 

2 

52 

ii ii 

2 

143 


Weights. 


Engines. 

Propeller 

and 

Shafting. 

Boilers 

and 

Water. 

Total. 

Per 

IH 3 

Tons. 

Tons. 

Tons. 

Tons. 

Lbs. 

121 

47 

257 

425 

465 

223 

85 

303 

6ll 

338 

165 

48 

144 

357 

781 

ZI 7 

43 

135 

295 

560 

4-25 

•75 

7-25 

12.25 

60 

497 

167 

656 

1320 

368 

55 

15 

no 

180 

351 

130 

27 

162 

3i9 

3 C 9 


Per 

Cube Ft. 
Cylinder. 


Tons. 

1.85 

1.6 

•9 

•7 

•52 

1.87 

3-44 

2.2.3 

































































STEAM-ENGINE.—•WEIGHT OF BOILERS. 


759 


Land-engines.— (Non-condensing.) 


Engine. 

Volume 

of 

Cyl’r. 

Engine. 

Spur-wheel 

and 

Connections. 

Sugar-Mill 

Complete. 

Boilers, 
Grates, etc. 

Engine per 
Cube Foot 
of Cylinder. 



Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Vertical) 18 ins. X4 feet 

7 

67 200 

37 800 

89 600 

26 880 

9600 

beam f 30 ins. X5 feet 

24-5 

105 000 

137179 

265879 

75000 

4290 

Horizon’l, 14 ins. X 2 feet 

2.2 

10914 

— 

— 

8 200 

5100 

“ 22 ins. X 4 feet 

10.6 

56 000 

— 

— 

30140 

5600 


To Compute Weigh, t of a, Vertical Beam and. Side—wheel 
Jet Condensing Engine. (T. F. Rowland, A.S.C.E.) 

Including cdl Metals , Boiler and A itachments , Smoke-pipe, Grates , Iron Floors , 
and Iron in Wooden Water-wheels, omitting Coal-bunkers. 

For a Pressure per Mercurial Gauge of 40 lbs. per Sq. Inch. 

For surface condenser add 10 to 15 per cent. 

Rule.—M ultiply volume of cylinder in cube feet by Coefficient in follow¬ 
ing- table corresponding to length of stroke, and product will give rough 
weight in lbs. For finished weight deduct 6 per cent. 


Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Feet. 

5 

2467 

Feet. 

7 

2213 

Feet. 

9 

1865 

Feet. 

11 

1619 

6 

2340 

8 

2000 

IO 

I 73 ° 

12 

1546 


Example i. —What are the rough and finished weights of a vertical beam engine, 
cylinder 80 ins. in diameter and 12 feet stroke of piston ? 

Area of 80 ins. = 5026.56, which x 12 feet = 419 cube feet , and X 1546 for 12 feet 
stroke = 647 774 lbs. rough weight. 

Then 647 774 X • 06 — 38 866, and 647 774 — 38 866 = 608 908 lbs. finished weight. 


WEIGHTS OF BOILERS. 


Weights of Iron Boilers (including Doors and Plates, and exclusive of Smoke- 
pipes and Grates) per Sq. Foot of Heating Surface. 

Surface Measured from Grates to Base of Smoke-pipe or Top of Steam Chimney. 


Boiler. For a Working Pressure of 40 Lbs. 


Single return, Flue*.water bottom 

U It tl . . 

“ “ “ Multi-flue*.water bottom 


Horizontal return, 

u it 

ft u 

Vertical “ 

Horizontal direct, Tubular*. 

u tt tt * 


Tubular f.water bottom 

“ t. — 

u * . . 

“ t.water bottom 


Weight.. 


Lbs. 

25.6 to 32.9 

24 to 30 
27 to 45 

25 to 43 

22.5 to 35 
21 to 33 

17.7 to 26.7 

18.5 to 26.5 

19.8 to 23.8 
17 to 21 

23.5 to 24 
18.1 to 18.6 
16.3 to 17.3 
24 to 26 


Cylindrical, external furnace,$ 36 ins. in diam., .25 inch thick.. 

“ Flue “ +361042 “ .25 “ “ 

Horizontal direct, Tubular.Locomotive. 

Vertical Cylinder direct, Tubular. — 

Weight of Cylindrical Furnace and Shell Boilers, all complete for Sea Service and 
for a pressure of 60 lbs. steam, 200 lbs. per IIP. 

tt Section of furnace square. Shell cylindrical. f Section of furnace and shell square. 

t Wrought-iron heads, .375 inch thick, flues, .25 iuch, and surface computed to half diameter of shell. 

Notes. —1. The range in the units of weight arises from peculiarities of construc¬ 
tion, consequent upon proportionate number of furnaces, thicknesses of metal, vol¬ 
ume of shell compared with heating surface, character of staying, etc. 

2. If pressure is increased the above units must be proportionately increased. 



















































y 60 STEAM-ENGINE.-BOILER-POWER, COMBUSTION. 


33 oiler-power. 


The power of a boiler is the volume or weight of steam alone (indepen¬ 
dent of any water that it may hold in suspension) that it will generate at its 
operating pressure in a unit of time. 

Marine boilers of the ordinary type and proportions, with natural draught, burn¬ 
ing anthracite coal, produce 3.5 to 5.5 IIP per sq. foot of grate per hour; with a 
free burning or a semi-bituminous coal, 5 to 7.5 IIP; and with a forced draught, 
with 25 to 30 lbs. best coal per sq. foot of grate per hour, 8 to 10 IIP. 

Marine engines, operating with a steam-pressure of 35 lbs. (m. g.), and with mod¬ 
erate expansion, consume 30 lbs. steam per IIP per hour, and with a high rate of 
expansion, under a pressure of 70 lbs., 20 lbs. steam. 

With a blast draught and consuming 30 to 40 lbs. of a fair quality of coal per sq. 
foot of grate per hour, 7 to 10 IP per hour can be attained. 

In locomotive boilers, having from 50 to 90 sq. feet of heating surface per sq. foot 
of grate, and at a rate of combustion of from 45 to 125 lbs. of coke, an average evap¬ 
oration of 9 lbs. of water per lb. of coke has been attained at ordinary temperatures 
and pressure. 


To Compute Volume of Air and Gas in a Furnace. 

When Volume at a Given Temperature is known. Rule.—M ultiply given 
volume by its absolute temperature, and divide product by the given abso¬ 
lute temperature. 

Note.—A bsolute temperature is obtained by adding 461° to given or acquired 
temperature. 

Example.—A ssume volume of air entering a furnace at 1 cube foot, its tempera¬ 
ture 6o°, and temperature of furnace 1623°; what would be the increase of volume? 


1 X 1623° -j- 461° 
6o° -j- 461° 


2084 

-= 4 times. 

521 


Volume of Furnace Gas per Tfb. of Coal. (Rankine.) 


Tempera¬ 

ture. 

12 Lbs. 

Air Supplied 
r 3 Lbs. 

24 Lbs. 

Tempera¬ 

ture. 

12 Lbs. 

Air Supplied 
18 Lbs. 

24 Lbs. 

32 0 

150 

225 

300 

752 ° 

369 

553 

738 

68 

l6l 

241 

322 

1112 

479 

718 

957 

104 

172 

258 

344 

1472 

588 

882 

1176 

212 

205 

307 

409 

1832 

697 

1046 

1 395 

57 2 

314 

47 1 

628 

2500 

906 

1357 

1812 


Temperature of ordinary boiler furnaces ranges from 1500 0 to 2500 0 . 

The opening of a furnace door to clean the Are involves a loss of from 4 to 7 per 
cent, of fuel. 


For other illustrations, see ante, page 744-6. 

IRate of Combustion. 

The rate of combustion in a furnace is computed by the lbs. of fuel consumed per 
sq. foot of grate per hour. 

In general practice the rate for a natural draught is, for anthracite coal from 7 to 
16 lbs., for bituminous, from 10 to 25 lbs., and with artificial or forced draught, as by 
a blower, exhaust-blast, or steam-jet, the rate may be increased from 30 to 120 lbs. 

The dimensions or size of coal must be reduced and the depth of the fire increased 
directly, as the intensity of the draught is increased. 

Temperature of gases at base of chimney or,pipe should be 6 oo°, and frictional 
resistance of surface of chimney is as square of velocity of current of gases. 

Ordinarily from 20 to 32 per cent, of total heat of combustion is expended in the 
production of the chimney draught in a marine boiler, to which is to be added the 
losses by incomplete combustion of the gaseous portion of the fuel and the dilution 
of the gases by an excess of air, making a total of fully 60 per cent. (Steam-boilers , 
Wm. H. Mock, U. S. N., 1881.) v ’ 


















STRENGTH OF MATERIALS.-ELASTICITY. 76 1 


STRENGTH OF MATERIALS. 

Strength of a material is measured by its resistance to alteration of 
form, when subjected to stress and to rupture, which is designated as 
Crushing, Detrusive, Tensile, Torsion, and Transverse, although trans¬ 
verse is a combination of tensile and crushing, and detrusive is a form 
of torsion at short lengths of application. 

ELASTICITY AND STRENGTH. 

Strength of a material is resistance which a body opposes to a per¬ 
manent separation of its parts, and is measured by its resistance to 
alteration of form, or to stress. 

Cohesion is force with which component parts of a rigid body adhere to 
each other. 

Elasticity is resistance which a body opposes to a change of form. 

Elasticity and Strength, according to manner in which a force is exerted 
upon a body, are distinguished as Crushing Strength, or Resistance to Com¬ 
pression ; Detrusive Strength, or Resistance to Shearing; Tensile Strength , 
or Absolute Resistance; Torsional Strength , or Resistance to Torsion; and 
Transverse Strength , or Resistance to Flexure. 

Limit of Stiffness is flexure, and limit of Resistance is fracture. 

Neutral Axis , or Line of Equilibrium , is the line at which extension ter¬ 
minates and compression begins. 

Resilience , or toughness of bodies, is strength and flexibility combined; 
hence, any material or body which bears greatest load, and bends most at 
time of fracture, is toughest. 

Stiffest bar or beam that can be cut out of a cylinder is that of which 
depth is to breadth as square root of 3 to i; strongest, as square root of 2 to 
1; and most resilient, that which has breadth and depth equal. 

Stress expresses condition of a material when it is loaded, or extended in 
excess of its elastic limit. 

General law regarding deflection is, that it increases, cceteris paribus, di¬ 
rectly as cube of length of beam, bar, etc., and inversely as breadth and cube 
of depth. 

Resistance of Flexure of a body at its cross-section is very nearly .9 of its 
tensile resistance. 

Coefficient of* Elasticity. 

Elasticity of any material subjected to a tensile or compressive force, 
within its limits, is measured by a fraction of the length, per unit of force 
per unit of sectional area, termed a constant, and coefficient of elasticity is 
usually defined as the weight which would stretch a perfectly elastic bar of 
uniform section to double its length. 

Unit of force and area is usually taken at one lb. per sq. inch. E represent¬ 
ing denominator of fraction. 

Example.— If a bar of iron is extended one 12000000th part of its length per lb. 
of stress per sq. inch of section, z T 

12000000 E 

The bar would, therefore, be stretched to double its normal length by a force of 
12000000 lbs. per sq. inch, if the material were perfectly elastic. 

3 S* 



7 62 


STRENGTH OF MATERIALS.-ELASTICITY. 


The same method of expressing coefficient of elasticity is applied to re¬ 
sistance to compression. That is, coefficient, in weight, is expressed by de¬ 
nominator of fraction of its length by which a bar is compressed per unit of 
weight per sq. inch of section. 

Ultimate extension of cast iron is 500th part of its length. 

Extension of Oast-iron Bars,when suspended Vertically. 
1 Inch Square and 10 Feet in Length. Weight applied at one End. 


Weight. 

Extension. 

Set. 

Weight. 

Extension. 

Set. 

Weight. 

Extension. 

Set. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

529 

.0044 

— 

2117 

.OI9O 

.coo 059 

8468 

.0871 

.00855 

1058 

.0092 

.000015 

4234 

•°397 

.002 65 

14 820 

. 1829 

•02555 


Woods. —MM. Chevaudier and Wertheim deduced that there was no 
limit of elasticity in woods, there being a permanent set for every extension. 
They, however, adopted a set of .00005 of length as limit of elasticity. 
This is empirical. 

MODULUS OF ELASTICITY. 


Modulus or Coefficient of Elasticity of any material is measure of its 
elastic reaction or force, and is height of a column of the material, 
pressing on its base, which is to the weight causing a certain degree of 
compression as length of material is to the diminution of its length. 

It is computed by this analogy : As extension or diminution of length 
of any given material is to its length in inches, so is the force that pro¬ 
duced that extension or diminution to the modulus of its elasticity. 

p 

Or, x : P :: l: w = —. x representing length a substance 1 inch square and 1 foot 

•C 

in length icould be extended or diminished by force P, and w iveight of modulus in lbs. 

To Compute Weight t of Modulus of Elasticity. 

Rule.— As extension or compression of length of any material 1 inch 
square, is to its length, so is the weight that produced that extension or com¬ 
pression, to modulus of elasticity in lbs. 

Example. —Tf a bar of cast iron, 1 inch square and 10 feet in length, is extended 
.008 inch, with a weight of 1000 lbs., what is the weight of its modulus of elasticity? 

.008 : 120 (10 X 12) 1000 : 15000000 lbs. 

To Compute IVEodvilns of Elasticity. 

When a Bar or Beam is Supported at Both Ends and Loaded in Centre. 
Rui.k.—M ultiply weight or stress per sq. inch in lbs. by length of material 
in ins., and divide product by modulus of weight. 


Or, 


l W 
M 


E; 


iw 


— M; 


E M 


= W. 


I representing length in ins., M modulus, 


W weight in lbs. per sq. inch, and E compression or extension. 

Example i. —If a wrought-iron rod, 60 feet in length and .2 inch in diameter, is 
subjected to a stress of 150 lbs., what will it be extended? 

Modulus of elasticity of iron wire is 28 230 500 lbs. (see following table), and area 
of it. 2 2 x .7854 = .314 16. 


UQ 

.314 16 


— 477.46 lbs. per sq. inch, and 60 X 12 = 720 ins. 


Then 477.46 X 


720 


343771-2 


; .0x2 18 inch. 


28 230 500 28 230 500 

-Take elements of preceding case under rule for weight of modulus. 


120 X 1000 
15000000 


= .008 inch. 


.008 X 15 000000 


= 1000 lbs. 























STRENGTH OF MATERIALS.-COHESION. 763 


HVIoduilms of Elasticity and 'Weight of Various ^Materials. 


Substances. 

Height. 

Weight. 

Substances. 

Height. 

Weight. 

Ash. 

Feet. 

Lbs. 

1 656 670 

1 345 OOO 

8 464 000 

Larch . 

Feet. 

A A 1 q OOO 

Lba. 

1 074 ooo 
720 ooo 

1 080 400 

Eeeeh. 

4 600 000 

2 460 000 

Lead,cast . 

146 ooo 

1 850 ooo 

Brass, yellow. 

Lignum-vitse. 

“ wire. 

Copper, cast. 

Elm. 

4 112 OOO 

4 800 000 

5 680 OOO 

14 632 720 

18 240 ooo 

1 499 500 

2 016 ooo 

Limestone. 

Mahogany. 

Marble, white. 

2 4OO OOO 

6 570 ooo 

2 150 OOO 

4750 ooo 

8 700 ooo 

8 070 ooo 

3 300 ooo 

2 071 OOO 

2 508 OOO 

1 710 ooo 

2 430 OOO 

1 830 OOO 

26 650000 

Fir red . 

8 330 000 

4 440 OOO 

2 790 OOO 

5 OOO OOO 

Oak. 

Glass. 

Gun-metal. 

5 550 ooo 

8 844 300 
170 OOO 

Pine, pitch. 

“ white. 

Hempen fibres.... 

Steel, cast. 

8 530 ooo 

Ice. 


2 37O OOO 

I? 968 500 

“ wire. 

9 ooo ooo 

1 672 ooo 

28 689 ooo 

I 718 800 

Iron, cast. 

5 750 OOO 

Stone, Portland ... 

“ wrought. 

7 550000 

25 820 ooo 

Tin. cast. 

1 053 ooo 

3 510000 

“ wire. 

8 377 000 

28 230 500 

Zinc. 

4 480 ooo 

13 440 ooo 


Weight a ^Material Avill "bear per Sep Inch, 'witliou.t 
Permanent Alteration of its Length. 


Material. 

Lbs. 

Material. j Lbs. 

Material. 

Lbs. 

Metals. 

Brass. 

6 700 
10000 

15 OOO 

17 800 

1 500 
45000 

Stones , etc. 
Marble. 

4900 

2000 

1500 

Woods. 

Beech . 

2360 

3240 

4290 

2060 

3000 

3960 

Gun-metal. 

Limestone*. 

Elm. 

Trnn cast. 

Portland. 

Fir, l ed. 

wrought... 
Lead. 

Woods. 

Larch. 

Mahogany. 

Steel. 

Ash. 

3540 

Oak. 


* Tensile strength 2800. 


Comparative Resilience of "Woods. 


Ash. 


Chestnut.... 

•• -73 

Larch. 

. .84 

Spruce. 


Beech.... 

... .86 

Elm. 

•• -54 

Oak. 

. .63 

Teak. 

• -59 

Cedar.... 


Fir. 

.. .4 

Pitch Pine... 

- 57 ^ 

Yellow Pine.. 

.64 


MODULUS OF COHESION. 

To Compute Length of a Prism of a Material which would 
he Severed L>y its own "Weight when Suspended. 

Rule. —Divide tensile resistance of material per sq. inch by weight of a 
foot of it in length, and quotient will give length in feet. 

Illustration. —Assume tensile resistance of a wrouglit-iron rod to he 60000 lbs. 
per sq. inch. Weight of 1 foot = 3.4 lbs. 

Then 60 000 -f- 3.4 = 17 647.06 feet. 

Length in Feet required to Tear Asunder the following Substances: 

Rawhide.15 375 feet. | Hemp twine... 75000 feet. | Catgut.25000 feet. 

Elasticity of Ivory as compared with Glass is as .95 to 1. 

When Height is given. Rule. —Multiply weight of i foot in length and 
1 inch square of material by height of its modulus in feet, and product will 
give weight. 

To Compute Height of Modulus of Plasticity. 

Rule. —Divide weight of modulus of elasticity of material by weight of 
1 foot of it, and quotient will give height in feet. 

Example. — Take elements of preceding case (page 762), weight of 1 foot being 
3 lbs.; what is height of its modulus of elasticity? 

15 000 000 -7-3 = 5 000 000 feet 



















































































764 STRENGTH OF MATERIALS.-CRUSHING. 


From a series of elaborate experiments by Mr. E. Hodgkinson, for the 
Railway Structure Commission of England, lie deduced following formulas 
for extension and compression of Cast Iron: 


c 

Extension: 13934040 —-290743200 — — W. 


Compression: 12 931 560 


c 

- -522979200 



e and c representing extension 


and compression , and l length in ins. 

Illustration.—W liat weight will extend a bar of cast iron, 4 ins. square and 10 
feet in length, to extent of .2 inch? 


.2 ,2 ^ 

13 934 040 x - -290 743 200 — - = 23 223. 4 — 8076.2 rrxs 147.2, which X 4 ins. 

120 120 


= 60 588.8 lbs. 


CRUSHING STRENGTH. 


Crushing Strength of any body is in proportion to area of its section, 
and inversely as its height. 

In tapered columns, it is determined by the least diameter. 

When height of a column is not 5 times its side or diameter, crushing 
strength is at its maximum. 

Cast Iron .—Experiments upon bars give a mean crushing strength of 
100000 lbs. per sq. inch of section, and 5000 lbs. per sq. inch as just sufficient 
to overcome elasticity of metal; and when height exceeds 3 times diameter, 
the iron yields by flexure. When it is 10 times, it is reduced as 1 to 1.75; 
■when it is 15 times, as 1 to 2; when it is 20 times, as 1 to 3; when it is 30 
times, as 1 to 4; and when it is 40 times, as 1 to 6. 

Experiments of Mr. Hodgkinson have determined that an increase of 
strength of about one eighth of destructive weight is obtained by enlarging 
diameter of a column in its middle. 


In columns of same thickness, strength is inversely proportional to the 
i - 6 3 power of length nearly. 

A hollow column, having a greater diameter at one end than the other, 
has not any additional strength over that of an uniform cylinder. 

Wrought Iron .—Experiments give a mean crushing stress of 47000 lbs. 
per sq. inch, and it will yield to any extent with 27000 lbs. per sq. inch, 
while cast iron will bear 80000 lbs. to produce same effect. 

Effects— A wrought bar will bear a compression of of its length, with¬ 
out its utility being destroyed. 

With cast iron, a pressure beyond 27000 lbs. per sq. inch is of little, if 
any, use in practice. 

Glass and hard Stones have a crushing strength from 7 to 9 times greater 
than tensile; hence an approximate value of their crushing strength may be 
obtained from their tensile, and contrariwise. 

Various experiments show that the capacity of stones, etc., to resist effects 
of freezing is a fair exponent of that to resist compression. 

Seasoning .—Seasoned woods have nearly twice crushing strength of un¬ 
seasoned. 


JElastic Limit compared to Crushing Resistance. 


Wrought-iron Commerce 

Bessemer steel. 

Cast steel. 


•545 

.615 

•473 


Cast steel. 

Fagersta steel 


.692 












STRENGTH OF MATERIALS.-CRUSHING. 765 


Crushing Strength of various Materials, deduced from 
Experiments of JVIaj. Wade, Hodgkin son, Capt. NEeigs, 
TJ• S. Stevens Institute, and L>y Gr. 1 , j . Yose. 


Reduced to a Uniform Measure of One Sq. Inch. 


Cast Iron. 


Figures and Material. 

Crushing 

Weight. 


Lbs. 

Gun-metal, American. 

174 803 

It U t 

85 000 

. 1 

125 000 

“ mean. 

IOO 000 

Low Moor, No. 1, English. 

62450 

“ No. 2, “ . 

92 330 

Clyde, No. 3, “ . 

106 039 


Figures and Material. 


Clyde, average, English. 

Stirling, mean of all, English .. 
“ extreme, English. 

Extreme, English.j 

Average (Hodgkinson), English 
Blaenavou No. 2. 


Crushing 

Weight. 

Lbs. 

82 000 
122 39;, 
134 400 
53 7 6 0 
153 200 
84 24O 
I09 7OO 


Wrought Iron. 


American, extreme.j 

127 720 

83 500 

English. j 

“ mean. 

47040 

1 ‘ average. 


65 200 
40 000 
37 8 5 o 


Various Metals. 


Aluminium bronze, 95 cop. 

Fine brass. 

Cast copper. 

Steel, cast. 

“ Fagersta. 


129 920 

Steel. Bessemer. 

164 800 

“ “ soft. 

117 000 

“ tempered. 

105 000 

“ Siemens. 

250 000 

Tin, cast. 

154500 

Lead. 


50 000 
66 200 
335000 

15500 
7 730 


Elastic Crushing Strength of Wrought Iron and Crucible Steel is equal to its ten¬ 
sile, of Bessemer Steel, 50 per cent, of its transverse strength. 


Woods. 


Ash. 

Beech. 

Birch. 

Box. 

Cedar, red. 

“ seasoned .. 

Chestnut. 

Elm. 

“ seasoned. 

“ English.... . 
Hickory, white.... 

Larch. 

Locust. 

Mahogany, Spanish 


6 663 
6963 

3300 

7 900 
10513 

5968 
6 500 
5 350 
6831 

IO OOO 
IO 3OO 

8925 
3 200 
5 500 

9 xl 3 

8 198 


Maple. 

Oak, American white. 
“ Canadian white. 
“ “ live.. 

“ English. 

“ Dantzic, dry.... 

Pine, pitch. 

“ white. 

“ yellow. 

“ Deal, Christiana 

Spruce, white. 

Teak. 

Walnut. 

Willow, seasoned. 


8 100 
10 000 

7 500 
5982 
6850 

9 5 oo 
6 484 
7700 
8947 

5 775 

8 200 
5850 

5 950 

12 IOO 

6645 

6 000 


Crosswise of Fibre. 

Oak. 2300 | Larch.. 1300 | Pines. 550 

Increase in Strength of Cubes of Sandstone, pa' Sq. Inch (under Blochs 
of Wood), as Area of Surface is increased. ( Gen'l Gillmore, U. S. A.) 

Inches. 


Stone. 

•5 

I 

i -5 

2 

2.25 

2-75 

3 

4 

Yellow Berea sandstone .. 
Blue “ “ 

Lbs. 

6080 

Lbs. 

6990 

9500 

Lbs. 

8 226 
10730 

Lbs. 

8 955 
12 000 

Lbs. 

9 x 3 ° 
12 500 

Lbs. 
9838 
13 200 

Lbs. 

IOI25 

Lbs. 

11 720 






























































































STRENGTH OF MATERIALS.—CRUSHING, 


Stones, Cements, etc. (Per Sq. Inch.) 


Figures and Material. 


Basalt, Scotch. 
“ Welsh. 


Beton, N. Y. S. Concreting Co. | 


Brick, pressed. 

“ Gloucester, Mass. 
“ hard burned. 


common. 


u 

c; 


“ yellow-faced burned, Eng. 

“ Stourbridge fire-clay, u 

“ Staffordshire blue, “ 

“ stock, English. 

“ Fareham, English. 

“ red, English. 

“ Sydney, N. S. 

Caen, France. 

Cement, Hydraulic, pure, Eng. j 

“ Portland, sand i. 

“ “ sand 3. 

“ “ 3 mos. 

“ “ i sand, 3 mos. 

“ “ 9 mos. 

i sand, 9 mos.... 
12 inch cubes. 

12 mos. 

i sand and gravel 

u ^ “ “ 

.Roman. 

“ pure, Eng... 

Rosendale. 

Sheppey, Eng. 

gravel 3.... 

Freestone, Belleville, N. J_ 

“ Connecticut. 

“ Dorchester, Mass.. 

“ Little Falls, N. Y.. 

Glass, crown. 

Gneiss. 

Granite, Aberdeen 
“ Cornish, 

“ Dublin, 

“ Newry, 


Concrete, lime 


Eng. 


Crushing 

Weight. 


Lbs. 

8 300 

16 800 
800 

1 400 
6 222 

10 219 
14 216* 
3630 
800 

4 000 
1440 
1650 
7 200 
2250 

5 boo 
808 

2 228 
I 543 

17 000 
32 000 

1 280 
600 

3 800 

2 464 
598° 

2330 

2650 

1 800 
342 
75o 
3270 
1 280 
460 
775 

3 522 

3 319 
3 o6g 
2991 
31 000 
19 600 
10 760 

6 339 
10450 
12 850 


Figures and Material. 


Granite, Patapsco, Md.. 
“ Portland, Eng. 

“ Quincy, Mass.. 
Greenstone, Irish. 

Li mestone ... 


compact, Eng. 
Magnesian," . 
Anglesea “ . 
Irish “ . 


Marble, Baltimore, Md 

u 


East Chester, N. Y.f_ 

Hastings, N. Y. 

Irish. 

Italian. 

“ white. 

Lee, Mass. 

Montgomery Co., Pa.... 

Statuary . 

Stockbridge, Mass. t.... 

Symington, large. 

“ fine crystal. 

“ strata horizontal 


Masonry, brick, common... 

“ ' “ in cement. 

Mortar, good. 


lime and sand. 

“ " “ beaten.. 

“ common. 

Oolite, Portland. 

Pottery-pipe, Chelsea. 

Sandstone, Aquia Creek §. 

“ Arbroath, Eng. 

" Connecticut. 

“ Craigleth, Eng. 

“ Derby grit “ . 

“ Holyh’d quartz, Ens; 

“ Seneca ||. 

“ Yorkshire, Eng. 


Slate, Irish. 

Terra Cotta. 

Whinstone, Scotch. 


Crushing 

Weight. 


Lbs. 

5 340 
4 570 

15 583 

15 000 
18 800 

4 000 
9 000 
7 800 
3 130 
3 600 

14 000 
8057 
18 061 

C 9 W 
18 941 
17440 
12 624 
9630 

22 702 
8950 

3 3 6 ° 
10382 
11156 
18 248 
10 124 
500 
800 
760 
240 
460 

595 

120 

3850 

12 000 

5 340 
7850 
3 no 
5825 
3136 

25 54 ° 
10 762 
57 io 

13 890 

23 744 
5000 
8300 


* Tested by author at Stevens’ Institute, N. J. f Post-office, Wash. 
§ Capitol, Treasury Department, and Patent Office, Washington, D. C. 


% City Hall, New York. 
|j Smithsonian Institute. 


Safe Load of Hollow, Cylindrical, and Solid Columns, 
Arches, Cliords, etc., of Cast Iron. 

IIolloio Columns. Per Sq. Inch. (F. W. Shields , M. I. C. E.) 


Length. 

Thick¬ 

ness. 

Load. 

Length. 

Thick¬ 

ness. 

Load. 

Length. 

Thick¬ 

ness. 

Load. 

Length. 

Thick¬ 

ness. 

Load. 

20 to 24 
diam’s. 

Inch. 

•375 

•5 

Lbs. 

2800 

33 6 ° 

20 to 24 
diam’s. 

Inch. 

.625 

•75 

Lbs. 

3920 

4480 

25 to 30 
diam’s. 

Inch. 

•375 

•5 

Lbs. 

2240 

2800 

25 to -io 
diam's. 

Inch. 

.625 

•75 

Lbs. 

336 o 

3920 


Solid Columns, etc.—3360 lbs. per sq. inch. [Brunei.) 
.A.relies.— 5600 lbs. per sq. inch. 






























































































STRENGTH OF MATERIALS.-CRUSHING. 


76; 


Chords and IPosts.—1 inch diameter and not more than 15 diameters in 
length .2 of breaking weight of metal. 

.625 inch diameter and not more than 25 diameters in length .5 of breaking weight 
of metal, and when more than 25 diameters in length from .1 to .025 of breaking 
weight of metal. [Baltimore Bridge Co.) 


Wrought-iron Cylinder's and Rectangular Tubes. 




External 



Internal 

Thickness. 

Area. 

Crushing Weight 

Length. 

Diameter. 



Diameter. 

per Sq. Inch. 

Cylinders. 

Ins. 



Ins. 

Ins. 

Sq. Ins. 

Lbs. 

IO 

feet 

1-495 



1.292 

. I 

•444 

14 661 

IO 


2.49 



2.275 

. 107 

.804 

29 779 

10 

U 

6.366 



6.106 

•13 

2-547 

35 886 

Rectangular Tubes. 








10 feet 


4.1 

X 

4.1 

•03 

•504 

10 980 

5 “ 


4.1 

X 

4.1 

•03 

•504 

11 5 H 

10 “ 

'd 

4.1 

X 

4 - 1 

.06 

1.02 

19 261 

10 “ 

0 

4-25 

X 

4-25 

•134 

2-395 

21 585 

7-5 “ 

I-* 

4-25 

X 

4-25 

•134 

2-395 

23 202 

10 “ 

- ^ 

05 

8.4 

X 

4-25 

(. 26 

I . 126 

6.89 

29 981 

10 “ 


8.1 

X 

8.1 

.06 

2.07 

13276 

7.66 “ 


8.1 

X 

8.1 

.06 

2.07 

13 3 °° 

10 ‘ ‘ 

) internal 

8.1 

X 

8.1 

.0637 

3 - 55 i 

19 732 

5 “ J 

j diapbrag’s 

8.1 

X 

8.1 

.0637 

3 - 55 i 

23 208 


Strength, per Sep Inch of* 2-Incli Cubes under Blochs 
of Wood. (Gerdl Gilimore, U. S. A.) 

• Surfaces Worked to a Clear Bed. 


Granite. Lbs - 

Staten Island blue. 22 250 

Maine. 15000 

Quincy, dark. 17 750 

“ light. 14750 

Westchester Co., N. Y. 18250 

Millstone Point, Conn. 16 187 

New London, Conn. 12500 

Richmond, Ya. 21 250 

“ “ gray. 14 100 

Cape Ann, Mass.{ *^23 

Westerly, R. T., gray. 14 937 

Fall River, Mass., gray. 15937 

Garrisons, Hudson River, gray.. 13 370 

Duluth, Minn., dark. 1775° 

Keene, N. H., bluish gray. 12875 

Used in Central Park, N. Y., red 17 500 

Jersey City, N. J., soap. 20750 

Passaic Co., “ gray. 24040 


Limestone. 

Glen’s Falls, N. Y. H 475 

Lake Champlain, N. Y. 25000 

Canajoharie, N. Y. 20700 

Kingston, “ . 139 00 

Garrisons, “ . 18500 

Marblehead, 0 ., white. 12600 

Joliet, Ill., white. 16900 


Lime Island, Mich., drab- j 25000 

Sturgeon Bay, Wis., bluish drab 21 500 


Limestone. 

Bardstown, Ky., dark. 

Cooper Co., Mo., dark drab. 

Erie Co., N. Y., blue. 

Caen, France. 

Marble. 

East Chester, N. Y. 

Italian, common. 

Dorset, Vt. 

Mill Creek, Ill., drab. 

North Bay, Wis., drab. 

Sandstone. 

Little Falls, N. Y., brown. 

Belleville, N. J., gray. 

Middletown, Conn., brown. 

Haverstraw, N. Y., red. 

Medina, N. Y., pink. 

Berea, 0 ., drab. j 

Vermillion, 0 .. drab. 

Fond du Lac, Wis., purple. 

Marquette, Mich., “ . 

Seneca, 0 ., red brown. 

Cleveland, 0 ., olive green. 

Albion, N. Y., brown. 

Kasota, Minn., pink. 

Fontenac, Minn., light buff.- 

Craigleth, Edinburgh. 

Dorchester, N. B., freestone.... 

Massillon, 0 ., yellow drab. 

Warrensburg, Mo., bluish drab. 


Lbs. 

16 250 
6650 
12 250 
3650 


13 504 
13 062 
7 612 
9 687 
20 025 


9850 

11 700 
6950 
4 350 

17 725 
7250 

IO 250 

8 850 

6 250 

7 450 

9 687 
6 800 

13500 
10 700 
6250 

12 000 

9 I 5 ° 
8750 

5000 































































STRENGTH OF MATERIALS.-CRUSHING, 



To Compute Cruisliiiig 'Weiglit of Oolnxxixis. 
Deduced by Mr. L. D. B. Gordon from Results of Experiments of various Authors. 

Cast Iron. (Hodgkinson.) 

36 a 


Round Solid or Hollow. 


Rectangular Solid or Holloiv. 


: W. For rectangular put 500. 

1 

400 

— 3 - 6 ~ _ W. For L, T, U, etc., put 19 a . 

7.2 


i + — 

500 


1 + 


900 


Round Solid. 


~WI’oxigHt Iron 
16 a 


W. 


(, Stoney.) 
Rectangular Solid. 


16 a 


■ = W. 


2400 


x + - 


3000 


Steel. (Baker.) 


Round Solid. —Strong steel, - 5I a ■ — W; mild steel, — 

r* , r 

1 -\ - 1 

900 

Rectangular Solid .—Strong steel, — 


W. 


IV: mild steel, 


1400, 
30 a 


= W. 


1 + 


1600 2480 

a representing area of metal in sq. ins., r ratio of length to least external diameter 
or side, and W crushing weight in tons. 

Illustration.—W hat is the crushing weight‘of a hollow cylindrical column of 
cast iron 10 ins. in diameter, 24 feet in length, and 1 inch in thickness? 

Area of 10 ins. ==78.54. ^-^- = 28.8, and 28.8 2 == 829.44. Area of 10 ins.= 


78.54 — area 1X2 ins. = 50.26. Then, 


80640 X 28.28 2280499.2 


829.44 


1 -f- 2.07 


— 74 2 ^33-6 lbs. 


400 


"Weight "borne with Safety "by Solid Cast-iron Columns. 
In 1000 Lbs.—(New Jersey Steel and Iron Co.) 


engtli. 

Feet. 

2 

IllS. 

3 

Ids. 

4 

Ins. 

5 

Ins. 

6 

Ins. 

7 

Ins. 

Dia 

8 

Ins. 

METER 

9 

Ins. 

IO 

Ins. 

11 

Ins. 

12 

Ins. 

13 

Ills. 

14 

Ins. 

15 

Ins. 

5 

I2.4 

44 

102 

184 

288 

414 

560 

728 

916 

1126 

1354 

— 

— 

— 

6 

9.4 

36 

88 

164 

264 

386 

532 

698 

884 

1082 

1320 

157° 

— 

— 

7 

7.2 

30 

76 

146 

242 

V360 

502 

660 

850 

1056 

1282 

1530 

1798 

2086 

8 

— 

24 

66 

130 

218 

332 

470 

630 

812 

1016 

1240 

i486 

1754 

2040 

9 

— 

20 

56 

1x4 

198 

306 

440 

59 6 

774 

974 

1196 

1440 

1706 

x 99 2 

IO 

— 

18 

48 

102 

180 

282 

410 

560 

739 

932 

1152 

X392 

1656 

iq 4 o 

12 

— 

— 

33 

80 

136 

238 

354 

494 

658 

846 

1056 

1292 

1550 

1828 

14 

— 

— 

28 

64 

122 

200 

304 

432 

586 

774 

966 

1192 

144° 

1712 

l6 

— 

— 

— 

52 

IOO 

170 

262 

373 

520 

686 

878 

1094 

1332 

i59 6 

18 

— 

— 

— 

44 

84 

144 

226 

332 

462 

616 

796 

IOOO 

1228 

1482 

20 

— 

— 

— 

— 

72 

124 

196 

292 

410 

552 

720 

912 

1130 

1372 


IVor Tubes or Hollow Columns. 

Subtract weight that may be borne by a column, of diameter of internal 
diameter of tube from external diameter, and remainder will give weight 
that may be borne. Thickness of metal should not be less than one twelfth 
diameter of column. 

Illustration. —Required the safe load of a solid cast-iron column 6 ins. in diam¬ 
eter and 20 feet in length. 

Under 6 and in a line with 20 is 72, which X 1000 = 72 000 lbs. 

Note.—T his is about one sixth of destructive weight. 










































STRENGTH OF MATERIALS.-CRUSHING. 


769 


Safe Loads as determined by Preceding Formulas. 

Cast Iron , one fifth to one sixth. Wrought Iron , one fourth to one fifth. 
Woods, one seventh to one tenth. 

WOODS. 

To Compute Destructive "Weight of Column. 

^4 5 £ 4 

Cylinder, —A- C = W. Rectangle. — C = W. Short Columns , or less 
L L 

"W^ CL S 

than 30 diameters in length. - - — W. d representing diameter and s 

W' —(— . 75 a S 

side in ins ., a area of section in sq. ins ., l length in feet, S crushing strength of ma¬ 
ternal, C coefficient of material , and W' destructive weight, as ascertained by compu¬ 
tation for a long column of like dimensions in lbs. 


Coefficients. 


Ash. 22000 

Elm,rock.. 


Red Pine. 

17 500 

“ Canadian. 17000 

Fir, Dantzic 


Yellow pine. 

12 OOO 

Beech. 17 500 

Oak, white. 


White “ . 


Cedar. 14000 

“ ‘ Eng... 


Spruce. 

14 OOO 

Elm. 17 500 

Pitch Pine.. 


Walnut. 

12 500 

Illustration. —What is 

destructive weight of a column of yellow pine 

10 ins. 

square and 12 feet in length or height? 




io 4 

10 oco 




— - X 12000=- 

X 12000 = 833333 Lbs. 


12 

144 





For long square columns of the following: Hodgkinson put C = Dantzic oak, dry, 
24 528; red deal, dry, 17 472; and French oak, dry, 15 456. 


To Compute Safe Weight in Tons.* 

Rectangular Oak Columns. Secured at Both Ends. Rule. —Divide length 
of column by thickness or least dimension, multiply unit in column C, cor¬ 
responding to quotient of length of column, divided by this least dimension 
and by width of column, all dimensions in ins. 


L 

T 

C 

L 

T 

c 

L 

T 

C 

L 

T 

c 

L 

T 

C 

L 

f" 

C 

I 

•43 

7 

• 36 

13 

. 26 

19 

.18 

25 

. 12 

3 i 

•°93 

2 

•43 

8 

•35 

14 

.24 

20 

•17 

26 

. 12 

32 

.089 

3 

.42 

9 

•33 

15 

•23 

21 

•17 

27 

.11 

33 

.084 

4 

•4 

IO 

• 3 1 

l6 

.21 

22 

•15 

28 

. I 

34 

.08 

5 

•39 

II 

.29 

17 

.2 

23 

.14 

29 

.098 

35 

.077 

6 

•38 

12 

.27 

l8 

.19 

24 

•13 

30 

•°97 

36 

•073 


Illustration.— Assume a white-oak column, secured at both ends, 12 by 8 ins., 
and 20 feet in length. 


20X12-4-8 = 30. C for which = .097. Hence, 12 X 8 X -097 = 9.312 tons. 

For other woods take the values in following table. Thus, if an oak column, as 
above, will sustain 9.312 tons, the strength of one of yellow pine is thus obtained: 
As 5.8 : 9.312 :: 3 : 4.816 tons. 


Relative Value of various Woods, their Crushing 
Strength and Stiffness "being Combined. 


Teak. 


Elm. 

• 5 

Mahogany... 

•• 3-7 

Yellow pine... 3 

English oak .. 

. 5-8 

Beech . 

• 4-4 

Spruce. 

..3.6 

Sycamore.2.6 

Ash. 

• 5 -i 

Quebec oak... 

. 4.1 

Walnut. 

•• 3-4 

Cedar. 1 


Comparative Value of Long Solid Columns of various 
[Materials. (Hodglcinson.) 

Cast Iron.1000 | Cast Steel_2318 | Oak.108.8 | Pine.78.5 


Hence, To compute destructive weight of an Oak or Pine column, take weight for 
one of Cast iron of like dimensions, and if for Oak divide by 9, and for Pine by 12.7. 

* All tons, except when otherwise designated, are 2240 lbs. 

3T 



































































y /0 STRENGTH OF MATERIALS.—DEFLECTION. 


DEFLECTION. 

Deflection, of Bars, Beams, Grirders, etc. 

Experiments of Barlow upon deflection of wood battens determined, 
that deflection of a beam from a transverse strain varied directly as 
breadth, and as cubes of both depth and length, and that with like beams 
and within limits of elasticity it was directly as the weight. 

In bars, beams, etc., of an elastic material, and having great length com¬ 
pared to their depth, deductions of Barlow will apply with sufficient accu¬ 
racy for all practical purposes; but in consequence of varied proportions of 
depth to length, of varied character of materials, of irregular resistance of 
beams constructed with scarphs, trusses, or riveted plates, and of unequal 
deflection at initial and ultimate strains, it is impracticable to deduce any 
exact laws regarding degrees of deflection of different and dissimilar figures 
and proportions. 

From an experiment of Mr. Tredgeld it was shown that deflection of cast 
iron is exactly proportionate to load until stress reaches a certain magnitude, 
when it becomes irregular. 

In experiments of Hodgkinson, it was further shown that sets from de¬ 
flections were very nearly as squares of deflections. 

In a rectangular bar, beam, etc., position of neutral axis is in its centre, 
and it is not sensibly altered by variations in amount of strain applied. In 
bars, beams, etc., of cast and wrought iron, position of neutral axis varies in 
same beam, and is only fixed while elasticity of beam is perfect. When a 
bar, beam, etc., is bent so as to injure its elasticity, neutral line changes, and 
continues to change during loading of beam, until its elasticity is destroyed. 

When bars, beams, etc., are of same length, deflection of one, weight being 
suspended from one end, compared with that of a beam Uniformly Loaded , 
is as 8 to 3; and when bars, etc., are supported at both ends, deflection in like 
case is as 5 to 8. Whence, if a bar, etc., is in first, case supported in middle, 
and ends permitted to deflect, and in second, ends supported, and middle 
permitted to descend, deflection in the two cases is as 3 to 5. 

Of three equal and similar bars or beams, one inclined upward, one down¬ 
ward, at same angle, and the other horizontal, that which has its angle up¬ 
ward is weakest, the one which declines is strongest, and the one horizontal 
is a mean between the two. 

When a bar, beam, etc., is Uniformly leaded, deflection is as weight, and 
approximately as cube of length or as square of length; and element of de¬ 
flection and strain upon beam, weight being the same, will be but half of that 
when weight is suspended from one end. 

Deflection of a bar, beam, etc., Fixed at one End , and Loaded at other , 
compared to that of a beam of twice length, Supported at both Ends, and 
Leaded in Middle, strain being same, is as 2 to 1 ; and when length and 
loads are same, deflection will be as 16 to 1, for strain will be four times 
greater on beam fixed at one end than on one supported at both ends; there¬ 
fore, all other things being same, element of deflection will be four times 
greater; also, as deflection is as element of deflection into square of length, 
then, as lengths at which weights are borne in their cases are as 1 to 2, de¬ 
flection is as 1 : 2 2 x 4 = 1 to 16. 

Deflection of a bar, beam, etc., having section of a triangle, and supported 
at its ends, is .33 greater when edge of angle is up than when it is down. 

In order to counteract deflection of a beam, etc., under stress of its load, 
where a horizontal surface is required, it should be cambered on its upper 
surface, equal to computed deflection. 


STRENGTH OF MATERIALS.-DEFLECTION. yy\ 


Safe Deflection. —One fortieth of an inch for each foot of span, with a 
factor of safety for load of .33 of destructive weight = y^o> but for ordinary 
loads and purposes, 

Cast Iron , yy ( j y f° 2 uou ? an ^ Wrought Iron , yyyy to or ygyy, 

after beam, etc., has become set. 

When Length is uniform , with same weight, deflection is inversely as 
breadth and square of depth into element of deflection, which is inversely as 
depth. Hence, other things being equal, deflection will vary inversely as 
breadth and cube of depth. 

Illustration.— Deflections of two pine battens, of uniform breadth and depth, and 
equally loaded, but of lengths of 3 and 6 feet, were as 1 to 7.8. 

Deflection of different bars, beams, etc., arising from their own weight, 
having their several dimensions proportional, will be as square of either of 
their like dimensions. 

Note. — In construction of models on a scale intended to be executed in full di¬ 
mensions, this result should be kept in view. 

When a continuous girder, uniformly loaded, is supported at three points 
by two equal spans, middle portion is deflected downwards over middle bear¬ 
ing, and it sustains by suspension the extreme portions, which also have a 
bearing on outer bearings. Middle portion is, by deflection, convex up¬ 
wards, and outer portions are concave upwards; and there is a point of 
“contrary flexure,” where curvature is reversed, being at junction of con¬ 
vex and concave curves, at each side of middle bearing. This point is dis¬ 
tant from middle bearing, on each side, one fourth of span. Of remaining 
three fourths of each span, a half is borne by suspension by middle portion, 
and a half is supported by abutment. Hence, distribution of load on bear¬ 
ings is easily computed, as given above. Deflection of each span is to that 
of an independent beam of same length of span as 2 to 5. 

In a beam of three equal spans, deflection at middle of either of side spans 
is to that of an independent beam as 13 to 25. 

In a long continuous beam, supported at regular intervals, deflection of 
each span is to that of an independent beam of one span as 1 to 5. 

Cylinder. —If a bar or beam is cylindrical, deflection is 1.7 times that of a 
square beam, other things being equal. 


Formulas for Deflection of Beams of Rectangular Sec¬ 
tion, etc. 

r l 3 w , b d 3 C D 

Fixed I Loaded at One End. - - 7 , = D; and -—- = W. 


at -< 
One End. # 


Uniformly. £ 


bd 3 C 
3 I 3 W 


I 3 

T 8 & cZ 3 C D m 

— D; and -—— — W. 

3 


8 bd 3 C 

r __ . 24 b d 3 C D „ r 

Fixed l Loaded m Middle. u = D ; and -— - = W. 


at 

Both Ends. 


X 


Uniformly. 


24 b d 3 C 
5 Z 3 W 

8 X 24 b d 3 C 


— D; and 8 ' X '* 4 “.f C D = W. 
5 t 3 


C Loaded in Middle. 


l 3 W _ J 16 b d 3 C D ttt 
= D; and -— -= W. 


Supported j 
at -[ 
Both Ends. \ 

l 


Uniformly. 


16 b d 3 C ’ l 3 

5UW 8 X 16 b d 3 C D 

- = D; and -—-— W. 


8x 16 bd 3 C 5 I 3 

„ . m 2 n~ W _ , l b d 3 CD 

at any one Point. = D ; and --—— = W. 

J l b d 3 C m 2 n 2 


Supported in Middle. 

3 I 3 W „ J 5 X 16 b d 3 G _ 

Ends Uniformly loaded. g x l6 b d3 c ' = D; and - fli 3 - = W ’ 

l representing length, b breadth, and d depth, all in ins., W weight or stress in lbs. or 
tons, m n distances of weight from supports, C a constant, and D deflection, in ins. 




















772 STRENGTH OF MATERIALS.-DEFLECTION. 


Deflection, of Beams of Rectangular Section. 

Fixed at f , 7 , . ^ „ , Z 3 W 3 l 3 W 

One End. ( 6 d 3 C ’ ’ 8 Z> h 3 C 

Fixed at f, , , . I 3 W ^ xr . 5 Z 3 W 

< Loaded m Middle. T “* ^- 7 - 


2 >oZA | . ' 24 6 d 3 C 

/- Z 3 W 

Supported l Loaded m Middle. — — 

rtZ < 

7 >oZA Ends. 




at any one Point. 
C a Constant as follows. 

Cast Iron. 


= D; Uniformly, 
= D; Uniformly, 

D 


8 X 24 A d 3 C 
5 Z 3 W 


?n. 2 » 2 W 


Ibd 3 C 


8 X 16 6 d 3 C 
W weight in tons. 


= D. 
= D. 


Rectangular. - 
Round. 


Rectangular Bars. —Loaded at One End. 875. 

“ at the Middle.... 28 000. 

Round Bars. —Loaded at One End. 594. 

“ at the Middle.19000. 

Wrought Iron. Cast Iron. 

—For tons and Z in ins. put C = 47 000. 28 000. 

“ “ Z in feet “ C = 27. 16. 

“ “ Zinins. “ 0 = 32000. 19000. 

“ “ Z in feet “ C= 18. 10.7. 

Hence, in order to preserve same stiffness in bars, beams, etc., depth must 
be increased in same proportion as length, breadth remaining constant. 

"W oocls. 

Mean of LasletVs, Barlow, etc. (D. K. Clark.) 

Z 3 W 

Supported at Both Ends. Loaded in Middle. 


l> d 3 C 


= D. 


Ash, Canadian. 1476 

“ Eng. 2722 

Beech. 2418 

Blue Gum. 2559 

Elm. 1227 

Fir, Dantzic. 2490 

“ Memel. 3630 

“ Riga. 2920 

Greenheart. 1888 

Iron Bark. 4378 


Iron-wood. 4228 

Larch. 2100 

Mahogany, Honduras 2x18 
“ Mexican. 3608 
“ Spanish.. 3360 

Norway spar. 2465 

Oak, Baltimore. 2761 

“ Canadian. 3445 

“ Dantzic. 2080 

“ Eng. 1848 


Oak, French. 2656 

“ white.2114 

Pitch pine.2968 

Red u . 2434 

Rock-elm. 


a 2 3 J 9 

Spruce. 3300 

“ Amer. 2669 

“ Scotch. 1583 

Teak. 1804 

Yellow pine. 2084 


Application of Table : To Compute Deflection of a Rectangular Beam of Wood. 
Illustration. —What is the deflection of a floor beam of yellow pine, 3 by 12 ins., 
12 feet between its supports, under a uniformly distributed load of 3000 lbs. ? 


C — 2084. 


5 X 12 3 X 3000 15000 


=. 299 inch. 


8 X 3 X 12 3 X 2084 50016 

Hence , To compute weight that may be borne by a given deflection of such a beam, 
8 X 3 X 12 3 X 2084 X -299 _ 14955 
5 X 12 3 — 5 


: 2991 IbS. 


Defiectioiv of Conti it axons Girders or Beams. 
Beams of Uniform Dimensions , Supported at Three or More Bearings. 

(D. K. Clark.) 


1. Two Equal Spans or 3 Bearings. 
Weight on 1st and 3d bearing = .375 W Z 
“ “ 2d bearing.=1.25 W Z 


2. Three Equal Spans or 4 Bearings. 
Weight on 1st and 4th bearing = .4 W Z 
“ “ 2d “ 3 d “ =1.1 WZ 


3. Four Equal Spans or 5 Bearings. 

Weight on xst and 5th bearing = .39 W Z | Weight on 2d and 4th bearing: 

Weight on 3d bearing = .93 W Z. 

Z 3 W „ . cZ* C D 


1.14 W l 


Cylindrical Beam. 


(D C 


= D; and 


I 3 


W. 

















































STRENGTH OF MATERIALS.-DEFLECTION. 


To Compute Maximum Load tliat may "be Dome by a 

Rectangular 33earn. 

Deflection not to exceed Assigned Limit of one hundred and twentieth of an 

Inch for each Foot of Span. 

Supported at Both Ends. Loaded in Middle. 

b d 3 

—-— = W. b and d representing breadth and depth in ins., I length in feet, C con- 
stant, and W weight or load in lbs. 


Cast Iron...0003 

Wrought Iron.0021 

Hickory.018 

Teak.024 


Constants. 


Oak, white.027 

Ash, white.03 

Fine, pitch.033 

“ yellow.036 


Oak, red.039 

Hemlock.039 

Pine, white.039 

Chestnut, horse.051 


Illustration. —What is maximum load that may be borne by a beam of white 
pine, 3 by 12 ins., 20 feet between its supports, and loaded in its middle? 

3 X 12 3 5184 


C = . 


039. 


Then 


X-039 15.6 


332.3 


lbs. 


WROUGHT IRON. 

Deflection of Wrought-iron Bars. 
Supported at Both Ends. Weight applied in Middle. 






1 

u 

a 




Weight and Deflection 


S 4 u 

II 

No 


Form. 

. 

0 bi 

.5 

bo 

Breadth. 

Depth. 

by Actual 

at one sixth 
of Destruc- 

at . » -th of 
12 5 

an Inch for 

Constant e 

educed Wei 

nd Deflect! 

W 13 

a 

n 

-O 

O 





<v 



Observation. 

tive Weight. 

each Foot of 

O 

O 












Span. 

(A a 

5 





Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

C 


I. 

American 


■ 

a 


1.83 

I 

I 

600 

.06 

266 

.027 

00 

H 

.015 

I 


2. 

English.. 

• 

U 

2-75 

2 

2 

4480 

.08 

131° 

.022 

1310 

.022 

I .29 


0 

O* 

U 

. • 

1 


2-75 

i -5 

2-5 

8960 

rh 

O 

H 

2128 

.025 

1873 

.022 

1.25 


4 - 

u 

. . 

u 

2.75 

i -5 

3 

8960 

.088 

3800 

•°37 

2259 

.022 

00 

00 



To Compute Deflection of, and Weig-lit tliat may be Dome 
Dy, a Rectangular Bar or Beam of Wrouglit Iron. 


W l 3 W l 3 p 60000 b d 3 C D 

60000 b d 3 I) 60000 b d 3 C l 3 

Illustration. —What weight will a beam 2 ins. in breadth, 5 ins. in depth, and 
15 feet between its supports, bear with safe deflection of y-Lg- of an inch for each 
foot of space, or T gVo of its len S th ? 


C from table = .88. D 


-- of 15 = .12 inch. 


T2 5 

60000 X2X5 3 X-88 X-i2 


1 800 000 

—- = 533-33 lbs - 


I 5 3375 

D. K. Clark gives for Elastic deflection, 47000 for Rectangular bars, and 32000 for 
Cylindrical. 


Note. —Deflection of to of the length may be allowed under special cir¬ 

cumstances; but under ordinary loads the deflection should not exceed one fourth 
of these, as yg 1 ^ to 

Practice in U. S. is to allow after girder has taken its permanent set. 

In small bridges there is a slight increase in deflection from high speeds, about 
.166 or . 144 of the normal deflection, with the same load moving at slow speed. 

In large girders there is no perceptible difference between the deflection at high 
and low speeds. 

3T* 









































STRENGTH OF MATERIALS.-DEFLECTION, 


Deflection of Wrought-iron Rolled. Beams. 

Supported at Both Ends. Weight applied in Middle. 

W is 

- ■ ■ — = C at Reduced Weight and Deflection. 

70000 d 2 (4 0 + 1-155 a 0 D 





Flanges. 



Wei 

ght and Deflection 


No. 

Form. 

fcX) 

p 

<v 

Width. 

Mean 

Thick¬ 

ness. 

Web. 

Depth. 

by Actual 
Observation. 

at one sixth 
of Destructive 
Weight. 

C 



Feet. 

Ins. 

Inch. 

Inch. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Inch. 


I. 

I 

IO 

3 

•485 

•5 

7 

12 000 

•4 

3800 

. 127 

1.05 

2. 

u 

20 

4.6 

.8 

•5 

9-85 

16 000 

i-i 5 

6300 

•453 

.92 

3 - 

u 

20 

5-7 

•643 

.6 

n-75 

20 OOO 

.85 

8000 

•34 

.98 


To Compute Deflection of, and "Weiglit tliat may be Borne 
by, a, W rouglit-iron Rolled 13 earn of Uniform aird Sym¬ 
metrical Section. 


Supported at Both Ends. Weight applied in Middle. ( D. K. Clark.) 

W Z3 __ 70000 d 2 (4 a + 1.155 a') D _ 

-. J-/ * 7 3 ’’ * 

70000 d 2 (4 a -f- i. 155 a') L 

l representing span in feet, d reputed depth, or depth less thickness of lower flange 
in ins., a area of section of lower flange, a’ area of section of web for reputed depth 
of beam, both in sq. ins., and W weight or stress in lbs. 

Illustration. —What is deflection of a wrought-iron rolled beam of New Jersejr 
Steel and Iron Co., 10.5 ins. in depth, flanges 5 by .5 ins., and width of web .47 
inch, when loaded in its middle with 8000 lbs., and supported over a span of 20 feet? 

d = 10.5 — . 5 ^ 10 ins., a — 5 X • 5 = 2.5 sq. ins. , and a' — 10 X -47 = 4-7 sq. ins. 

8000 X 20 3 64000000 . , 

Then - — = — r — - =. 59 mch. 

70000 X io 2 X (4 X 2.5 + i-i 55 X 4-7) 107 °99 5 00 

If weight is uniformly distributed, divide by ri2 500 instead of 70000. 

A like beam 6 ins. in depth, loaded with 2608 lbs., and supported over a span of 
12 feet, gave by actual test a deflection of .3 inch, and by above formula it is also 
.3 inch. 

Note. —Deflection for such a beam, for a statical weight or stress of 17 100 lbs., 
uniformly distributed, by rules of N. J. Steel and Iron Co., would be .54 inch, which, 
with difference in weights, will make deflections alike. 


Deflection of Wronght-iron Riveted. Reams. 
Supported at Both Ends. Weight applied in Middle. 
w is 

— C at Reduced Weight and Deflection. 


/a a' a"\ 
168000 ^-^ 


No. Form. 


1. 


2 . 


Length. 


Feet. 

7 I 

11.66 j 

22.5 \ 


Flanges. 


Ins. 


4 - 5 X 

•5 

4 - 5 X 

•375 

4 - 5 X 

•5 

7 X 
•5 


Angles. 


Ins. 

2.125X2 
X-28 
2.125X2 
X-29 
2X2 
X- 3 I2 5 
2X2 
X.3125 
2X2 
X -375 
3X3 
X -4375 


Web. 


Inch. 

u 

) 

H 

r-375 


rC 

-♦J 

PU 

a> 

Q 


Ins. 


12.5 


16.5 


Weight and Deflection 

at one sixth 
of Destructive 
Weight. 


by Actual 
Observation 


Lbs. 

4 216 

77 280 

115584 


Inch. 

. 1 

.46 

•875 


Lbs. 

4062 

12 880 

19265 


Inch. 

.096 

•075 

. 148 


• 6 3 

1.96 

3.86 
















































STRENGTH OF MATERIALS.-DEFLECTION. 


775 


To Compute Deflection of, and. Weight that may he home 
by, a Riveted Beam of Wrought Iron. 


W 1 3 


, n (a 4 -a . a \ , „ 
168 ooo ^-j- J d 2 G 


= D. 


168 ooo 


(^+f) 


d 2 CD 


l 3 


W. 


a , a', and a" representing areas of upper and lower flanges with their angle pieces , 
and of Web for its entire depth , all in sq. ins. 

Note.—I f there are not any flanges, as in No. i, angle pieces alone are to be computed for flange 
area. 

Illustration.— What weight will a riveted and flanged beam of following dimen¬ 
sions sustain, at a distance between its supports of 25 feet, and at a safe deflection 
of .2 inch or of its length? 


15 0 0 

Top flange. 6X-5 ins. | Web. 

Bottom flange.. 6X-5 “ | Depth.. 

Angles.2.25 x 2.25 X .5 ins. 


. 5 ms. 


W 


a and a' each — 6 X - 5 — 3 -(- 2.25 -f- 2.25 — .5 x -5 X 2=7 sq. ins. 
a" = .5 x 17 = 8.5 sq. ins. C, as per No. 2,= .43, but inasmuch as flanges in this 
case are much heavier, assume . 5. 

^7 + 7 , 8.5^ 


168 ooo 


Then 


C-F+tO 


17 X .2 x .5 


25 ‘ 


44 303 720. 

15625 


: 2835.4 MS- 


Strength of a Riveted beam compared to a Solid beam is as 1 to 1.5, while for 
equal weights its deflection is 1.5 to 1. 

Tubular Grirders. ’Wrought Iron. 

Supported at Both Ends. Weight applied in Middle. 


No. 


Section. 


3 - 

4 - 

5 - 

6 . 

7 - 


Thickness .03 inch 

(< *525 “ 

top .3^2 “ ) 
bottom .244 “ / 
sides .125' “ ) 
Thickness .75 “ 

Thickness .0375“ 


.0416“ 

•143 “ 


X [*> 

C O 

01 01 

1-1 n 

Breadth. 

De] 

Inter¬ 

nal. 

ith. 

Ex¬ 

ternal. 

Weight. 

Deflection. 

Deflection at 
.008 inch for 
each Foot of 
Span. 

1 

£ 

n 

O 

CO 

•>0 

VO 

Feet. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Inch. 



3-75 

1.9 

2.94 

3 

448 

. I 

•03 

288 

3 ° 

G -5 

22.95 

24 

33 685 

•56 

.24 

473 

3 ° 

l6 

23.28 

24 

32 53 8 

I. II 

.24 

224 

45 

24 

34-25 

35-75 

128 850 

1.85 

•36 

362 

17 

12 

11.925 

12 

2 755 

•65 

.136 

62.8 

17 

9- 2 5 

13-535 

13.62 

2 262 

.62* 

.136 

47-9 

17 

9- 2 5 

I 4 - 7 I 4 

15 

l6 80O 

I.39* 

.136 

119 


* Destructive weight. 


To Compute Deflection of, and Weight that may he 
borne by, a Wrought-iron Tubular Grirder. 

16 6 d 3 C D „ W l 3 


: W. 


— D. 


1 3 ' 16 b d 3 C 

Illustration. —What weight may be safely borne by a wrought-iron tube, alike 
to No. 3 in preceding table, for a length of 40 feet, and a deflection of .32 inch? 

16 X 16 X 24 3 X 224 X-24 190253629 


3 °- 


27 ooo 


= 7046 lbs. 


Flanged Rivets. 

Deflection of Iron and Steel Flanged Rails within their elastic limit, compared 
with their transverse strength, is as 17 to 20, and with double-headed it is as n to 23. 





































STRENGTH OF MATERIALS.—DEFLECTION, 


RAILS. 


Supported at Both Ends. Weight applied in Middle. 

Iron. 


No. 

Form. 

Length 

of 

Bearing. 

Head. 

Bottom. 

Weight 

per 

Yard. 

Depth. 

Area. 

Observed 
Weight and 
Deflection. 

Destructive 

Weight 

and 

Deflection. 



Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Sq. Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

I. 

1 

2-75 

2.25X1 

2.25X1 

60 

4-5 

6.166 

13 440 

•034 

26680 

.065 

2. 

u 

4-5 

2.3 Xi 

2.3 Xi 

65 

4-5 

6.68 

II 200 

.11 

24 640 

.204 

3 - 

<< 

5 

2.3 Xi 

2.3 Xi 

82 

5-4 

8.25 

25 76° 

.2 

5 i 520 

•378 

4 - 

T 

2-75 

3-5 X -8 

2.25X1 

60 

4 

6.7 

II 200 

•035 

26 680 

.065 

5 - 

i 

2.58 

2.23X1 

3-5 X .6 

57 

3-5 

5-85 

II 200 

.097 

20 160 

.128 


Steel. 



A ti 

. 




Depth. 



Destructive 

No. Form. 

C 0 CS 

•3 M 

Heat 

Bottom. 

Weig 

per 

Yard 

Web. 

Centres 

of 

Heads. 

Total. 

Area. 

Weight and 
Deflection. 

Weight 

and 

Deflection. 


Feet. 

Ins. 

Ins. 

Lbs. 

Inch. 

Ins. 

Ins. 

S.Ins. 

Lbs. 

In. 

Lbs. 

Inch. 

6 ‘ I 

5 

— 

— 

78 

•75 

4.2 

5-4 

7.67 

36 086 

•25 

80 192 

•55 

7 - “ 

Bessemer. 

3.62 

— 

— 

86 

— 

— 

5-5 

8-43 

22 4OO 

.14 

26 680 

.165 

8 -X 

5 

2-5 

6-375 X^ 7 

84 

•65 

3-37 

4-5 

Tt- 

N 

00 

27 29O 

.24 

27 290 

.24 


To Compute Deflection, of Donble-lieaded Rails -within 
Elastic Limit. (D. K. Clarlc.) 


Supported at Both Ends. Weight applied at Middle. 


IRON. 


W 1 3 

-________ D. a representing area of one head, less portion per- 

57000 (4 a d' 2 - f-1.155 t d 3 ) 

taining to web, d whole depth of rail, d' vertical distance between centres of heads, 
t thickness of web, all in ins., I length in feet, and W weight in lbs. 


STEEL. 


For 57 000 put 67 400. 

Illustration. —Take case No. 3 (Iron), in preceding table, with a weight of 26000 
lbs.; what will be its deflection between bearings 5 feet apart ? 

a = 1.911. d' = 4.2. <2 = 5.4. t~. 82. 


Then 


26 000 X 5 3 


3 250000 


57000 (4 X 1-911 X 4-2 2 + 1.155 X .82 X 5 - 4 3 ) 57 00 °X 284 


— .2 inch. 


To Compute Deflection of Iron and Steel Rails of Un- 
symmetrical Section -within Elastic Limits. 

Elastic Deflection of Steel Flanged Rails of Metropolitan Railway of London, as 
determined by Mr. Kirkaldy, at a span of 5 feet, and loaded in middle, was .02 inch 
per ton. (See Manual of D. K. Clark,pp. 667-670.) 













































STRENGTH OF MATERIALS.-DEFLECTION. 


CAST IRON. 


Deflection of Rectangular Bars and. Beams of various 
Sections, etc., by LT. S. Ordnance Corps, Barlow, 
Hodgkinson, and CnDitt. 


Supported at Both Ends. Weight applied in Middle. 





• 

t -4 

CS 




Weight and Deflection. 


A A ,, 

No. Form. 

0 fee 

a a 
+-> 

Breadth. 

Depth. 

By Actual 

At one sixth 

At Tlyff th of 

an inch for 

Constant a 

;duced We' 

nd Deflecti 

W 13 

ft 

CO 

'IS 

-0 



fee 

a 


Observation. 

W eight. 

each foot of 

0 



t-i 







span. 

(5 oS 

0 



Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

C 


1. American_ 

HI 

mi 

1.66 

2 

2 

5000 

.036 

1666 

.012 

M 

OO 

O 

Ln 

• 013 

OJ 

00 

H 


2. English. 

u 

4 

I 

I 

212 

•32 

80 

.12 

22 

•033 

4.11 


O. U . 

L l 


4 


1008 

.4 

5333 

2 . II 

337 ° 

i -33 

3-89 



a 





4. “ . 

i 

4-5 

3 


1120 

1.42 

215 

.27 

30 

. 037 

2-37 



y 




5. “ . 

H 

4-5 


I 2 ' 5 

2231 

• 5 i 

422 

. I 

156 

•037 

2-33 



y 


\ -5 




To Compute Deflection of, and 'W'eiglit that may- he 
home by, a Rectangular Bar or Beam of Cast Iron. 

W l 3 WP _ D 10400 b d 3 C D _ w 

10400 b d 3 D 10400 b d 3 C l 3 


Illustration. — What weight will a beam 2 ins. in breadth, 5 ins. in depth, and 
16 feet between its supports, bear with safe deflection of yU- of an inch for each 
foot of span, or of its length? 

C from table = 3. 89. D: yU of 16 = 1. 33 ins. 

10400 X 2 X 5 s X 3.89 X 1.33 134S1620 

- — -— -- — = 3284 lbs. 

i6 d 4096 

Clark gives C uniform for Rectangular bars of 2.69, and 1.85 for Cylindrical. 
FLANGED BEAMS. Cast Iron. 


Supported at Both Ends. Weight applied in Middle. 


To Compute Deflection of, and Weight that may he 
home by, a Flanged Beam of Cast Iron of Uniform 
and Symmetrical Section. 


W l 3 


27000 d 2 (4 a -)- 1.155 a 



27 000 d 2 (4 a -(- 1.155 a' 2 ) D 

T 3 


= W. 


Illustration. —What is deflection of a cast-iron beam (Hodgkinson’s) 7.15 ins., 
flanges 2.6 X .86 ins. and 5 x 1.6 ins., and width of web 1 inch, when loaded in its 
middle with n 200 lbs., over a span of 15 feet? 

d — 7.15 — 1-6 = 5.55 MM.> <* = 5 X i.6 = 8 ins., and a' — 7.15 —1.6 = 5.55 ins. 


Then 


11 200 X 15 3 

27000 5.55 s (4X 8 + 1.155 X 5 - 55 2 ) 


37 800 000 

27000 30.8 (32 + 35.57) 


= 1.48 ins. 


Note i.—T he observed deflection of this beam was 1.28 ins., at one sixth of its de¬ 
structive weight it was .3, and at U of an inch for each foot of span it was 
.125 inch. 


2.—The mean ratio of elastic to destructive stress is 73 per cent. 

Formulas for value of deflection signify that deflection varies directly as weight, 
and as cube of length; and inversely as breadth, cube of depth, and coefficient of 
elasticity. 


































778 


STRENGTH OF MATERIALS.-DEFLECTION. 


Elastic Strength of Beams of Unsymmetrical Section .—Elastic strength is 
approximately deducible from ultimate strength, according to ordinary ratio 
of one to the other, ascertained experimentally. Elastic strength and de¬ 
flection of a homogeneous beam of any section is same, whether in its nor¬ 
mal position or turned upside down. 

Comparative Strength, and. Deflection of Cast-iron 
Flanged Beams. 


Description of Beam. 

Comp. 

Strength. 

Description of Beam. 

Comp. 

Strength. 

earn of equal flanges . 

•58 

Beam with flanges as 1 to 4.5.. 

.78 

“ with only bottom flange. 

.72 

“ “ “ 1 to 5.5.. 

.82 

“ “ flanges as 1 to 2... . 

.63 

“ “ “ 1 to 6 .. 

I 

“ “ “ 1 to 4. ... 

•73 

“ “ 1 to 6.73. 

.92 


SHAFTS. 

To Compute Deflection and. Distributed "Weight for 
Limit of Deflection. 

Wrought Iron. 

Deflection. Weight. 


Supported at Ends. 
Round. 

Square. 


Fixed at Ends. 


Supported at Ends. 


W 1 3 

and 

W 1 3 

D 

664 d 4 

and 

1330 d 4 

66400 d 4 

133000 d 4 


£2 

1 2 

W l 3 

and 

W l 3 

j) 

975 s 4 

and 

1950 s 4 

97 500 s 4 

195 000 s 4 


L 2 

1 2 


Round. 


Square. 


Round. 


Square. 


W 1 3 
39 400 d 4 
W l 3 
58 000 s 4 


W is 
78 800 d 4 
W l 3 

116 000 s 4 


and 


and 


and 


and 


Cast Iron. 
WZ3 


79000 d 4 
W 1 3 


= D. 


116 000 s 4 

Steel. 
= D. 


W 1 3 


158 000 d 4 
W l 3 


232 OOO s 4 


= D. 


394 d 4 

580 S 4 
l 2 


788 d'f 
Z 2 

1160 b 4 


Fixed at Ends. 

= w. 
= w. 


and 

and 

and 


790 d 4 
t2 

1160 S 4 

: 


1576 d 4 


f2 


— and 


2320 ft 4 

P : 


w. 

w. 

: W. 
: W. 


d representing diameter and s side of shaft, in ins., I length between centres of bear¬ 
ings, in feet, and W weight in lbs. 

Deflection of a Cylindrical Shaft from its Weight alone, 
when Supported at Both Ends. 

I 4 

.007318 — D- I representing length in feet, d diameter in ins., and C con¬ 

stant, ranging from 475 to 550. 

The greatest admissible deflection for any diameter is .001 67 — = D. 
Admissible Distances between Bearings. ■5 / .qi 28 d C = Z. 


Diam. 
of Shaft. 

Distance. 


Wrought 

Iron. 

Steel. 

Diam. 
of Shaft. 

Ins. 

Feet. 

Feet. 

Ins. 

I 

12.27 

12.61 

5 

2 

15.46 

15.84 

6 

3 

* 7-7 

18.19 

7 

4 

19.48 

20.02 

8 


Distance. 

Diam. 
of Shaft. 

Distance. 

Wrought 

Iron. 

Steel. 

Wrought 

Iron. 

Steel. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

2O.99 

21-57 

9 

25-53 

26.24 

22.3 

22.92 

IO 

26.44 

27.18 

23.48 

24 -I 3 

II 

27-3 

28.05 

2 4 - 55 

25.23 

12 

28.1 

28.88 


When Ends of Shaft are rigidly connected at Ends. 

Barlow gives D = .66 of results obtained by above formula; but when deflection 
of attached length is considerable, Navier gives D = .25 of above. 



















































STRENGTH OF MATERIALS.-DEFLECTION. 


779 


Deflection of NPill and. Factory Shafts. 

V W 

- -- = D. I representing length between supports in ins., W weight at middle 

O 7T (X* C 

in lbs., d diameter of shaft in ins., and C as follows : 

Bessemer steel.. 3800000 | Wrought iron. 3500000 


To Compute Deflection of a Cylindrical Shaft. 


Rule. —Divide square of three times length in feet by product of follow¬ 
ing Constants and square of diameter in ins., and quotient will give deflection. 

Cast iron, cylindrical. 1500 I Wrought iron, cylindrical. 1980 

“ “ square. 2560 | “ “ square. 3360 


Example.— Length of a cast-iron cylindrical shaft is 30 feet, and its diameter in 
centre 15 ins.; what is its deflection? 


_ 2 

30 X 3 
1500 X 15 2 


8100 
337 5oo 


.024 ins. 


SPRINGS. 

Flexure of a spring is proportional to its load and to cube of its length. 
Deflection of a Carriage Spring. 

A railway-carriage spring, consisting of 10 plates .3125 inch thick, and 2 
of .375 inch, length 2 feet 8 ins., width 3 ins., and camber or spring 6 ins., 
deflected as follow's, without any permanent set: 

.5 ton.5 inch. I 1.5 tons. 1.5 ins. I 3 tons. 3 ins. 

1 “ . 1 “ I 2 “ . 2 “ | 4 “ . 4 “ 

Compression of an India- rubber Buffer of 3 Ins. Stroke. 

1 ton. 1.3 ins. I 2 tons. 2 ins. I 5 tons. 2.75 ins. 

1.5 tons. 1.75 “ I 3 “ . 2.375 “ I 10 “ .3 “ 


GJ-eneral Deductions. 

Deflection depends essentially upon form of Girder, Beam, etc. 

A continuous weight, equal to that a beam, etc., is suited to sustain, will 
not cause deflection of it to increase unless it is subjected to considerable 
changes of temperature. 

Heaviest load on a railway girder should not exceed .16 of that of de¬ 
structive weight of girder when laid on at rest. 

Semi-girders or Beams .—Deflection of a beam, etc., fixed at one end and 
loaded at other, is 32 times that of same beam supported at both ends and 
loaded in middle. 

Deflection consequent upon Velocity of Load .—Deflection is very much in¬ 
creased by instantaneous loading; by some authorities it is estimated to be 
doubled. 

Momentum of a railway train in deflecting girders, etc., is greater than 
effect from dead weight of it, and deflection increases with velocity. 

When motion is given to load on a beam, etc., point of greatest deflection 
does not remain in centre of beam, etc., as beams broken by a travelling load 
are always fractured at points beyond their centres, and often into several 
pieces. 

Heaviest running weight that a bridge is subjected to is that of a loco¬ 
motive and tender, which is equal to 2 tons per lineal foot. 

Girders should not, under any circumstances, be deflected to exceed .025 
inch to a foot in length. 























STRENGTH OF MATERIALS.-DEFLECTION. 


A carriage was moved at a velocity of io miles per hour ; deflection was 
.8 inch, and when at a velocity of 30 miles deflection w r as 1.5 ins. 

In this case, 4150 lbs. would have been destructive weight of bars if ap¬ 
plied in their middle, but 1778 lbs. would have broken them if passed over 
them with a velocity of 30 miles per hour. 


Relative Elasticity- of various [Materials. (Trumbull.) 


Ash. 


. 1 I Pine, white... 

. 2.4 I Pine, pitch... 

Beech. 


. 2.9 1 “ yellow.. 

. 2.6 | Wrought Iron 


Cast Iron .—Permanent deflection is from .33 to .5 of its breaking weight, 
and deflection should never exceed .125 of ultimate deflection, and it is not 
permanently affected but by a stress approaching its destructive weight. 

By experiments of U. S. Ordnance Corps (Report, 1852), set or permanent deflec¬ 
tion was .38 of its breaking weight, ultimate deflection .133 ins. Deflection for 
of span = .013, or . 1 of ultimate deflection. 

By experiments of Mr. Hodgkinson (See Rep. of Commas on Railway Structures, 
London, 1849), set for English iron bore a much greater proportion to its breaking 
weight. 

A beam, etc., will bend to .33 of its ultimate deflection with less than .33 
of its breaking weight, if it is laid on gradually, and but .16 if laid on 
rapidly. 

Chilled bars deflect more readily than unchilled. 

Results of Experiments on tlae Su. Ejection. of Cast-iron 
Bars to continued Strains. 

(Rep. of Comm's on Railway Structures , London, 1849.) 

Cast-iron bars subjected to a regular depression, equal to deflection due to 
a load of .33 of their statical breaking weight, bore 10000 successive de¬ 
pressions, and when broken by statical weight, gave as great a resistance as 
like bars subjected to a like deflection by statical weight. 

Of two bars subjected to a deflection equal to that carried by half of their 
statical breaking weight, one broke with 28 602 depressions, and the other 
bore 30 000, and did not appear weakened to resist statical pressure. 

Hence, Cast-iron bars will not bear continual applications of .33 of their 
breaking weight. 

Mr. Tredgold, in his experiments upon Cast Iron, has shown that a load of 300 
lbs., suspended from middle of a bar 1 inch square and 34 ins. between its sup¬ 
ports, gave a deflection of .16 of an inch, while elasticity of metal remained unim¬ 
paired. Hence a bar 1 inch square and 1 foot in length will sustain 850 lbs., and 
retain its elasticity. 

Wrought Iron .—All rectangular bars, having same bearing, length, and 
loaded in their centre to full extent of their elastic power, will be so deflect¬ 
ed that their deflection, being multiplied by their depth, product will be a 
constant quantity, whatever may be their breadth or other dimensions, pro¬ 
vided their lengths are same. 

A bar of Wrought Iron, 2 ins. square and 9 feet in length between its sup¬ 
ports, was subjected to 100000 vibratory depressions, each equal to deflec¬ 
tion due to a load of .55 of that which permanently injured a similar bar, 
and their depressions only produced a permanent set of .015 inch. 

Greatest deflection which did not produce any permanent set was due to 
rather more than .5 statical weight, which permanently injured it. 

A wrought-iron box girder, 6x6 ins. and 9 feet in length, -was subjected 
to vibratory depressions, and a strain corresponding to 3762 lbs., repeated 
4337° times, did not produce any appreciable effect on the rivets. 





STRENGTH OF MATERIALS.-DEFLECTION. ^8 I 

Deflection of Solid rolled beams compared to Riveted beams is as i to 1.5. 

Wrought-iron Girders of ordinary construction are not safe when sub¬ 
jected to violent impacts or disturbances, with a load equal to .33 of their 
destructive weight. 

Wood.— -In consequence of wood not being subjected to weakening by the 
effect of impact, a factor of safety of 5 for single pieces is held to be suffi¬ 
cient, but for structures, in consequence of loss of strength in its connections, 
a factor of from 8 to 10 becomes necessary. 

Working Strength, or Factors of Safety.* 

Elastic strength of materials is, in general terms, half of its ultimate de¬ 
structive or breaking strength. If a working load of .5 elastic strength, or 
.25 of ultimate strength, be accepted, equal range for fluctuation within 
elastic limit is provided. But, as bodies of same material are not all uni¬ 
form in strength, it is necessary to observe a lower limit than .25 where 
material is exposed to great or to sudden variations of load or stress. 

Cast Iron. — Mr. Stoney recommends .25 of ultimate tensile strength, for 
dead weights ; .16 for bridge girders; and .125 for crane posts and machin¬ 
ery. I11 compression, free from flexure, cast iron will bear 8 tons (17920 
lbs.) per sq. inch ; for arches, 3 tons (6720 lbs.) per sq. inch; for pillars, 
supporting dead loads, .16 of ultimate strength; for pillars subject to 
vibration from machinery, .125 ; and for pillars subject to shocks from 
heavy-loaded wagons and like, .1, or even less, where strength is exerted in 
resistance to flexure. 

Wrought Iron. —For bars and plates, 5 tons (11 200 lbs.) per sq. inch of 
net section is taken as safe working tensile stress; for bar iron of extra 
quality, 6 tons (13440 lbs.). In compression, where flexure is prevented, 
4 tons (8960 lbs.) is safe limit; in small sizes, 3 tons (6720 lbs.). For col¬ 
umns subject to shocks, Mr. Stoney allows .16 of calculated breaking weight; 
with quiescent loads, .25. For machinery, .125 to .1 is usually practised; 
and for steam-boilers, .25 to .125. 

Mr. Roebling claims that long experience has proved, beyond shadow of 
a doubt, that good iron, exposed to a tensile strain not above .2 of its ulti¬ 
mate strength, and not subject to strong vibration or torsion, may be de¬ 
pended upon for a thousand years. 

Steel. —A committee of British Association recommended a maximum 
working tensile stress of 9 tons (20 160 lbs.) per sq. inch. Mr. Stoney rec¬ 
ommends, for mild steel, .25 of ultimate strength, or 8 tons (17920 lbs.) per 
sq. inch. Limit for compression must be regulated very much by nature of 
steel, and whether it be annealed or unannealed. Probably a limit of 9 tons 
(20160 lbs.) per sq. inch, same as limit for tension, would be safe max¬ 
imum for general purposes. In absence of experience, Mr. Stoney further 
recommends that, for steel pillars, an addition not exceeding 50 per cent, 
should be made to safe load for wrought-iron pillars of same dimensions. 

Wood. —One tenth of ultimate stress is an accepted limit. Piles have, in 
some situations, borne permanently .2 of their ultimate compressive strength. 

Foundations. —According to Professor Rankine, maximum pressure on 
foundations in firm earth is from 17 to 23 lbs. per sq. inch; and, on rock, it 
should not exceed .125 of its crushing load. 

Masonry. — Mr. Stoney asserts that working load on rubble masonry, 
brick-work, or concrete rarely exceeds .16 of crushing weight of aggregate 
mass; and that this seems to be a safe limit. In an arch, calculated pressure 
should not exceed .05 of crushing pressure of stone. 

* Essentially from Manual of D. K. Clark, London, 1877. 

3 U 



782 


STRENGTH OF MATERIALS.-DETRUSIVE. 


Ropes .—For round, working load should not exceed .14 of ultimate strength, 
andforflat.n. T , T . T , 

Dead Load. Live Load. 

Perfect material and workmanship . 2 4 

Dr. Rankine gives (Good ordinary material £ wood. 4- to 5 8 to o 

following factors: ) and workmanship ... ( Maso nry ....... 4 8 

A Dead Load is one that is laid on very gradually and remains fixed. 

A Live Load is one that is laid on suddenly, as a loaded vehicle or train 
passing swiftly over a bridge. 


DETRUSIVE OR SHEARING STRENGTH. 

Detrusive or Shearing Strength of any body is directly as its strength, 
or thickness, or area of shearing surface. 


Results of* Experiments upon Detrusive Strength of 
Aletals -with a Punch. 



Diameter 

Thickness 

Power 

exerted. 

Power required for a 

Metals. 

of 

Punch. 

of 

Metal. 

Surface of Metal of One 
. Sq. Inch. 


Ins. 

Ins. 

Lbs. 

Lbs. 

f E.-2 2. 

Brass. 

I 

•045 

5 448 

37 000 

Cast iron. 


30 000 


Copper. | 

•5 

1 

.08 

•3 

3 9 8 3 

21 25O 

30 000 

22 300 

CD O 2 
P P. 3 
a . B 

<J O I 

Steel.. 

•5 

•25 

34720 

90 000 

CD CD | 

11 Bessemer. 

•875 

( IO3 600 

51800 


•75 

{184 800 

O CD 


92 400 


( 

•5 

•17 

11950 

45000 

H ® S 

Wrought iron. < 

1 

.615 

82 870 

43 9°° 

O ^ <D 

( 

2 

1.06 

297 4OO 

44300 

l T-? % 


To Compute Power to Punch Iron, Brass, or Copper. 

Rule. —Multiply product of diameter of punch and thickness of metal by 
150000 if for wrought iron, by 128000 if for brass, and by 96000 if for 
cast iron or copper, and product will give power required, in lbs. 

Example.— What power is required to punch a hole .5 inch in diameter in a plate 
of brass.25 inch thick? , 5 X .25 X 128000 = 16000 lbs. 


Comparison between Detrusive and. Transverse 
Strengths. 

Assuming compression and abrasion of metal in application of a punch of 
one inch in diameter to extend to .125 of an inch beyond diameter of punch, 
comparative resistance of wrought iron to detrusive and transverse strain, 
latter estimated at 600 lbs. per sq. inch, for a bar 1 foot in length, is as 3 to 1. 


WOODS. 


Detrusive Strength of Woods. 


Lbs. 


Spruce.470 

Pine, white.490 


Lbs. 

Pine, pitch... 510 Ash. 

Hemlock.540 Chestnut. 


Per Sq. Inch. 

Lbs. 

650 Oak ... 
690 Locust. 


Lbs. 

780 

X180 


To Compute Bength of Surface of Resistance of Wood 
to Horizontal Thrust. 

Rule. —Divide 4 times horizontal thrust in lbs. by product of breadth of 
wood in ins., and detrusive resistance per sq. inch in lbs. in direction of fibre, 
and quotient will give length required. 

Example.— Thrust of a rafter is 5600 lbs., breadth of tie beam, of pitch or Georgia 
pine, is 6 ins.; what should be length of beyond score for rafter ? 

Assume strength 510 as above. Then 4X5600 _ 22 400 _ . 

3 6 X 510 3060 ' 3 



































STRENGTH OF MATERIALS.—DETRUSIVE. 


Shearing. 

Wrought Iron. 

Resistance to shearing of American is about 75 per cent., and of English 
80 per cent., of its tensile strength. 

Resistance to shearing of plates and bolts is not in a direct ratio. It ap¬ 
proximates to that of square of depth of former, and to square of diameter 
of latter. 

Results of Experiments upon Shearing Strength of 
Various JVIetals by ^Parallel Cutters. 

Wrought Iron .—Thickness from .5 to 1 inch, 50000 lbs. per sq. inch. 

Made by Inclined Cutters , angle = 7 0 . 


Plates. 

Thickness. 

Power. 

Bolts. 

Diam. 

Power. 

Brass . 

Ins. 

•05 

.207 

Lbs. 

549 
11 196 
14 930 

Brass. 

Ins. 

1. II 

. 77 ^ 

Lbs. 

29 700 

Copper. 

Copper. 

Steel. 

.24 

•51 

1 

Steel. 


28 720 
3°93 
35 4 zo 

Wrought iron .j 

39 I 5 ° 
44 800 

Wrought iron .j 

* / / j 

•32 

1.142 


Result of Experiments in Shearing, made at t J. S. INTavy- 
Yard, Washington, on Wrought-iron Bolts. 


Diam. 

Minimum. 

Stress. 

Maximum. 

Per Sq.Inch. 

Diam. 

Minimum. 

Stress. 

Maximum. 

Per Sq.Inch. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

•5 

8 900 

9 400 

44149 

• 8 75 

25 5°° 

27 600 

4i 503 

•75 

18 400 

19650 

39 553 

I 

32 9OO 

35 8o° 

40 708 


Mean 41033 lbs. 


Result of Experiments on .ST'B Inch Wrought-iron 
Bolts. ( E. Clark.) 


Lbs. 

Single shear.54096 

Double “ .46904 


Tons. 
24.15 
22.1 


Lbs. 

Double shear of two .625-inch plates 
riveted together (one section) .... 45 696 


Tensile strength.50176 lbs. 


Tons. 


20.4 


Riveted. Joints. 

Experiments on strength of riveted joints showed that while the plates 
were destroyed with a stress of 43 546 lbs., the rivets were strained by a 
stress of 39088 lbs. 

Cast Iron. 

Resistance to shearing is very nearly equal to its tensile strength. An 
average of English being 24 000 lbs. per sq. inch. 


Steel. 

Shearing strength of steel of all kinds (including Fagersta) is about 72 per 
cent, of its tensile strength. 

Treenails. 


Oak treenails, 1 to 1.75 ins. in diameter, have an average shearing strength 
of 1.8 tons per sq. inch, and in order to fully develop their strength, the planks 
into which they are driven should be 3 times their diameter. 

W oods. 

When a beam or any piece of wood is let in (not mortised) at an inclina¬ 
tion to another piece, so that thrust will bear in direction of fibres of beam 
that is cut, depth of cut at right angles to fibres should not be more than .2 
of length of piece, fibres of which, by their cohesion, resist thrust. 

Ash.650 lbs. I Deal. 625 lbs. I Pine.650 lbs. 

Chestnut.600 “ j Oak.2300 “ | Spruce.625 “ 














































STRENGTH OF MATERIALS.-TENSILE. 


• TENSILE STRENGTH. 

Tensile /Strength is resistance of the fibres or particles of a body to 
separation. It is therefore proportional to their number, or to area of 
its transverse section, and in metals it varies with their temperature, 
generally decreasing as temperature is increased. In silver, tenacity 
decreases more rapidly than temperature; and in copper, gold, and plat¬ 
inum less rapidly. 

Cast Iron. 

Experiments on Cast-iron bars give a tensile strength of from 4000 to 
5000 lbs. per sq. inch of its section, as just sufficient to balance elasticity of 
the metal ; and as a bar of it is extended the 12300th part of its length for 
every 1000 lbs. of direct strain, or one sixteenth of an inch in 64.06 feet per 
sq. inch of its section, it is deduced that its elasticity is fully excited when 
it is extended less than the 2400th part of its length, and extension of it at 
its limit of elasticity, which is about .5 of its destructive weight, is esti¬ 
mated at 1500th part of its length. 

Average ultimate extension is 500th part of its length. 

A bar will contract or expand .000006173 inch, or the 162000th of its 
length, for each degree of heat; and assuming extreme of moderate range 
of temperature in this country 140° (— 20° + 120°), it will contract or ex¬ 
pand with this change .0008642 inch, or the 1157th part of its length. 

It follows, then, that as 1000 lbs. will extend a bar the 12300th part of 
its length, contraction or extension for 1157th part will be equivalent to a 
force of 10648 lbs. (4.75 tons) per sq. inch of section. It shrinks in cooling 
from one eighty-fifth to one ninety-eighth of its length. 

Mean tensile strength of American, as determined by Maj. Wade for U. S. 
Ordnance Corps, is 31 829 lbs. (14.21 tons) per sq. inch of section; mean of 
English, as determined by Mr. E. Hodgkinson for Commission on Applica¬ 
tion of Iron to Railway Structures, 1849, is 19484 lbs. (8.7 tons) ; and by Col, 
Wilnrot, at Woolwich, in 1858, for gun-metal, is 23257 lbs. (10.35 tons), 
varying from 12320 lbs. (5.5 tons) to 25 520 lbs. (10.5 tons). 

Mean ultimate extension of four descriptions of English, as determined 
for Commission above referred to, was, for lengths of 10 feet, .1997 inch, 
being 600th part of its length; and this weight would compress a bar the 
775th part of its length. 

Tensile strength of strongest piece ever tested—45970 lbs. (20.52 tons). 
This was a mixture of grades 1, 2, and 3 from furnace of Robert P. Parrott 
at Greenwood, N. Y., and at 3d fusion. 

At 2.5 tons per sq. inch it will extend same as wrought iron at 5.6 tons. 


From experiments of Maj. Wade he deduced the following mean results : 


Density. 

Tensile. 

Transverse. 

Torsion. 

Crushing. 

Hardness. 

7.225 

31 829 

8182 

8614 

| 144916 1 

22.34 


Tensile per sq. inch of section; Transverse per sq. inch, one end fixed, 
load applied at other end at a distance of 1 foot; and Torsion per sq. inch, 
stress applied at end of a lever 1 foot in length. 

Green sand castings are 6 per cent, stronger than dry, and 30 per cent, 
stronger than chilled; but when castings are chilled and annealed, a gain of 
115 per cent, is attained over those inade in green sand. 

Resistance to crushing and tensile stress is for American as 4.55 to 1, and 
for English as 5.6 to 7 to 1. Strength increasing with density. 


STRENGTH OF MATERIALS.-TENSILE. 


785 


Remelting. —Strength, as well as density, are increased by repeated re¬ 
meltings. The increase is the result of the gradual abstraction of the con¬ 
stituent carbon of the iron, and the consequent approximation of the metal 
to wrought iron. 

Result of the 4th melting of pig iron, as determined by Major Wade, was 
to increase its strength from 12880 lbs. (5.75 tons) to 27888 lbs. (12.45 
tons), and its specific gravity from 6.9 to 7.4. 

Three successive meltings of Greenwood iron, N. Y., gave tensile strength 
of 21300,30100, and 35 700 lbs. 

Result of 5th melting by Mr. Bramwell was to increase strength of Acadian 
iron from i6’8oo lbs. (7.5 tons) to 41 440 lbs. (18.5 tons). 

Remelting increases its resistance to a crushing stress from 70 to 80 tons 
(14 per cent.) per sq. inch of section. 

Hot and. Cold Blast. 

Mr. Hodgkinson deduced from experiments that relative strength of 1.2 
and 3 ins. square was as 100, 8o, and 77, and that hot blast had less tensile 
strength than cold blast, but greater resistance to a crushing stress. 

Captain James ascertained that tensile strength of .75 inch bars, cut out 
of 2 and 3 inch bars, had only half strength of a bar cast 1 inch square. 

Mr. Robert Stephenson concluded, from experiments of recent date, that 
average strength of hot blast was not much less than that of cold blast; but 
that cold blast, or mixtures of cold blast, were more regular, and that mixt¬ 
ures of cold blast and hot blast were better than either separate. 


Stirling’s NLixed or Toughened Iron. 

By mixture of a portion of malleable iron with cast iron, carefully fused 
in a crucible, a tensile strain of 25 764 lbs. has been attained. This mixt¬ 
ure, when judiciously managed and duly proportioned, increases resistance 
of cast iron about one third; greatest effect being obtained with a propor¬ 
tion of about 30 per cent, of malleable iron. 


Malleable Cast Iron. 

Tensile strength of annealed malleable is guaranteed by some Manufact¬ 
urers of it at 56000 lbs.; it is capable of sustaining 22400 lbs. without per¬ 
manent set. 

‘W'rcmglit Iron. 


Experiments on English bars gave a tensile strength pf from 22 000 lbs. 
to 26 400 lbs. per sq. inch of its section, as just sufficient to balance elasticity 
of the metal; and as a bar of it is extended the 28000th part of its length 
for every 1000 lbs. of direct strain, or one sixteenth of an inch in 116.66 feet 
per sq. inch of its section, it is deduced that its elasticity is fully excited 
when it is extended the 1000th part of its length and extension of it at its 
limit of elasticity, which is from .45 to .5 of its destructive weight, is esti¬ 
mated at 1520th part of its length. 

A bar will expand or contract .000006614 inch, or 151 200 part of its length 
for each degree of heat; and assuming, as before stated for 
extreme range of temperature in air in this country is 140°, it will contract 
ofexpand with this change .000 926, or 1080th of its length, which is equiva¬ 
lent to a force of 20 740 lbs. (9.25 tons) per sq. inch of sec ion. 

Mean tensile strength of American bars and plates (45 ow to 76000), 
60500 lbs. (27 tons) per sq. inch of section; as determined by 1 
in 1836, is 55900 lbs ; and mean of English, as determined by Capt. Brown, 
Barlow, Brunei, and Fairbairn, is 53 900 lbs.; and by Mr. Kirkaldy, bars and 
plates (47040 to 55 9x0) 51475 lbs; (22.97 tons). 

3 U* 


•TENSILE. 


; 86 


STRENGTH OF MATERIALS.- 


Greatest strength observed 73 449 lbs. (32.79 tons). 

Ultimate strength, as given by Mr. I). K. Clark, 59 732 lbs. (26.66 tons). 

Average ultimate extension is 600th part of its length. 

Strength of plates, as determined by Sir William Fairbairn, is fully 9 per 
cent, greater with fibre than across it. 

Resistance of wrought iron to crushing and tensile strains is, as a mean, 
as 1.5 to 1 for American; and for English 1.2 to 1. 

Reheating. —Experiments to determine results from repeated heating and 
laminating, furnished following: 

From 1 to 6 reheatings and rollings, tensile stress increased from 43 904 
lbs. to 61 824 lbs., and from 6 to 12 it was reduced to 43904 again. 

Effect of Temperature. — Tensile strength at different temperatures is as 
follows: 6o°, 1; 114 0 , 1.14; 212°, 1.2; 250°, 1.32; 270°, 1.35; 325 0 , 1.41; 
435 °, 1 - 4 - 

Experiments of Franklin Institute gave at 

8o°.56000 lbs. I 720 0 .55 000 lbs. I 1240 0 .22 000 lbs. 

570 0 .66500 “ | 1050 0 .32000 “ I 1317 0 . 9000 “ 

Annealing. —Tensile strength is reduced fully 1 ton per sq. inch by an¬ 
nealing. 

Cold Rolling. —Bars are materially stronger than when hot rolled, strength 
being increased from one fifth to one half, and elongation reduced from 21 
to 8 per cent. 

Hammering increases strength in some cases to one fifth. 

Welding. —Strength is reduced from a range of 3 to 44 per cent. 20 per 
cent., or one fifth, is held to be a fair mean. 

Temperature. —From o° to 400° strength is not essentially affected, but at 
high temperature it is reduced. When heated to redness its strength is re¬ 
duced fully 25 per cent. 

Tensile strength at 23 0 was found to be .024 per cent, less than at 64°. 

Cutting Screw Threads reduces strength from 11 to 33 per cent. 

Hardening in water, oil, etc., reduces elongation, but does not essentially 
increase the strength. 

Case Hardening reduces strength fully 10 per cent. 

Galvanizing does not affect strength of plates. 

Angled Bars, etc. —Their strength is fully 10 per cent, less than for bolts 
and plates. 


Elements connected -with. Tensile Resistance of* various 

Substances. 




Substances. 

Stress per Sq. 

I Inch for limit 

of Elasticity. 

Ratio of Stress 
to that causing 
Rupture. 

Substances. 

Stress per Sq. 
Inch for limit 
of Elasticity. 


Lbs. 

3 355 

4 000 

5 000 
2856 

52 000 
75700 

3 33 2 


Wrought iron, ordinary.... 

Lbs. 

17 600 



“ “ American. 

Oak. 

.2 

•23 

“ “ English. 

( 18 850 


“ American 

15 000 
47 532 
36300 

“ wire. 

Yellow pine. 

•5 

•23 

li wire, No. 9, unannealed 
“ “ " annealed.. 




Turning .—Removing outer surface does not reduce the strength of bolts. 



























STRENGTH OF MATERIALS.-TENSILE. 


787 


TIE-RODS. 


IR.esu.lts of Experiments on Tensile Strength, of Wrouglit— 

iron Tie-rods. 

Common English Iron, 1.1875 Ins. in Diameter. 


Description op Connection. 


Semicircular hook fitted to a circular and welded eye. 

Two semicircular hooks hooked together. 

Right-angled hook or goose-neck fitted into a cylindrical eye 

Two links or welded eyes connected together.. 

Straight rod without any connective articulation. 


Breaking Weight. 

Lbs. 

14000 
16 220 
29 120 
48 160 
56 000 


Ratio of Ductility and IVIalleaDility of NEetals. 


In order of 
Wire-drawing 
Ductility. 

In order of 
Laminable 
Ductility. 

In order of 
Wire-drawing 
Ductility. 

In order of 
Laminable 
Ductility. 

In order of 
Wire-drawing 
Ductility. 

In order of 
Laminable 
Ductility. 

Gold. 

Silver. 

Platinum. 

Iron. 

Copper. 

Zinc. 

Tin. 

Lead. 

Nickel. 

Gold. 

Silver. 

Copper. 

Tin. 

Platinum. 

Lead. 

Zinc. 

Iron. 

Nickel. 


Relative resistance of Wrought Iron and Copper to tension and compres¬ 
sion is as 100 to 54.5. 


Steel. 


Experiments of Mr. Kirkaldy, 1858-61, give an average tensile strength 
for bars of 134400 lbs. (60 tons) per sq. inch for tool-steel, and 62720 lbs. 
(28 tons) for puddled. Greatest observed strength being 148 288 lbs. (66.2 
tons). Plates, mean, 86800 lbs. (32 to 45.5 tons) with fibre, and 81 760 lbs. 
(36.5 tons) across it. 

Its resistance to crushing compared to tension is as 2.1 to 1. 

Hardening .—Its strength is very materially increased by being cooled in 
oil, ranging from 12 to 55 per cent. 

Crucible. —Experiments by the Steel Committee of Society of Civil 
Engineers, England, 1868-70, give a tensile strength of 91 571 lbs. per sq. 
inch (40.88 tons), with an elongation of .163 per cent., or 1 part in 613, and 
an elastic extension of .0000347th part for every 1000 lbs. per sq. inch, or 
1 part in 28818. 

Bessemer. —Experiments by same Committee give a tensile strength 
of 76653 lbs. per sq. inch (34.22 tons) with an elongation of .144 per cent., 
or 1 part in 695, and an elastic extension of .oooo34 82d part for every 1000 
lbs. per sq. inch, or 1 part in 28 719. 


Result of Experiments by Committe of Society of Civil 
Engineers of England, 18 G 8 -' 70 , and NEr. Daniel Kir- 
Raldy, lST'G. 

Per Sq. Inch. 


Steel. 

Elastic Strength. 

Elastic E 

in Parts 
of 

Length. 

xtension 

per 

1000 Lbs. 

Ratio of 
Elastic 
to 

Ultimate 

Strength. 

Destructive 

Weight. 


Lbs. 

Tons. 

Per Cent. 

In Length. 

Per Cent. 

Lbs. 

Tons. 

Crucible. 

49 840 

22.25 

.225 

.0005 

58.2 

86 464 

38.6 

Bessemer. 

44 800 

20 

.204 

.00045 

59 

75 757 

33'82 

Fagersta, unannealed . 

48 608 

21.7 

— 

— 

59- 2 

78 176 

34-9 

“ annealed.... 

200 

17-5 

— 

— 

5i-5 

72 576 

32.4 

Siemens, unannealed.. 

32 080 

14.56 

— 

— 

46.4 

69 888 

31.2 

“ annealed.... 

28 784 

12.85 

— 

— 

44.4 

64 512 

28.8 




































STRENGTH OF MATERIALS.-TENSILE 


Average Tensile Elasticity of Steel Bars and. Blates. 

[Com. of Civil Engineers, 1870.) 


Description. 

Elasticity per 
Sq. Inch. 

Elastic Exten¬ 
sion in Parts of 
Length. 

Ratio ofElas 
tic to Destruc 
tive Strength 

Bars. 

Lbs. 

Parts. 

Per Cent. 

Crucible, hammered and rolled. 

Bessemer, “ “ . 

50 557 
43814 

i in 4^5 

1 in 675 

50.2 

55 

Fagersta, rolled. 

56 560 

— 

64.8 

“ unannealed. 

34048 

— 

55-6 

“ hammered and rolled. 

55 574 

— 

64.7 

“ “ “ annealed. 

40858 

— 

54 

“ plates, unannealed. 

30710 

1 in 980 

59 - 2 

“ “ annealed. 

26 940 

1 in 1020 

56.5 

Siemens, “ unannealed. 

32 500 

— 

46.4 

“ “ annealed. 

28 780 

— 

44.4 

“ tires. 

40174 

— 

58.8 

Krupp’s shaft. 

42 112 

1 in 185 

— 


Tensile strength of steel increases by reheating and rolling up to second 
operation, but decreases after that. 


Tensile Strength, of Various Materials, deduced from 
Experiments of II. S. Ordnance Department, Fair- 
bairn, Hodgkinson, Kirkaldy, and by the Author. 

Power or Weight required to tear asunder One Sq. Inch, in Lbs. 

Metals. Lbs. 

Steel, Pittsburgh, moan. 94450 


Lbs. 


u 

11 

u 


Metals. 

Antimony, cast. 1053 

Bismuth, cast. 3248 

Cast Iron, Greenwood. 45970 

mean, Major Wade... 31 829 

gun-metal, mean. 37 232 

malleable, annealed.. 56000 

“ Eng., strong. 29000 

“ “ weak. 13400 

“ “ averages.I 15 ^°° 

0 l 21280 

“ “ gun-metal. 23257 

“ “ mean*. 19484 

“ “ Low Moor, No. 2 14076 

“ “ Clyde, No. 1_ 16125 

“ “ “ No. 3.... 23468 

“ “ Stirling, mean.. 25764 

Copper, wrought. 34000 

“ rolled. 36000 

“ cast. 24250 

“ bolt. 36800 

“ wire. 61 200 

Gold.. 20384 

Lead, cast. 1 800 


pipe. 


encased. 


. 2 240 

. 3 759 

“ rolled sheet. 3320 

Platinum wire. 53000 

Silver, cast. 40000 

Steel, cast, maximum.142000 

“ mean. 88560 

puddled, maximum. 173817 

Amer. Tool Co.179980 

! 210000 
300000 

plates, lengthwise. 96 300 

“ crosswise.. 93700 

Chrome bar.180000 


U 

u 


Whitworth’s.j 

Siemens’s plates.. j 


Bessemer, rolled.•! 76 ^ 5 ° 

( 125000 

“ hammered.152900 

Eng., cast.134000 

“ “ plates, mean.... 93500 

“ plates. 86800 

“ puddled plates. 62720 

“ crucible. 91 570 

“ homogeneous. 96280 

“ blistered, bars.104000 

“ Fagersta bars. 89600 

“ “ plates. 98560 

89 600 
152 000 
64 900 
69 880 

“ “ Krupp’s shaft.'. 92243 

Tin, cast. 5000 

“ Banca. 2100 

Wire rope, per lb. w’t per fathom 4480 
“ “ galvanized steel, “ - 6 720 

Wrought Iron, boiler plates.. . j 45 5°o 

“ rivets. 65000 

“ bolts, mean. 

“ “ inferior 30000 

“ hammered. 54000 

“ shaft. 44750 

“ wire. 73600 

“ No. 9.100000 

“ No. 20.120000 

“ “ diam. .0069 inch 301168 

“ “ galv’ized .058 “ 64960 

“ Eng., heavy forging. 33600 

“ “ plates, lengthw’e 53800 

“ “ “ crosswise 48800 


* By Comm’s on application of Iron to Railway Structure. 











































































STRENGTH OP MATERIALS.—TENSILE. 


7 89 


Metals. Lbs. 

Wrought Iron, Eng., mean.51000 

“ Eng., Low Moor.57600 

“ “ Lancashire.48800 

“ “ Thames.65920 

“ “ armor-plates_40000 

“ “ bar.. f 313 °° 

( 50 000 

“ “ charcoal.63000 

“ “ rivet, scrap.5x760 

“ Russian, bar, best.... 59 500 

“ “ “ 49000 

“ Swedish, “ best.... 72 000 

“ “ “ 48900 

Zinc. 3 500 

“ sheet.'.. j T 000 

( 10000 

Alloys or Compositions. 


Alloy,Cop.6o,Iron 2,Zinc 35,Tin2. 85 120 

“ Tin 10, Antimony 1.uoco 

Aluminium, Cop. 90.71 6co 

“ maximum.96320 

Bell-metal. 3 670 

Brass, cast.18000 

“ wire.49000 

Bronze, Phosphor., extreme.50915 

“ “ mean.34464 


“ ordinary. 

“ Cop. 10, Tin 1., 

“ “ 9, “ 1., 

“ “ 8, “ x. , 

“ “ 2, Zinc 1 

Gun-metal, ordinary.. 

“ mean. 

“ bars. 

Speculum metal_ 

Yellow metal. 


23500 
33 000 
38 080 
36 000 
29 000 
18 000 
33600 
42 040 
7 000 
48 700 


Woods. 

Ash, white. 

“ American. 

“ English. 

Bamboo. 

Bay. 

Beech, English. 

Birch. 

“ Amer., black. 

Box, African. 

Bullet. 

Cedar, Lebanon. 

“ West Indian. 

“ American.. 

Chestnut.. 

“ horse.. 

Cypress.. 

Deal, Christiana. 

Ebony. 

Elm. 

Gum, blue. 

“ Alabama. 

Hackmatack. 

Hickory . 

Holly. 

Lance . 


! 


I 


14 000 
9500 
16 000 

6 300 

14 000 
11 500 

15 000 

7 000 
23 000 
19 000 

11 400 
7 5 oo 

xi 600 

12 500 
10000 

6 000 

12 400 
27 000 

6 000 

13 000 
18 000 

15 860 
12 oco 
11 000 

16 000 

17 350 
23 000 


Woods. 

Larch. 

Lignum vitae. 

Locust. 

Mahogany, Honduras.. 

“ Spanish. 

Oak, Pa., seasoned. 

“ Va., “ . 

“ white. 

“ live, Ala.. 

“ red.. 

“ African.. 

“ English. 

“ Dantzic.. 

Pear.... 

Pine, Va. 

“ Riga.. 

“ yellow.. 

“ white.. 

“ red. 

Poon.. 

Poplar.. 

Redwood, Cal. 

Spruce, white.. 

Sycamore.. 

Teak, India.. 

“ African. 

Walnut, Eng.. 

“ black. 

“ Mich. 

Willow. 

Yew. 

Across Fibre. 

Oak. 

Pine.. 


( 

1 


I 


{ 


! 

I 


Miscellaneous. 
Basalt, Scotch. 


Beton, N. Y. Stone Con’g Co.... j 


Blue stone. 
Brick, extreme. 

“ inferior.. 


! 


Cement, Portland, 7 days. 


“ pure, 1 mo. 

sand 2, 320 days 

U U J u 

“ pure, “ 

“ sand 1, in water 
1 mo 

(( U j j y’r 

“ “ 3, I year.. 

< 4 4 4 _ T 44 

“ “ 7,1 “ .. 

Hydraulic. 

Rosedale,Ulst. Co., 7 days 
“ sand i, 30 “ 


9 mos. 


Lbs. 

4 2 CO 
9 5 co 

11 800 
16000 

20 500 
21000 

8 000 

12 000 
20333 
25 222 
16 500 

16 380 

10 250 
9500 
4 5 oo 

7 57i 
4 200 

9 860 
19 200 
14000 

13 oco 

11 8co 
13 oco 
13 300 

7 000 
10833 
10 290 

12 400 
9 600 

13 000 
15 000 

21 000 

7 800 
16633 

17 580 

13 000 

8 000 

2 300 
55 o 

1469 
300 
500 
77 
75o 
100 
290 
400 
860 
393 
7i3 
948 
1 152 

201 

3i9 

310 

2I 4 

163 

284 

104 

102 

560 

700 






































































































790 


STRENGTH OF MATERIALS.-TORSION. 


Miscellaneous. 

Cement, Roman, in water 7 days 
“ “ “ 1 mo,, 

u u “ j year 

“ “ sand 1,42 days, 

a a a 2 a 

a a a „ a 

O) ? 

Flax. 

Glass, crown. 

Glue. 

Granite. 

Gutta Percha. 


Hemp rope. 

Ivory . 

Leather belting.. 

Limestone. 

Marble, statuary. 

“ Italian.. 
Marble, white ... 
“ Irish.... 


Lbs. 

90 
115 
286 
284 
199 
160 
25 000 
2546 
4 000 
57 8 
3 5 oo 
12 000 

16 000 

1 000 
330 
670 

2 800 

3 200 
5200 
9 000 

17 600 


Miscellaneous. 
Mortar, 1 year. 

“ hydraulic. 

“ 'ordinary_•.. 

Oxhide. 

Rope, Manila. 

“ tarred hemp. 

Sandstone. 

“ fine green.. 

‘ ‘ Arbroath. 

“ Caithness__ 

“ Portland. 

“ Craigleth. 

Silk fibre. 

Slate.. 

Whalebone. 


Lbs. 

60 

150 

S 5 

13° 

35 

6 300 
9 000 

15 000 

150 

1 260 

563 

x 261 

473 

1054 

8 57 
1 000 

453 
52 000 
9 600 
12 800 

7 000 


TORSIONAL STRENGTH. 


SHAFTS AND GUDGEONS. 

Shafts are divided into Shafts and Spindles , according to their mag¬ 
nitude, and are subjected to Torsion and Lateral Stress combined, or to 
Lateral Stress alone. 

A Gudgeon is the metal journal or Arbor upon which a wooden shaft 
revolves. 


Lateral Stiffness and Strength. —Shafts of equal length have lateral stiff¬ 
ness as their breadth and cube of their depth, and have lateral strength as 
their breadth and square of their depths. 

Shafts of different lengths have lateral stiffness directly as their breadth 
and cube of their depth, and inversely as cube of their length; and have 
lateral strength directly as their breadth and as square of their depth, and 
inversely as their length. 

Hollow Shafts having equal lengths and equal quantities of material have 
lateral stiffness as square of their diameter, and have lateral strength as their 
diameters. Hence, in hollow shafts, one having twice the diameter of an¬ 
other will have four times the stiffness, and but double the strength; and 
when having equal lengths, by an increase in diameter they increase in stiff¬ 
ness in a greater proportion than in strength. 

When a solid shaft is subjected to torsional stress, its centre is a neutral 
axis, about which both intensity and leverage of resistance increase as radius 
or side; and the two in combination, or moment of resistance per sq. inch, 
increase as square of radius or side. 

Round Shaft. —Radius of ring of resistance is radius of gyration of sec¬ 
tion, being alike to that of a circular plate revolving on its axis, viz., .7071 
radius. The ultimate moment of resistance then is expressed by product 
of sectional area of shaft, by ultimate shearing resistance per sq. inch of 
material by radius, and by .7071. 


Or, .7854 d 2 r S X -7071 = .278 d ! S = RW. (D. K. Clark.) 
d representing diameter of shaft and r radius , S ultimate shearing stress of mate¬ 
rial in lbs. per sq. inch, R radius through which stress is applied , in ins., and W 


moment of load or destructive stress, in lbs. 


Hence, 


.278 d 3 S 


R W _ , , /R W 

= W; ^8^ = S; and V“S _Xl,534==d 

































STRENGTH OF MATERIALS.-TORSION. 


79 1 


Round Shaft. —Strength, compared to a square of equal sectional area, 
is about as i to .85. Diameter of a round section, compared to side of 
square section of equal resistance, is as 1 to .96. 


Square Shaft .—Moment of torsional resistance of a square shaft exceeds 
that of a round of same sectional area., in consequence of projection of cor¬ 
ners of square; but inasmuch as material is less disposed to resist torsional 
stress, the resistance of a square shaft, compared to a round one of like area 
of section, is as 1 to 1.18, and of like side and diameter, as 1.08 to 1. 


Hence. 


.278 x 1.08 s 3 S 
R 


— W. Hollow Round Shafts. 


.278 (d* — d' i) S 

1fd 


:W. 


When Section is comparatively Thin. 


1.57 d 3 t S 
it 


= W. s representing side, 


d and d' external and internal diameters, and t thickness of metal in ins. 


Torsional Angle of a bar, etc., under equal stress, will vary as its length. 
Hence, torsional strength of bars of like diameters is inversely as their 
lengths. 

Stress upon a shaft from a weight upon it is proportional to product of the parts 
of shaft multiplied into each other. Thus, if a shaft is 10 feet in length, and a weight 
upon centre of gravity of the stress is at a point 2 feet from one end, the parts 2 
and 8, multiplied together, are equal to 16; but if weight or stress were applied in 
middle of the shaft, parts 5 and 5, multiplied together, would produce 25. 

When load upon a shaft is uniformly distributed over any part of it, it is consid¬ 
ered as united in middle of that part; and if load is not uniformly distributed, it is 
considered as united at its centre of gravity. 

Deflection of a shaft produced by a load which is uniformly distributed over its 
length is same as when .625 of load is applied at middle of its length. 

Resistance of body of a shaft to lateral stress is as its breadth and square 
of its depth; hence diameter will be as product of length of it, and length 
of it on one side of a given point , less square of that length. 

Illustration. —Length of a shaft between centres of its journals is 10 feet; what 
should be relative cubes of its diameters when load is applied at 1, 2, and 5 feet 
from one end? and what when load is uniformly distributed over length of it? 

I X l 1 — I 3 = d 3 \ and when uniformly distributed, < 33 - 4 - 2 —d 1 . 

10 X 1 = 10 — i 2 — g = cube of diameter at 1 foot ; 10X2 = 20 — 2 2 = 16 = cube 
of diameter at 2 feet ; 10 X 5 — 50 — 5 2 = 25 =cube of diameter at 5 feet. 

When a load is uniformly distributed, stress is greatest at middle of length, and 
is equal to half of it; 25 -1- 2 = 12.5 z= cube of diameter at 5 feet. 


Torsional Strength of any square bar or beam is as cube of its side, and 
of a cylinder as cube of its diameter. Hollow cylinders or shafts have great¬ 
er torsional strength than solid ones containing same volume of material. 


To Compute Diameter of a Solid. SHaft of Cast or 
"Wrong-lit Iron to Idesist Lateral Stress alone. 

When Stress is in or near Middle. Rule. —Multiply weight by length of 
shaft in feet; divide product by 500 for cast iron and 560 for wrought iron, 
and cube root of quotient will give diameter in ins. 

Example. —W'eight of a water-wheel upon a cast-iron shaft is 50000 lbs., its length 
30 feet, and centre of stress of wheel 7 feet from one end; what should be diameter 
of its body? 

3 / ( 5 ° 000 X 3 °\ _ ,. , 2 i ns if weight was in middle of its length. 

V \ 500 / 

Hence diameter art 7 feet from one end will be, as by preceding Rule, 30 x 7 — 
7 2 = i6i = relative cube of diameter at 7 feet; 30 X 15 — i 5 2 = 23 S — relative cube 
of diameter at 15 feet, or at middle of its length. 

Then, as -^225 : 14.42 :: -^161 : 12.91 ins., diameter of shaft at 7 feet from one end. 






792 


STRENGTH OF MATERIALS.-TORSION. 


For Bronze, 420; Cast steel, 1000 to 1500; and Puddled steel, 500. 

When Stress is uniformly laid along Length of Shaft. Rule. — Divide 
cube root of product of weight and length by 9.3 for Cast iron and 10.6 
for Wrought iron, and quotient will give diameter in ins. 

.... ,• ^50000X30 

Example. —Apply rule to preceding case. - = 12.31 ms. 

9-3 

For Bronze, 8.5 ; Cast steel, 18.6 to 27.9; and Puddled steel, 9.3. 

When Diameter for Stress applied in Middle is given . Rule. —Take cube 
root of .625 of cube of diameter, and this root will give diameter required. 

Example. — Diameter of a shaft when stress is uniformly applied along its length 
is 14.42 ins. ; what should be its diameter, stress being applied iu middle? 

V .625 X 14 42 3 = ^.625 X 3000 = 12.33 ins. 

To Compute Diameter of a. Solid. Shaft of Cast Iron to 
Resist its "Weight alone. 

Rule. —Multiply cube of its length by .007, and square root of product 
will give diameter in ins. 

Example.— Length of a shaft is 30 feet; what should be its diameter in body? 
V( 3 o 3 X .007) = V i8 9 = * 3-75 ins 

HOLLOW SHAFTS. 


To Compute Diameter of a Hollow Shaft of Cast Iron 
to Sustain its Load in Addition to its Weight. 

When Stress is in or near Middle. Rule. —Divide continued product of 
.012 times cube of length, and number of times weight of shaft in lbs., by 
square of internal diameter added to 1, and twice square root of quotient 
added to internal diameter will give whole diameter in ins. 

Example.— Weight of a water-wheel upon a hollow shaft 30 feet in length is 2.5 
times its own weight, and internal diameter is 9 ins.; what should be whole diam¬ 
eter of shaft? 

.012 X 30 3 X 2. 


s/(~ 


I + 9 : 


—^ -f- 9 = 2 J -— = 6.28 ins ., and 6.28 -j- 9 = 15-28 ins. 


To 


to 


Or, 


Compute Diameter of a Round or Square Shaft 
Resist Combined Stress of Torsion and "Weight. 

Rule. —Multiply extreme of pressure upon crank-pin, or at pitch-line of 
pinion, or at centre of effect upon the blades of a water-wheel, etc., that a 
shaft may at any time be subjected to; by length of crank or radius of 
wheel, etc., in feet; divide the product by Coefficient in following Table, and 
cube root of quotient will give diameter of shaft or its journal in ins. 

3/™-d 

v C 

Example.— What should be diameter for journal of a wrought-iron water-wheel 
shaft, extreme pressure upon crank-pin being 59400 lbs., and crank 5 feet in length? 
/59 400 X 5 

= 2475, and V2475 = 13.53 ins. 


When T100 Shafts are used , as in Steam-vessels, etc., tenth One Engine. 
Rule. —Divide three times cube of diameter for one shaft by four,' and 
cube root of quotient will give diameter of shaft in ins. 

or ’vV =<i ' 

Example.—A rea of journal of a shaft is 113 ins.; what should be diameter, two 
shafts being used ? 

0 ^ 12*^ 

Diameter for area of 113 = 12. Then -:= 1296, and '-\/i2g6 —10.9 ins. 


4 









STRENGTH OF MATERIALS.-TORSION. 


793 


Torsional Strength, of Various HVIetals. 

(Maj. Wm. Wade , TJ. S. Ordnance Corps, 1851, Steel Committee [ England , 1868], and 
Stevens Institute , N. J ., 1878.) 

Reduced to a Uniform Measure of One Inch in Diameter or Side. 

Stress applied at One Foot from Axis of Body and at Face of Axis. 


♦ 


Destructive Stress 

Torsional 

Coefficient ^= W. 

Bars and Metals. 

Tensile 

Strength. 

at 

Computed 

at 

Strength 

WR 

2 

S 

10 

15 

20 




12 Ins. 

d 3 x ‘ 

Ins. 

Ins. 

Ins. 

IllS. 

Ins. 

Cast Iron. 
\Diam. H ins - ) 

Lbs. 

Lbs. 

Lbs. 

1082 







i ( .65 m. 1 

Is Area 1 sq. inch ) 

45 000 

520 

492 

IOO 

95 

90 

85 

80 

Diam. {3^5 in?..) 

u 

3800 








(2.61 “ > 

79°4 

230 

45 

40 

35 

30 

25 

Area 2.97 sq. ins. ) 





^ Diam. ) Least... 

9 000 

1550 

3664 

53o) 






J§ = 1.9 f Mean ... 

31829 

2145 

4462 

650 

130 

125 

120 

115 

IIO 

* ins. ) Greatost. 

45000 

2840 

59°7 

850) 



q Side 1 inch .... j 



728 

728 






| Area 1 sq. inch j 


350 

125 

120 

115 

no 

105 

Wrought Iron. 










- Diam. f Least ... 

38027 

1250 

2600 

37 6 ) 






f| = 1.9 ' Mean.... 

56 300 

1375 

2860 

416 ( 

120 

115 

no 

105 

IOO 

ins. (Greatest. 
Area 2.83 sq. ins. 

74592 

1500 

3120 

452) 






Bronze. 










Diam.= (Least.... 

17 698 

500 

1040 

152 

30 

28 

26 

_ 

_ 

1.9 ins. (Greatest. 

56 786 

650 

1352 

197 

38 

36 

34 

— 

— 

Area 2.83 sq. ins. 






Cast Steel. 
Diam.= (Least_ 

42 000 

2600 

5408 

788 

160 

155 

550 



1.9 ins. I Greatest. 

128 000 

7760 

16140 

2353 

475 

470 

465 

— 

— 

Area 2.83 sq. ins. 







Bessemer Steel. 










Diam. = i.382 ins. ) 
Area 1.5 sq. ins.) 

36 960 

1568 

3261 

1236 

245 

240 

235 

230 

225 


To Compute Diameter of Shafts of Oak and. Pine. 

Multiply diameter ascertained for Cast Iron as follows: Oak by 1.83, 
Yellow Pine by 1.716. 

IVTetals and. Woods. 

Ultimate Torsional Strength. —Of Cast Iron may be taken as equal to its 
transverse strength for American and .9 for English, or as .26 of its tensile 
strength for American and .23 for English. Of Wrought Iron, as .7 to .8 of 
its transverse strength for American and .7 to 1 for English, and of Steel, as 
.72 of its tensile strength. 

Elastic Torsional Strength. —Of Cast Iron may be taken as equal to its 
transverse strength, of Wrought Iron 40 per cent, of its ultimate torsional 
strength, of Steel 44 per cent, of its tensile strength, and 45 per cent, of its 
ultimate torsional strength. 

Bessemer Steel. —Has a torsional strength of 6670 lbs. per sq. inch at a ra¬ 
dius of one foot, being somewhat less than that of Cast Iron, Fagersta has 50 
per cent, of its ultimate transverse strength, and Siemens 44.5 per cent, of 
its ultimate tensile. v 

3 x 















794 


STRENGTH OF MATERIALS.-TORSION". 


Note.— Examples here given are deduced from instances of successful practice; 
where diameter has been less, fracture has almost universally taken place, stress 
being increased beyond ordinary limit. 

2.—When shafts of less diameter than 12 ins. are required, Coefficients here given 
may be slightly reduced or increased, according to quality of the metal and diame¬ 
ter of shaft; but when they exceed this diameter, Coefficients may not be increased, 
as strength of a shaft decreases very materially as its diameter increases. • 

Order of shafts, with reference to degree of torsional stress to which they 
may be subjected, is as follows: 

1. Fly-wheel. | 2. Water-wheel. | 3. Secondary shaft. | 4. Tertiary, etc. 

Hence, diameters of their journals may be reduced in this order. 

To Compute Diameter of a Wrought-iron Centre Shaft 
for connecting Two Engines at a Eight ./Angle. 

Conditions of such a shaft are as follows: 

Greatest stress that it is subjected to is when leading engine is at .75 of 
its stroke, and following engine .25 of its stroke; hence, position of each 
crank is as sin. 22 0 30' X 2 = -7071 of length of crank or radius of power. 

/2 P.707 R 


Consequently. 


v- 


125 


: d. P representing extreme pressure on piston. 


Note.— In computing P it is necessary to take very extreme pressure that piston 
may be subjected to, however short the period of time. Average pressure does not 
meet requirement of case. 

Illustration.— Extreme pressure upon each piston of two engines connected at 
a right angle was m 592 lbs., and stroke of pistons 10 feet; what should have been 
diameter of centre shaft ? and what of each wheel or driving shaft ? 


*/(- 


592 X 2 X 


•707 yy 


125 




'7 88 955 
125 


= 18.48 ins. centre shaft. 


For ordinary mill purposes, driving shafts should be as cube roots of .25 of 3 
times cube of centre shaft. ;o iC3v , 

o x 3 , 

5 1 - -= 16.79 ms - 


Thus 


•25 

To Compute Torsional Strength of Hollow Shafts and. 

Cylinders. 

Rule.—F rom fourth power of exterior diameter subtract fourth power of 
interior diameter, and multiply remainder by Coefficient of material; divide 
this product by product of exterior diameter and length or distance from axis 
at which stress is applied in feet, and quotient will give resistance in lbs. 

di—d'i C „ 

0r > <n =R 

Example.— What torsional stress may be borne by a hollow cast-iron shaft, hav¬ 
ing diameters of 3 and 2 ins., power being applied at one foot from its axis? 

0 = 130. 3 4 — 2 4 X 130 = 8450, which-r- 3 X 1 = ^^ = 2816.6 lbs. 


To Compute Torsional Strength of Round and Square 

Shafts. 

Rule.— Multiply Coefficient in preceding Table by cube of side or of 
diameter of shaft, etc., and divide product by distance from axis at which 
stress is applied in feet; quotient will give resistance in lbs. 

Illustration.— What torsional stress may be borne by a cast-iron shaft of best 
material, 2 ins. in diameter, power applied at 2 feet from its axis. 

C from table = 130. — 0 X 23 = — 520 lbs. 

2 2 

For steamers, when from heeling of vessel or roughness of sea the stress may be 
confined to one wheel alone, diameter of journal of its shaft should be equal to 
that of centre shaft. 








STRENGTH OF MATERIALS.—TORSION - . 


795 


GUDGEONS. 

To Compute Diameter of a Single Gudgeon of Cast 
Iron, to Support a given 'Weiglxt or Stress. 

Rule.—D ivide square root of weight in lbs. by 25 for Cast iron, and 26 
for Wrought iron, and quotient will give diameter in ins. 

Example.— Weight upon a gudgeon of a cast-iron water-wheel shaft is 62 500 lbs.; 
what should he its diameter ? 

• -v/62500 250 

- - - -— — 10 ms. 

25 25 

To Compute Diameter of Two Gudgeons of Cast Iron, 
to Support a given Stress or 'Weiglxt. 

Rule.—P roceed as for two shafts, page 792. 


To Compute Ultimate Torsional Strength, of Round and. 
Square Shafts. (D. K. Clark.) 


Cast Iron. Round. 
.4 S 3 S 


.278 d 3 S 
R ' 


= W; 1.534,? 


Square. 


. , , , W R” 

W, and 1.36 3 / —— = s. 

o 


WB , , WR 

—— = and ——— = S. 

S ’ 278 d 3 


Hollow. 


.278 (cG — d'*) s 


: W. 


R ’ J V -S Rd 

S representing ultimate shearing strength , and W moment of load , both in lbs ., s side 
of square shaft, and R radius of stress, both in ins. 

Illustration. — What is ultimate torsional strength of a round cast-iron shaft 
4 ins. in diameter, stress applied at 5 feet from its axis ? 


Assume S = 20 000 lbs. 


.278 X 4 3 X 20000 
Then ——---= 5930 lbs. 


5 X 12 

By experiments of Major Wade, ordinary foundry iron has a torsional strength 
of 7725 lbs., or 644 lbs. per sq. inch at radius of one foot. 

Then 7725 X 48 = 8240 lbs. 

5 X 12 

.2224 d 3 S a 

—• W. Square. 


Thus, take preceding illustration. 


Wrought Iron. Round. 


W. 


When Torsional Strength per sq. inch for radius of 1 inch is ascertained, 
substitute C for .278, .4, .2224, or .32. 

Stress which will give a bar a permanent set of .5° is about .7 of that 
which will break it, and this proportion is quite uniform, even when strength 
of material may vary essentially. 

Wrought Iron, compared with Cast Iron, has equal strength under a stress 
which does not produce a permanent set, but this set commences under a less 
force in wrought iron than cast, and progresses more rapidly thereafter. 
Strongest bar of wrought iron acquired a permanent set under a less strain 
than a cast-iron bar of lowest grade. 

Strongest bars give longest fractures. 


Steel. Round. 


• 2 d 3 s __ When S is not known, substitute for 

R ~ ‘ S 72 s = 72 per cent, of tensile strength. 


Torsional Strength of Cast Steel is from 2 to 3 times that of Cast Iron. 

Following rules are purposed to apply in all instances to diameters of 
journals of shafts, or to diameter or side of bearings of beams, etc., where 
length of journal or distance upon which strain bears does not greatly ex¬ 
ceed diameter of journal or side of beam, etc.; hence, when length or distance 
is greatly increased, diameter or side must be correspondingly increased. 

Coefficients for torsional breaking stress of Iron, Bronze, and Steel, as de¬ 
termined by Major Wade, are: Wrought Iron, 640; Cast Iron, 560 ; Bronze, 
460; Cast Steel, 1120 to 1680. Puddled Steel does not differ essentially from 
that of cast iron. 










79 6 


STRENGTH OF MATERIALS.-TORSION. 


Formulas for Minimum ayul Maximum Flam, of Wrought-iron Shafts. 

(A. E. Seaton, London, 1883, and Board of Trade, Eng.) 


Compound Engines. 




D 2 p d 2 15 


S = diameter. D and d representing diam¬ 


eter of low and high pressure cylinders, and S half stroke, all in ins. , p pressure of 
steam in boiler, in lbs. per sq. inch, and C a coefficient, as follows : 


Angle 

of 

Crank. 

Ska 

Crank. 

'ts. 

Pro¬ 

peller. 

Angle 

of 

Crank. 

Shafts. 

c ™ k - |p3ir,. 

Angle 

of 

Crank. 

Shaf 

Crank. 

ts. 

Pro¬ 

peller. 

Angle 

of 

Crank. 

Sha 

Crank. 

'ts. 

Pro¬ 

peller. 

90 0 

( 2468 

(4OOO 

2880 

5400 

IOO° 

(2279 
(4000 

2659 

5400 

IIO° 

(2I 3 I 
( 4000 

2487 

5400 

120° 

( 2016 
( 4000 

2352 

5400 


/iH? 

- - C =: diameter. A. E. Seaton, London, 1883. 


Side-wheel Engines, Sea Service. —One cylinder crank journal, C = 8o; outboard 
100; Two cylinder crank journal 50; outboard 65; and centre shaft 58. 

Propeller Engines. —One cylinder crank journal 150; Tunnel 130; Two cylinder 
compound crank 130; Tunnel no; Two cranks, crank 100; Tunnel 85; Three cranks, 
crank 90; and Tunnel 78. 

River Service. —C may be reduced one fifth. 

Illustration. —With a compound propeller engine, steam cylinders 20 and 40 
ins. in diameter, by 40 ins. stroke, operating under a pressure of 80 lbs. steam 
(mercurial gauge), what should be the diameter of the shafts of wrought iron? 


f 


20 2 X 80-P40 2 X 15 


4000 

and 


X 4 ° : 


56 000 
4000 


X 40 = 8.24 ins. crank shaft; 




56 000 
5400 


X 40 = 7.46 ins. propeller shaft. 


Journals of Shafts, etc. 

Journals or bearings of shafts should be proportioned with reference to 
pressure or load to be sustained by the journal. Simplest measure of bear¬ 
ing capacity of a journal is product of its length by its diameter, in sq. ins.; 
and axial area or section thus obtained, multiplied by a coefficient of pressure 
per sq. inch, will give bearing capacity. 

Sir William Fairbairn and Mr. Box give instances of weights on bearings of 
shafts, etc., from which following deductions are made, showing pressure per sq. 
inch of axial section of journal : 

Crank pins, 687 to 1150 lbs. per sq. inch. 

Link bearings, 456 to 690 lbs. per sq. inch. 

Pressure on bearings, as a general rule, should not exceed 750 lbs. per sq. inch of 
axial area. 

Length of Journals should be 1.12 to 1.5 times diameter. 

Journals of Locomotives or Like Axles are usually made twice diameter, and to 
sustain a pressure of 300 lbs. per sq. inch of axial area, or 10 sq. ins. per ton’of load. 

Solid. Cylindrical Couplings or Sleeves. 

+ V 5 -S d = I); 3 d=L; -8 d — l\ .25 (Z-f- . 12 = 1c. d representing diameter 
and L length of sleeve, l length of lap or scarf of shaft, lc breadth of key, its depth be¬ 
ing half its breadth, and D diameter of coupling or sleeve, all in ins. 

Flanged Couplings. 

d-{-V3-5 d = D; 3d-j-i = F; . 3 -J- .4 = Z; d+i = L; l A-4 — s. T> repre¬ 

senting diameter of body of coupling, F diameter of flanges, l thickness of both flanges 
L length of each coupling , s projection of end of one shaft and retrocession of other 
from centre of coupling, and d diameter of shaft, all in ins. 

Supports for Shafts. (Molesworth.) 

5 $/d 2 — L. L representing distance of supports apart, in feet. 

































STRENGTH OF MATERIALS.-TORSION. 


797 


/L W 

To Resist Lateral Stress. 3 —— — d. W representing weight or pressure 
at centre of length in lbs., and D diameter or side, if square, in ins. 

Value of C.—Wrought Iron, 560; Cast Iron, 500; Cast Steel, 1000 to 1500; Bronze, 
420; and Wood, 40. When Weight is distributed put 2 C. 

Values of C for Shafting of Various Metals , as observed by different 
Authorities, and deduced from Formulas of Navier. 16 ^ f. — c. 


d 3 


Ultimate Resistance. 


Metal. 

c 

Metal. 

C 

Metal. 

Wrought Iron. 


Cast Iron. 


Steel. 

American, Pemb e ,Me. 

61 673 

( 

36846 

American, Conn.. 

11 Ulster.... 

61 815 

American, mean < 

3830° 

“ Spindle 

“ mean. 

66 436 

l 

42 821 

“ Nash. I. Co. 

English, refined. 

49^8 

“ 18 trials 

44 957 

English, Shear.... 

U U 

Swedish. 

54 585 
61 909 

English, mean.. | 

22 132 
38217 

Bessemer.j 


16 W R 


IVIiil and. Factory Sliafts. (J. B. Francis.) 
Cylindrical. Square. 

3 V 2 W R 


d 3 


= T. 


d 3 T 
16 R 


:W. 


T. 


Cast Iron 


( 22 000 
■" ‘ ( 65 000 

mean.35000 

“ Eng. 30000 


Mean value of T. 
Wrought Iron.... { 4 9 °°° 

( 94OOO 




mean.50000 

“ Eng. 45000 


82 926 
102 131 
95213 
in 191 
73060 
79662 


: W. 


76000 


Steel.f 

( in000 

“ mean. 86000 

“ “ Bessemer 78000 

Illustration. —What is the ultimate or destructive weights that may be borne 
by a Round Cast-iron shaft 2 ins. in diameter, and by a Square shaft 1.75 ins. side, 
stress applied at 25 ins. from axis? Assume T = 36000. 

Round. Square. 


3.1416 X 2 3 X 36000 


16 X 25 


2261.95 lbs. 


1.75 3 X 36 000 

25 


^3) a/ 2 = 1837. 


8 lbs. 


Their lengths should be reduced, and diameter increased, in following cases: 
1st. At high velocities, to admit of increased diameter of journals, thereby 
rendering them less liable to heating. 2d. As they approach extremity of a 
line of shafting. 3d. Attachment of intermediate pulleys or gearing. 


Prime Movers of Power. 
Wrought „ /100IH? 


Iron 
Steel , 3 


Cast 

Iron. 


3 J 


n 

62^5 HP 
n 

167 IIP 


; d, and .01 n d 3 =. IIP. 


n 


= d, and .016 nd 3 — IIP. 3 
— d, and .006 nd 3 — IIP. 3 


Transmitters of Power. 

IIP. 

= d, and .032 nd 3 = IIP. 
= d, and .012 n d 3 = IIP. 


3 / 5° I-EP _ j an( j Q2 n 
n 

31.25 IIP 


n 

83.5 HR 


IIP representing liorse-power transmitted, n number of revolutions, and d diameter 
of shaft in ins. 

Illustration l— What' should be diameter of a wrought-iron shaft, to simply 
transmit 128 IP at 100 revolutions per minute? 

/50 X 128 _ /6400 

— Q ' —: 4 ms. 


100 v 100 

2.—What HP will a steel shaft of 4 ins. diameter transmit at 100 revolutions per 
minute ? 

.032 X 100 X 4 3 — 204.8 horses. 

3X* 







































798 STRENGTH OF MATERIALS.—TRANSVERSE. 


TRANSVERSE STRENGTH. 

Transverse or Lateral Strength of any Bar , Beam , Rod , etc., is in propor¬ 
tion to product of its breadth and square of its depth; in like-sided bars, 
beams, etc., it is as cube of side, and in cylinders as cube of diameter of 
section. 

When One End is Fixed and the Other Projecting , strength is inversely as 
distance of weight from section acted upon; and stress upon any section is 
directly as distance of weight from that section. 

When Both Ends are Supported only , strength is 4 times greater for an 
equal length, when weight is applied in middle between supports, than if one 
end only is fixed. 

When Both Ends are Fixed , strength is 6 times greater for an equal length, 
when weight is applied in middle, than if one end only is fixed. 

When Ends Rest merely upon Two Supports, compared to one When Ends are 
Fixed, strength of any bar, beam, etc., to support a weight in centre of it, is 
as 2 to 3. 

When Weight or Stress is Uniformly Distributed , weight or stress that can 
be supported, compared with that when weight or stress is applied at one end 
or in middle between supports, is as 2 to 1. 

Nfetals. 

In Metals, less dimension of side of a beam, etc., or diameter of a cylinder, 
greater its proportionate transverse strength, in consequence of their having 
a greater proportion of chilled or hammered surface, compared to their ele¬ 
ments of strength, resulting from dimensions alone. 

Strength of a Cylinder , compared to a Square of like diameter or sides, is 
as 5.5 to 8. Strength of a IIolloiv Cylinder to that of a Solid Cylinder , of 
same area of section, is about as 1.65 to 1, depending essentially upon the 
proportionate thickness of metal compared to diameter. 

Strength of an Equilateral Triangle, Fixed at One End and Loaded at the 
Other , having an edge up, compared to a Square of the same area, is as 22 to 
27 ; and strength of one, having an edge down , compared to one with an edge 
up, is as 10 to 7. 

Note.—I n Barlow and other authors the comparison in this case is made when 
the beam, etc., rested upon supports. Hence the stress is contrariwise. 

Strongest rectangular bar or beam that can be cut out of a cylinder is one 
of which the squares of breadth and depth of it, and diameter of the cylinder, 
are as 1, 2, and 3 respectively. 

Oast Iron. 

Mean transverse strength of American, as determined by Major Wade, is 
681 lbs. per sq. inch, suspended from a bar fixed at one end and loaded at 
the other; and mean of English, as determined by Fairbairn, Barlow, and 
others, is 500 lbs. 

Experiments upon bars of cast iron, 1, 2, and 3 ins. square, give a result 
of transverse strength of 447, 348, and 338 lbs. respectively; being in the 
ratio of 1, .78, and .756. 

Woods. 

Beams of wood, when laid with their annular layers vertical, are stronger 
than when they are laid horizontal, in the proportion of 8 to 7. 

Relative Stiffness of Materials to Resist a Transverse 

Stress. 

Ash.089 I Cast Iron.... x 1 Oak .095 I Wrought iron 1.3 

Beech.073 | Elm.073 | White pine... .1 | Yellow pine.. .087 






STRENGTH OF MATERIALS.-TRANSVERSE. 799 


Strength of a Rectangular Beam in an Inclined position, to resist a vertical 
stress, is to its strength in a horizontal position, as square of radius to square 
of cosine of elevation; that is, as square of length of beam to square of dis¬ 
tance between its points of support, measured upon a horizontal plane. 


Transverse Strength, of* Various [Materials. 

( U. S. Ordnance Department , Hodglcinson , Fairbairn , KirJcaldy, and by the Author.) 

Power reduced to uniform Measure of One Inch Square , and One Foot in Length; 

Weight suspended from one End. 


Metals. 

Brass. 260 

Cast Iron, mean of 4 grades. 660 

“ “ (Maj. Wade).. 681 

“ ordinary. 575 

u extreme, West P’t F’dry 980 

“ gun-metal,* “ “ 740 

“ Eng., Low Moor,cold blast. 472 
• “ “ Ponkey, “ 581 

“ “ Ystalyfera “ 770 

“ “ mean, 65 kinds. 500 

“ “ “ 15 kinds, cold blast 641 

“ u planed bar. 518 

“ “ rough bar. 534 

Copper. 244 

Steel, hammered, mean.1500 

“ cast, soft.1540 

“ “ hard.4200 

“ hematite, hammered.1620 

“ Krupp’s shaft.2096 

“ Fagersta, hammered.1200 

Wrought Iron, mean. 600 

“ “ English. 475 

“ “ Swedish f. 665 


Cuba. 


Woods. 

Ash.168 

“ English.160 

“ Canada.120 

Balsam, Canada. 87 

Beech.130 

“ white.112 

Birch .{ 115 

Cedar, white.160 

f 63 

.1 105 

Chestnut.160 

Elm. 125 

“ Canada, red.170 

Fir, Baltic, mean.153 

“ Canada, yellow. j ^ 

<■ “ red.120 

“ Norway.123 

“ Dantzic.163 

Riga.112 

“ Memel .161 

“ “ red. 75 

Greenheart, Guiana. 160 

Gum, blue.136 

Hackmatack.102 

Hemlock.100 


Woods. 


Hickory. 


•I 


250 

Iron wood, Burmah.240 

Larch, Russian.118 

Lignumvitse. 162 

Locust.i-295 

Mahogany.112 

Mangrove. .. 162 

Maple..... 

Oak, white.150 

“ live.160 

“ red, black. 135 

1 ‘ African.207 

“ English.{ 

French.160 

“ Dantzic. 88 

u Canada.146 

“ Sardinia.142 

•“ Spanish.105 

Pine, white.125 

“ pitch.137 

t£ yellow.130 

“ Georgia.200 

Poon.184 

Poplar.112 

Spruce, Canada.125 

“ black. 87 

Sycamore.125 

Tamarack. j .ico 

Teak.165 

Walnut.112 

Willow. 87 

Whitewood.116 

Stones, Bricks, etc. 

Brick, common, mean.20 

“ pressed, “ .40 

“ English, stock.11.8 

“ “ fine .14 

Brick arch.15 

Cement, mean.15 

“ “ Portland. j IO ' 2 

l 37-5 

“ “ Sheppey. 5 

“ “ hydraulic, Portland. 5 

“ “ Roman.... 2 

“ Puzzuolana. 4.5 

“ Portland, 1 year. 8 

“ Roman, “ 2.5 

Concrete, Eng., fire-brick beam, ) 
cement.j 


3'i 


* This was with a tensile strength of 27000 lbs. 

t With 840 lbs. the deflection was x inch, and the elasticity of the metal destroyed. 


























































































800 STRENGTH OF MATERIALS.-TRANSVERSE. 


Stones, Bricks, etc. 

Concrete, Eng., fire-brick, sand 3 ,) 

lime 1. J ■' 

“ Eng., clay and chalk_ 5.4 

Flagging, blue, New York.31-25 

Freestone, Conn.13 

“ Dorchester.10.8 

44 New Jersey, mean.19 

“ New York.24 

44 Eng., Craigleth.10.7 

44 “ Darby, Victoria... 1.3 

“ “ Park Spring. 4.3 

Glass, flooring.42.5 

Granite, blue, coarse.18 

44 Quincy.26 

“ mean.25 

“ Eng., Cornish.22 

Limestone. 

44 English....11 

Marble. 


Stones, Bricks, etc. 

Marble, Adelaide. 

44 Italian. 

Mortar, lime, 60 days. 

44 1 lime, 1 sand. 

44 1 44 2 44 . ...... 

44 1 44 4 44 ... 

Oolite, English, Portland. 

Paving, Scotch, Caithness. 

44 Ireland, Valentia. 

44 Welsh. 

44 English, Yorkshire, blue.. 

4 4 4 4 Arbroath. 

Slate . 


4-5 

2-5 

2 

i -75 

1.25 

21.2 

68 

68.5 

i57 

10.4 

17 

81 


44 Bangor.90 

44 English, Llangollen.43 

Stones, English, Bath. 5.2 

“ 44 Kentish, Rag.35.8 

44 “ Yorkshire, landing 22.5 

“ Caen.12.5 


Elastic Transverse Strength of Woods , compared with their Breaking Weight , 

is as follows: 


Per Cent. 


Ash. 29 

Beech. 25 

Elm. 32 

Larch. 38 


Per Cent. 

Norway Spruce.... 30 


Oak, Dantzic. 36 

44 English. 33 

Pitch Pine. 24 


Per Cent. 

Red Pine. 29 

Riga Fir. 30 

Teak. 32 

Yellow Pine. 30 


Increase in Strength, of several Woods by Seasoning. 

Per Cent. 

Ash.44.7 | Beech.61.9 | Elm.12.3 | Oak.26.1 | White pine....9 


Concretes, Cements, etc. 


Materials. 

Breaking 

Weight. 

Materials. 

Breaking 

Weight. 

concretes (English). 

Lbs. 

bricks (English). 

Lbs. 

Fire-brick beam, Portl’d cement 
44 sand 3 parts, lime 1 part 
cements (English). 

Pine clay and chalk. 

*2. I 

Best stock. 

11.8 

.7 

Fire-brick. 


New brick. 


5-4 

37-5 

10.2 

Old brick. 

j-w. y 

Portland. j 

Stock-brick, well burned. 

5-8 

2.5 

44 inferior, burned... 

Sheppey. 

5 




Transverse Strength of Various Figures of Cast Iron. 
Reduced to Uniform Measure of Sectional Area of One Inch Square and One Foot in 
Length. Fixed at one End ; Weight suspended from the other. 


Form of Bar or Beam. 



Square 



Square, diagonal vertical... 



Cylinder. 


O Hollow cylinder; greater 
diameter twice that of 
lesser. 


Breaking 

Weight. 

Form of Bar or Beam. 

Breaking 

Weight. 

Lbs. 

673 

| Rectangular prism. 

I 2 X • 5 ins. in depth... 

44 3 X .33 44 in depth... 

44 4X.25 44 in depth... 

Lbs. 

1456 

2392 

2652 

568 

A Equilateral triangle, an 1 
ff/Kk. edge up.j 

560 

573 

WM Equilateral triangle, an 1 
V edge down.} 

958 


G|P 2 ins. in depth x 2 x 1 
.268 inch in width... j 

2068 

794 

i 2 ins. in depth X 2 X ) 
JL .268 inch in width ..) 

555 


















































































STRENGTH OF MATERIALS.-TRANSVERSE. SOI 


Solid, and Hollow Cylinders of various ^Materials. 


One Foot in Length. Fixed at one End; Weight suspended from the other. 


Materials. 

External 

Diam. 

Internal 

Diam. 

Breaking 

Weight. 

Materials. 

External 

Diam. 

Internal 

Diam. 

Breaking 

Weight. 

WOODS. 

Ash. 

Ins. 

2 

Inch. 

Lbs. 

685 

604 

METAL. 

Cast iron cold) 

Ins. 

Ins. 

Lbs. 

U 

2 

I 

blast.) 

3 

— 

12 000 

Fir*. 



STONE-WARE. 




White pine.. 

U u 

1 

2 

_ 

/ / ■* 

75 

610 

Rolled pipe of 1 
line clav.) 

K> 

OO 

1.928 

190 


* An inch-square batten, from same plank as this specimen, broke at 139 lbs. 


Formulas for Transverse Stress of Rectangular Bars, 
Beams, Cylinders, etc. 

Fixed at One End. Loaded at the Other. 


Bars, Beams, etc. 


I W 
ftd 2 


S; 


S ftd 2 
l 


m SfttZ 2 , 
:W; 


l W /IW_ 

’ Srf 2 “ ’ VS ft ~ ’ 


V 


it w 

and Cylinder 3/-—= ft and d. 


Fixed at Both Ends. Loaded in Middle. 


Bars, Beams, etc. 


I W 


6 bd 2 
'l W 


= S; 


6 S 6 d 2 


/* W __ 


l 

and Cylinder 


:W; 


6 S 6 d 2 


V 6S 


W 
6 and d. 


= *; 




6 S d 2 


Bars,. Beams, etc. 

7 


3 lb d 
2 to n W 


„ 3 l ft d 2 S _ 2 to n W 

; s ; --- — ”; .. , 2 

2 to 3 s 6d‘ 

2 to n W 




6; 


Fixed at Both Ends. Loaded at any Other Point than in Middle. 

2 to n W „ ilb d 2 S 2 to n W . /2 wnW 


3 S td‘ 


= &; 


3 SJ6 


_ /2 to n VV 

d; and Cylinder ^ ^ — bandd, 


Bars, Beams, etc. 


Supported at Both Ends. Loaded in Middle. 

IW „ 4 Sftd 2 TTT . 4 S ft d 2 


4 ft d 2 

'Z W 


S; 


Z 


W; 


VV 


Z w 


4 S d : 


! = &; 


d; and Cylinder — & anc ^ 


V 4 S ft 

Supported at Both Ends. Loaded at any Other Point than in Middle. 


Bars, Beams, etc. 


m n W 
Ibd 2 ' 


S; 


S Z 6 d 2 


W; 


TO n W 

S ft d 2 


= *; 


m n W 
S Z d 2 


= &; 


/to « W 7 ~ /to n W 

/ ^ ^ ■ — d; and Cylinder 3 /— — — bandd. 

W_ 

s - - 

ft d 2 put d 3 as above. 

IFAcn weight is uniformly distributed , same formulas will apply , W repre¬ 
senting onip /za//’ required or given weight. 

S representing stress in a Bar, Beam, or Cylinder, one foot in length, and one inch 
square, side, or in diameter ; and W weight, in lbs.; ft breadth, and d depth, in ins.; 
I length , to distance of weight from one end, and nfrom the other, all in feet. 


/I W 

Vs6 =t! - 


In Cylinders, for 


In Square Beams, etc., for 6 and d put 




B riclr-work. 

A brick arch, having a rise of 2 feet, and a span of 15 feet 9 ins., and 2 
feet in width, with a depth at its crown of 4 ins., bore 358 400 lbs. laid along 
its centre. 











































802 STRENGTH OF MATERIALS.—TRANSVERSE. 


Coefficient or Tractor of* Safety. 

Coefficient or factor of safety of different materials must be taken in view 
of importance of structure, or instrument, probable or required period of du¬ 
ration of it, and if it is to bear a quiescent, vibratory, gradual, or percussive 
stress, and to meet these varied conditions, it will range from .125 to .3 of 
the maximum or ultimate strength here given or ascertained. 

To Compute Transverse Strength, of a Rectangular Bar 

or Beam. 

When a Bar or Beam is Fixed at One End, and Loaded at the Other. 
Rule. —Multiply Coefficient of material in preceding Tables, or, as may be 
ascertained, by breadth and square of depth in ins., and divide product by 
length in feet. 

Note. —When a beam, etc., is loaded uniformly throughout its length, result must 
be doubled. 


Example.— What weight will a cast-iron bar, 2 ins. square and projecting 30 ins. 
in length, bear without permanent injury? 

Assume strength of material at 660, and its elasticity at one fifth or .2 of its 
strength. 


Then 


660 X .2 X 2 X 2 2 _1056 

2.5 — 2.5 


: 422.4 lbs. 


If Dimensions of a Beam or Bar are Required to Support a Given Weight 
at its End. Rule. —Divide product of weight and length in feet by Coeffi¬ 
cient of material, and quotient will give product of breadth and square of 
depth. » 

Example.— What is the depth of a wrought-iron beam, 2 ins. broad, necessary to 
support 576 lbs. suspended at 30 ins. from fixed end? 

Assume strength of iron at 150. 

Then 2 ‘ 57 - = 9.6, and ^/— — 2. 19 ins. depth. 

When a Beam or Bar is Fixed at Both Ends , and Loaded in the Middle. 
Rule. —Multiply Coefficient of material by 6 times breadth and square of 
depth in ins., and divide product by length in feet. 

Note.— When beam is loaded uniformly throughout its length, result must be 
doubled. 

Example.— What weight will a bar of cast iron, 2 ins. square and 5 feet in length, 
support in middle, without permanent injury? 

Assume strength of material as in a previous case at .2 of 660. 


Then 


660 X-2X2X6X2 2 6336 

5 ~ S 


= 1267.2 lbs. 


If Dimensions of a Beam or Bar are Required to Support a Given Weight 
in Middle , between Fixed Ends. Rule. —Divide product of weight and 
length in feet by 6 times Coefficient of material, and quotient will give prod¬ 
uct of breadth and square of depth. 

Example. —What dimensions will a square cast-iron bar, 5 feet in length, require 
to support without permanent injury a stress of 2160 lbs. ? 

Assume strength of material at . 2 of 660 or 132, as preceding. 


Then 


2160 x 5 10 800 


132 X 6 792 

and t/6. 82 = 2.61 ins. depth. 


— 13.64, which , divided by 2 for assumed breadth — 6.82, 


When Breadth or Depth is Required. Rule. —Divide product obtained 
by preceding rules by square of depth, and quotient is breadth; or by 
breadth, and square root of quotient is depth. 

Example.—I f 128 is the product, and depth is 8; then 128 ~ ^ 2, breadth. 

Also, 128 -4- 2 = 64, and -^64 = 8, depth. 








STRENGTH OF MATERIALS.-TRANSVERSE. 803 


When Weight is not in Middle between Ends. Rule. —Multiply Coefficient 
of material by 3 times length in feet, and breadth and square of depth in 
ins., and divide product by twice product of distances of weight, or stress 
from either end. 

Example. —What weight will a cast-iron bar, fixed at both ends, 2 ins. square and 
5 feet in length, bear without permanent injury, 2 feet trom one end? 

Assume strength of material at .2 of 660 or 132, as preceding. 

Then X 3 x 5 X Vx »» = m 

2 X (5 — 2 ) 12 

When a Beam or Bar is Supported at Both Ends , and Loaded in Middle. 
Rule. —Multiply Coefficient of material by 4 times breadth and square of 
depth in ins., and divide product by length in feet. 

Note.—W hen beam is loaded uniformly throughout its length, result must be 
doubled. 

Example. — What weight will a cast-iron bar, 5 feet between the supports, and 2 
ins. square, bear in middle, without permanent injury? 

Assume strength of iron at 132, as preceding. 

Then 132 X 2 X 4 X 2 2 = 4224 -f- 5 =• 844.8 lbs. 

If Dimensions are Required to Support a Given Weight. Rule. —Divide 
product of weight and length in feet by 4 times Coefficient of material, and 
quotient will give product of breadth, and square of depth. 

When Weight is not in Middle between Supports. Rule. —Multiply Coef¬ 
ficient of material by length in feet, and breadth and square of depth in ins., 
and divide product by product of distances of weight, or stress from either 
support. 

Example. —What weight will a cast-iron bar, 2 ins. square and 5 feet in length, 
support without permanent injury, at a distance of 2 feet from one end, or support? 

Assume strength of iron at 132, as preceding. 

Then £3?X5X 2X»» = 5£?o = 88o m 
2 X (5 — 2) 6 

To Compute Pressure upon Ends or upon Supports. 


Rule i.—D ivide product of weight and its distance from nearest end or 
support, by Avhole length, and quotient will give pressure upon end or sup¬ 
port farthest from weight. 

2.—Divide product of weight and its distance from farthest end, or sup¬ 
port, by whole length, and quotient will give pressure upon end or support 
nearest weight. 

Example.— What is pressure upon supports in case of preceding example? 


880 X 2 _ jp s U p 0n SU pp 0r t farthest from the weight; — ° X - = 528 lbs. upon 
support nearest to weight. 

When a Bar or Beam , Fixed or Supported at Both Ends, bears Tioo 
Weights at Unequal Distances from Ends. 

m W , l w , 7 , n w V W 

-j- —pressure at w end , and —— 4——- — pressure at W end. 

L L L L 


m and n representing distances of greatest and least weights from their nearest 
end , W and w greatest and least weights , L whole length , l distance from least weight 
to farthest end , and l' distance of greatest weight fi-om farthest end. 

Illustration.— A beam 10 feet in length, having both ends fixed in a wall, bears 
two weights—viz., one of 1000 lbs., at 4 feet from one of its ends, and the other of 
2000 lbs., at 4 feet from the other end; what is pressure upon each end? 


4 X 2000 , 6 X 1000 „ , 4 X 1000 6 X 2000 T 

-1-—-— 1400 lbs. at w ; - - 1 --1600 lbs. at W. 


10 


10 


10 


10 










804 STRENGTH OF MATERIALS.-TRANSVERSE. 


When Plane, of Bar or Beam Projects Obliquely Upward or Downward. 
When Fixed at One End and Loaded at the Other. Rule. —Multiply Co¬ 
efficient of material by breadth and square of depth in ins., and divide product 
by product of length in feet and cosine of angle of elevation or depression. 
Note.— When beam is loaded uniformly along its length, result must be doubled. 
Example. —What is weight an ash beam, 5 feet in length, 3 ins. square, and pro¬ 
jecting upward at an angle of 7 0 15', will bear without permanent injury? 

Assume breaking weight of ash at 160, and its elasticity at .25 of its strength, and 
cosine of 7 0 15" = .992. 

Then 1 ^° x - 2 5 X 3 X 3 3 = io8o = as 

5 X .992 

To Compute Transverse Strength, of an IE unilateral Tri¬ 
angle or T Beam. 


Rule.—P roceed as for a rectangular beam, taking following proportions 
of Coefficient of material: 


Fixed at One or 
Both Ends. 


Equilateral triangle, edge up... 
Equilateral triangle, edge down 
T beam, flange up. 


bXd 2 X. 2 C 
b X d 2 X -34 “ 
b X d 2 X .42 “ 


Supported at 
Both Ends. 


Equilateral triangle, edge up. b X d 2 X • 34 “ 

Equilateral triangle, edge down. b X d 2 X ■ 2 “ 

T beam, flange up.. b X d 2 X ■ 42 “ 


To Compute Transverse Strength of a Solid Cylinder. 

Rule. —Proceed as for a rectangular beam, and take .6 of Coefficient or 
of product. 

A mean of 18 results with cold blast gun-metal, gave a coefficient for 740 lbs. 


When Fixed at One End , and Loaded at the Other. Rule. —Multiply 
weight to be supported in lbs. by length of cylinder in feet; divide product 
by .6 of Coefficient of material, and cube root of quotient will give diameter. 


Note. —When cylinder is loaded uniformly throughout its length, cube root of 
half quotient will give diameter. 

Example.— What should be diameter of a cast-iron cylindrical beam of gun-metal, 
8 ins. in length, to break at 15000 lbs. ? 


15000 X 8-r 12 


= 3 


= 2.61 ins. 


. 6 X 740 "V 444 

When Fixed at Both Ends , and Loaded in Middle. Rule. —Multiply 
weight to be supported in lbs. by length of cylinder between supports in 
feet; divide product by .6 of Coefficient of material, and cube root of one 
sixth of quotient will give diameter. 

Note. —When cylinder is loaded uniformly abrng its length, cube root of half the 
quotient will give diameter. 

Example. —What is the diameter of a cast-iron cylinder of gun-metal, 2 feet be¬ 
tween supports, that will break at 35 964 lbs. ? 

35964 X 2 /162 

-- 2 - = 162, and 3 / _ _ — 3 ms . 

.6x740 V 6 

Mean results of cylinder and square bars gave 444 and 740 lbs. Hence, strength 
of a cylinder compared to a square is as 444 to 740 or .6 to 1. 

Then 4 X 3 3 X 444 _ ^ ^ ^ 


To Compute Diameter of a Solid. Cylinder to Support 
a given Weight. 

When Supported at Both Ends , and Loaded in Middle. Rule. —-Multiply 
weight to be supported in lbs. by length of cylinder between supports in 
feet; divide product by .6 of Coefficient of material, and cube root of one 
fourth of quotient will give diameter. 














STRENGTH OF MATERIALS.-TRANSVERSE. 


Note.— When cylinder is loaded uniformly along its length, cube root of half the 
quotient will give diameter. 

Example.— What is diameter of a cast-iron gun-metal cylinder, i foot between its 
supports, that will break at 48000 lbs. ? 

48 000 X 1 „ , „ /108 , . 

■ —-—■- = 108, and 3 / — = 3,61 ms. 

.6X740 V 4 


(1) Loaded at Middle. ^ =1 W. 


Rectangular. (D. K. Clark.) 

(2) Loaded at One End. 


zb d- 


l 


L 


W. 


(3) Loaded at Middle. 


5.5 b d 2 


Cylindrical. 


W. (4) Loaded at One End. 


1.375 b d 2 


l — ... J 

W representing ultimate stress in tons. 

Above Coefficients are for iron of a tensile strength of 7 tons per sq. inch. 


W. 


(1) 

(2) 

( 3 ) 

( 4 ) 



(0 

(2) 

( 3 ) 

(D 

9.2 

2-3 

6-3 

1.6 

For 12 

tons put 

U 

13.8 

3-4 

9.4 

2.4 

10.4 

2.6 

7 - 1 

1.8 

13 

14.5 

3-6 

10.2 

2.6 

ii -5 

2.9 

7-9 

2 

14 

u 

16 

4 

11 

2.8 

12.7 

3-2 

8.6 

2.2 

15 

k ( 

17. 2 

4-3 

11.8 

3 


9 

10 “ 

11 “ 

To Compute Destructive Weight, or Loads that may "be 
Horne by Wrouglit-iron Rolled Beams and Grirders, 
or Riveted Tubes of various Figures and Sections. 

Supported at Both Ends. Load applied in Middle. 

When Section of Beam or Girder is that of any of the Figures in follow¬ 
ing Table. Rule. —Divide product of area of section, depth, and Coefficient 
for girder, etc., from following Table, by length between supports in feet, 
and quotient will give destructive weight in lbs. 

Note. —The Coefficients given are based upon experiments with English iron. 

Solid Beams. 

Illustration. —What load will destroy a wrought-iron grooved beam of following 
dimensions, 10 feet in length between supports, and loaded in its middle? 

Flanges, 5.7 X -64 inch; Web, .6 inch; Depth, 11.75 ins.; Area, 13.34 S T ’ cs - 
Assume Coefficient 4638 as for like case (12) in following table, page 8c6. 

13 34X11.75X4638 = 726821 J 68z x Ws 

10 10 

Ultimate stress for such a beam by experiment was estimated at 97997 lbs. 
Formulas of Various A uthors give following Results: 


D. K. Clark. 


d (4 1.1555 a') 

.6 l 


. W. a representing area of section of lower 

flange, a' area of section of web, less one flange, d depth of beam, less average depth 
of one flange, all in ins., I length in feet, and W ultimate destructive weight in tons. 
This formula is based upon the assumption that the beam has lateral support. 


11.75 —.6 (4 X 5-7 X .64-1.155 X u-75 —-6 X - 6) _ 238.69 

.6 X 10 6 


139.78, which X 2240 


4 * d - =W. C = 7616 lbs., and for b d 2 put bd’ — zb'd' 2 . 


— 88 107 lbs. 

Molesworth. 

b and d representing exterior and b' and d' interior dimensions, and l length in ins. 
5.7 X u.75 2 —[5.7 —.6 X 11.75 —(.64 X 2 2 )] = 786.6 — 558.9 = 227.7. 

4 X 7616 X 227.7 _ 693 665 


Then 


57805.4 lbs. 


10 X 12 120 

Fairbairn’s formula would give a result less than half of the first, and Hodgkin- 
son’s alike to that of Molesworth. 

3 Y 


















8 o6 STRENGTH OF MATERIALS.-TRANSVERSE. 


WROUGHT IRON. 

'Transverse Strength, of Wrought-iron Boiled Beams 
and. Grirders. (Barlow, Fairbairn, Hughes, Kirkaldy , etc.) 

Reduced to Uniform Measure of One Foot in Length. 

Supported at Both Ends ; Stress or Weight applied in 'Middle. 


Section. 

Flanges. 

Web. 

Depth ( d ). 

Distance. 

Area 

(A). 

Destruct 

For 

ive Weight. 
Length of 

l ~- = C. 





Distance. 

One Foot(f) 

A d 


Ins. 

Iri 9 . 

Ins. 

Feet. Ins. 

Sq. Ins. 

Lbs. 

Lbs. (W). 


PM f 

— 

I 

I 

I 


I 

2 500 

2500 

2 500 

s* \ 

— 

2 

2 

2 

9 

4 

6 600 

18 150 

2 266 

1 

— 

i -5 

3 

2 

9 

4-5 

10 080 

27 720 

2 053 

a 

— 

I 

3 

3 


y 

7050 

21 150 

2 350 

# 

— 

I 

I 

5 


•78 

474 

2370 

2 370 

X 

3-5 X .6 

.8 

3-5 

2 

7 

5-65 

20 160 

52 480 

2654 

If 

2.5 Xi 

4 X -38 

} -325 

7 

2 

9 

5-9 

44 000 

121 OOO 

293O 

I 

2.6 X125 

•85 

5 

4 

6 

7-44 

19 000 

85 500 

2 298 

( 

3 X -49 

•5 

7-°7 

IO 


5-87 

24 200 

242 OOO 

5 830 

“ ) 

4.6 X .8 

•5 

9- 8 5 

20 


n -5 

38 080 

761600 

6724 

( 

5.7 X .64 

.6 

n -75 

IO 


13-34 

72 688 

726 880 

4638 

1 

K> 

OO 

Cn 

X 

CO 

OO 

• 3 i 

2-5 

4 


x -75 

3150 

12 600 

2 880 

T 1 

7 X- 5 

4 X -5 

} .38 

16.5 

22 

6 

18.9 

49 280 

1108 800 

3 556 

JL { 

4 - 5 X -375 

2X2X.3125 

} .38 

14.25 

l6 

5 

10.5 

47 000 

775 500 

5183 

1 < 

4.5X28 

4 - 5 X 3 

1 ' 2S 

7 

7 


6-35 

24380 

170660 

384O 

0 1 

3-9 

3-9 

1 - 13 

6 

7 

6 

2.62 

9976 

74820 

4766 

LJ ^ 

i 5-5 

i 5-5 

j -53 

24 

30 


41.4 

128 885 

3 866 550 

3 896 

“ ! 

24 

24 

•75 

•75 

| 35-75 

45 


87.38 

257 080 

11 568 600 

3 703 

O 

— 

•131 

{12.4 

1 12.138 

I 10 


5-05 

17885 

178 850 

2856 

0 

— 

• x 43 

} 15 
l 9-75 

}x° 


5-56 

26 250 

262 500 

3147 

Steel. 

— 

•75 

5-2 

5 


7.72 

102 480 

512 400 

12 760 


These results are very conclusive of the correctness of above formula, as 
will be seen in cases given, and they are deduced from beams and girders 
varying from i to 45 feet in length; hence, when length of a beam or girder 
of any of the sections given is less, relative breaking weight may be in¬ 
creased, in consequence of increased stability of beam or girder. 

For fall experiments on Tubes and Tubular Girders, etc., see Rep. of Commas on 
Railiuay Structures, London, 1849. 

Tensile strength of iron assumed at 45 000 lbs. per sq. inch. 

















STRENGTH OF MATEEIALS.-TRANSVERSE. 


Elements of 'W’romgh.t-iroia Rolled, lie a iris. 


With Safe Load Uniformly Distributed. For Length of One Foot. 
(Beams Supported Sideways.) 



Width. 


Area. 


Weight 


Depth. 






per 

Load. 


Web. 

Flange. 

Web. 

Flange.] 

Total. 

Foot. 


Ins. 

Inch. 

Ins. 

S.Ins. 

Sq.Ins. 

Sq.Ins. 

Lbs. 

Lbs. 

4 Light | 

•1875 

•25 

2 

2.75 

•75 

1 

1.02 

1. 9 I 

I .?? 

2.91 

6 

10 

l8 OOO 
30 IOO 

4 Heavy 

•3I 2 5 

3 

1.25 

2.41 

3.66 

12.3 

36 800 

4 Light 

.21 

2 

.68 

1.12 

1.8 

6 

21 600 

4 Heavy 

•25 

2-75 

•73 

2.27 

3 

IO 

36 000 

4 Light 

.156 

2.25 

•75 

1-25 

2 

6 

20 600 

4 Heavy j 

•25 

•5 

2-75 

3 

I 

1-25 

I 91 
2.41 

2.91 

3.66 

IO 

12.3 

30 200 
37OOO 

5 Light 

• 25 

2-75 

I. 2 

1.79 

2.99 

IO 

38 7 00 

5 Heavy 

•3 I2 5 

O 

J 

1.56 

2-34 

3-9 

13-3 

49 100 

5 Light 

•25 

2-75 

I 

2 

3 

IO 

42 000 

5 Heavy 

•3 

3 

1.2 

2.4 

3-6 

12 

50000 

5 Light 

•25 

2-75 

1.25 

1.79 

3-4 

IO 

38900 

5 Heavy 

•5 

3 

1.56 

2-34 

3-9 

13-3 

49 IOO 

6 Light 

•25 

3 

1.5 

2.51 

4.01 

!3-3 

62 000 

( 

•3 

3-5 

1.8 

3- 11 

4-9 1 

16.6 

76 800 

6 Heavy! 

•5 

5 

3 

5-7 

8.7 

30 

132 000 

( 

.625 

5-25 

3-75 

8.09 

11.84 

40 

172 000 

6 Light 

•25 

2-75 

I. l6 

2.84 

4 

13-3 

70 000 

6 Heavy 

•31 

3-5 

1.28 

3 - 7 i 

4.99 

16.6 

88 000 

6 Light 

•25 

3 

i-5 

2.51 

4.01 

13-3 

63 400 

6 Heavy 

■3 

3-5 

1.8 

3 -n 

4.91 

16.6 

78 000 

7 Light 

•3 

3-75 

2. I 

3-4 

5-5 

18.3 

IOI 000 

7 

•35 

3-5 

i-95 

3'55 

5-5 

18.3 

108 000 

7 Heavy 

•56 

3-69 

3 -n 

3-79 

6.9 

23 

124 000 

7 Light 

•375 

3-5 

2-59 

3-25 

5-84 

20 

102 OOO 

8 “ 

•3 

4 

2.4 

3-97 

6-37 

21.6 

135000 

8 Heavy 

•375 

4-5 

2.96 

5-07 

8.03 

26.7 

168 000 

8 Light 

•35 

4 

2.28 

4.22 

6-5 

21.6 

148 000 

8 Heavy 

• 55 

4.12 

3-4 6 

4-65 

8. ii 

27 

166 000 

8 “ 

•3 

4 

2.4 

3-97 

6.37 

21.6 

135 40° 

8 “ 

•37 

4-5 

2.96 

5-07 

8.03 

26.6 

168 300 

q Light 

•3 

3-5 

2.7 

3-83 

6-53 

23-3 

152 000 

o Heavy 

.38 

4 

3-4 6 

4.86 

8.32 

28.3 

189 000 

9 “ 

•57 

4-5 

5-i3 

7.2 

12.33 

41.7 

268 000 

9 Light 

•31 

3-5 

2.24 

4.76 

7 

23-3 

184 000 

9 Heavy 

•4 

4 

2.8 

5-6 

8.4 

28 

216 000 

9 “ 

.6 

5-375 

3-83 

11.17 

15 

50 

39400° 

9 Light 

•3 

3-5 

2.7 

3-83 

6-53 

23-3 

153200 

9 Heavy 

•384 

4 

3-46 

4.86 

8.32 

28.3 

189 500 

9 “ 

•58 

4-5 

5.22 

7.2 

12.42 

41.7 

268 400 

10.5 Light 

•375 

4-5 

3-93 

6.51 

10.44 

35 

286 000 

io-S He’vy 

•47 

5 

4-93 

8.43 

13-36 

45 

360 000 

10.5 Light 

•44 

4-5 

3-79 

6.71 

10.5 

35 

310 000 

10.5 He’vy 

•73 

4.87 

6.21 

7-3 

1351 

45 

362 000 

io. s Light 

•375 

4-5 

3-93 

6.51 

10.44 

35 

285 700 

10.5 He’vy 

•47 

5 

4-93 

8-43 

13.36 

45 

362 600 

12 Light 

•49 

4-75 

4.88 

7.62 

12.5 

41.7 

416 000 

12 Heavy 

•59 

5-5 

5-45 

n-55 

17 

56-7 

584000 

12.25 L 

•47 

4.8 

5-75 

6.58 

12.33 

41.7 

37700 ° 

12.5 H 

.6 

5-5 

7-39 

9-38 

16.77 

56-7 

511000 

12.25 L 

.48 

4-79 

5-88 

6.58 

12.46 

41.7 

378 400 

12.25 H 

.6 

5-5 

7-39 

9-3 8 

16.77 

56-7 

5115 00 

15 Light 

5 

4-75 

6-34 

8.66 

15 

50 

604 000 

15 Heavy 

•65 

5-3 I2 5 

7.72 

12.28 

20 

66.7 

820000 

iS- l8 75 L 

•5 

5 

7-59 

7-45 

15.04 

5o 

551000 

15.125 H 

.6 

5-75 

9.07 

10.95 

20.02 

66.7 

748 000 

1 5.1875 L 

.56 

5 

8.9 

7 

15.09 

50 

551600 

15.125 H 

•65 

5-5 

9-57 

io-45 

20.02 

66.7 

748 400 


Manufacturer. 


| N. J. Steel and Iron Co. 

u u u 

Phoenix Iron Co. 

it U 

Passaic River Mill Co. 

| :t tt tt 

N. J. Steel and Iron Co. 

U (4 44 

Phoenix Iron Co. 

44 a 

Passaic River Mill Co. 

44 44 44 

N. J. Steel and Iron Co. 

| u tt u 

Phoenix Iron Co. 

4 4 4 4 

Passaic River Mill Co. 

44 44 44 

N. J. Steel and Iron Co. 
Phoenix Iron Co. 

4 4 44 

Passaic River Mill Co. 
N. J. Steel and Iron Co. 

44 44 44 

Phoenix Iron Co. 

4 4 44 

Passaic River Mill Co. 

44 44 44 

N. J. Steel and Iron Co. 

44 44 44 

44 (4 44 

Phoenix Iron Co. 

4 4 4 4 

4 4 4 4 

Passaic River Mill Co. 

4 4 4 4 U 

44 44 44 

N. J. Steel and Iron Co. 

4 4 4 4 4v 4 

Phoenix Iron Co. 

tt tt 

Passaic River Mill Co. 

44 44 44 

Phoenix Iron Co. 

4 4 4 4 

N. J. Steel and Iron Co. 

44 44 44 

Passaic River Mill Co. 

44 44 44 

Phoenix Iron Co. 

4 4 4 4 

N. J. Steel and Iron Co. 

44 44 44 

Passaic River Mill Co. 

44 44 44 

























80S STRENGTH OF MATERIALS.-TRANSVERSE. 


Elastic Transverse Strength of Wrought-iron Bars is about 45 per cent, of 
their transverse strength, and of Plates 55 per cent., or 48 per cent, of their 
tensile strength; of solid rolled beams, 50 per cent.; and of double-headed 
rails, 46 per cent, of their transverse strength; of Fagersta Steel, 56 per cent, 
of its transverse strength; of double-headed Steel rails, 47 per cent.; of Bes¬ 
semer Steel, 37.5 to 48 per cent.; of Steel flanged, 68 per cent.; and of 
Wrought-iron Steel flanged, 62 per cent, of its transverse strength. 

Transverse strength of Solid Cast-iron Beams or Girders is about 50 per cent, 
of ultimate strength; of double-headed or flanged rails, 46 per cent.; and of single- 
llanged rails, 62 per cent, of its tensile strength. 

Note.— The actual breaking weight of a 10.5 ins. beam of New Jersey Steel and 
Iron Co., weight 35 lbs. per foot, for a length of span of 20 feet, is 30000 lbs. 


Channel and. IDeclv Beams and Strut Bars. 


With Safe Load Uniformly Distributed for Length of One Foot. 
(Beam supported Sidewise.) 


Depth. 

Designation. 

w 

Web. 

dth. 

Flange. 

Area. 

Section. 

Weight 
per Foot. 

Load. 

Strengtl 

Sidewise. 

as Strut. 
Edgewise. 

Ins. 

Channel. 

Inch. 

Ins. 

Sq. Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

3 

Extra Light 

.2 

1-5 

i -45 

5 

10 500 

51 

341 

4 


.2 

i -5 

1.65 

5-5 

15 700 

49 

597 

5 

(( U 

.2 

1.625 

1.92 

6-33 

22 800 

57 

93 ° 

6 

u u 

.18 

I -875 

2.25 

7-5 

33680 

77 

1403 

6 

Light 

.28 

2.25 

3-2 

II 

45 700 

IOI 

I 343 

6 

Heavy 

•4 

2-5 

4-32 

i 5 

58 300 

123 

1257 

7 

Extra Light 

.2 

2 

2-54 

8-33 

39 5oo 

82 

1700 

7 


•25 

2-5 

3-6 

12 

62 000 

136 

1883 

8 

L t U 

.2 

2.2 

3-3 

II 

65 800 

109 

2493 

8 

Light 

.26 

2-5 

4.48 

15 

88950 

142 

2480 

9 


•33 

2-5 

5.08 

16.33 

104 OOO 

124 

2892 

9 

10.5 

Heavy 

Light 

• 43 
•375 

3-125 

2.75 

7.02 

6 

23 -33 
20 

146 OOO 

134750 

190 

160 

2925. 

3685 

12.25 


.42 

3 

8.62 

28.33 

238 OOO 

172 

5275 

12.25 

Heavy 

.68 

4 

14. X 

46.66 

381 OOO 

317 

5WO 

i5 

Light 

•5 

4 

12 

40 

401 OOO 

301 

7833 

15 

Heavy 

•75 

4-75 

18.85 

63-33 

625 OOO 

428 

7762 

7 

1 Deck ( 

•3 1 

4-5 

5-35 

21.66 

63 500 

351 

36 

8 

j Beams. ( 
Strut Bars. 

•38 

4-5 

6.29 

18.38 

91 800 

547 

37 

5 

Light, Single 

— 

— 

i -55 

5-33 

9 100 

44 

457 

5 

Heavy, “ 

— 

— 

2-15 

7-33 

II 900 

48 

433 


Operation of Table. 

To Compute Deptli of a Beam to Support a TTniformly- 

Distributed. Load. 

Rule. —Multiply load in lbs. by length of span in feet, and take from 
table the beam, the load of which is nearest to and in excess of the product 
thus obtained. 

Example.— What should be depth of a beam to sustain with safety a uniformly 
distributed load of 30000 lbs., over a span of 15 feet? 

30000 X 15 = 450000, which is load for a heavy beam 12.25 ins. >n depth. 
Weight of beam should be added to load. 

Inversely .—If the load is required, divide load in table by span of beam in feet 
and subtract weight of beam. 


To Compute Deflection of Lilre Beams. 

Rule. — Divide square of span in feet by 70 times depth of beam in ins. 
Example. —Assume beam as preceding. 

15 2 225 


70X12.25 857.5 


. 262 ins. 
























STRENGTH OF MATERIALS.-TRANSVERSE. 809 


Comparative Strength, and. Deflection of Cast-iron 

Flanged Beams. 


Description of Beam. 

Comp. 

Strength. 

Description of Beam. 

Comp. 

Strength. 

Beam of equal flanges . 

•58 

Beam with flanges as 1 to 4.5... 

.78 

“ with only bottom flange.. 

.72 

“ with flanges as 1 to 5.5... 

.82 

“ with flanges as 1 to 2.... 

•63 

“ with flanges as 1 to 6 . 

I 

“ with flanges as 1 to 4.... 

•73 

“ with flanges as 1 to 6.73.. 

.92 


Dimensions and Proportions of Wrought-iron Flanged 

Beams. ( D. K. Clark.) 


Depth. 

Bread th 
of Flanges. 

Thicli 

Web. 

Jiess. 

Flanges. 

Weight per 
Lineal Foot. 

Ultimate 

Strength. 

Loaded 
in Middle. 

Safe Stress 

Uniformly 

Distributed. 

Ins. 

IllS. 

Inch. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

3 

2 

•1875 

.2187 

5-5 

2 800 

910 

3 

3 

•25 

• 3 I2 5 

IO 

5600 

1 860 

3- 12 5 

1.625 

W 

00 

Ln. 

. 2187 

5-5 

2 49O 

830 

4 

2 

• 25 

•3125 

8 

5 49 ° 

1 830 

4 

3 

■25 

•375 

12 

8 510 

2 830 

4-75 

2 

■25 

•3125 

8 

6 940 

2 310 

5 

3 

-3125 

■4375 

13 

13 440 

4 480 

5 

4-5 

•375 

• 5 

23 

19 270 

6 420 

5-5 

2 

•375 

•4375 

IO 

11 880 

3960 

6 

5 

•4375 

• 5625 

30 

23830 

7940 

6.25 

2 

• 3 I2 5 

•4375 

II 

13 440 

4440 

6.25 

2.25 

•3125 

•375 

18 

13 OOO 

4330 

6.25 

3- 2 5 

• 3 I2 5 

.4062 

12.5 

17470 

5820 

7 

2.25 

.281 

•375 

14 

14790 

4930 

7 

2.25 

•3125 

•4375 

14 

17 020 

5670 

7 

3.625 

•3125 

•4375 

*9 

23300 

7 700 

7 

3.625 

•3125 

•5 

19 

25980 

8 660 

8 

2-375 

• 3 I2 5 

•4375 

15 

20 830 

6940 

8 

2-5 

•375 

•375 

15 

21280 

7 090 

8 

4 

•375 

■5 

21 

34500 

11500 

8 

5 

•375 

•5625 

29 

44 800 

14930 

8 

5 -125 

•4375 

•5625 

29 

47040 

15 680 

9- 2 5 

3-75 

•4375 

•5 

24 

41560 

13850 

9-5 

4-5 

■375 

•6875 

30 

59 360 

19750 

10 

4-5 

•4375 

•5625 

32 

56000 

18 660 

IO 

4-75 

•4375 

•5625 

32 

58240 

19 410 

10 

4-75 

•75 

• 625 

36 

76160 

25390 

12 

5 

•5625 

•8175 

42 

100 800 

33 600 

12 

6 

•5625 

•9375 

56 

136 640 

45530 

14 

5-5 

•5625 

•875 

60 

150 020 

50000 

14 

6 

•5625 

•8175 

60 

152 260 

50750 

16 

5-625 

■75 

•8175 

62 

188160 

62 720 


Wrought-iron Rectangular Griuders or Tubes. (Riv'd.) 
Supported at Both Ends. Loaded in Middle. 

A ^. 9 . — W. A representing area of section in sq. ins., d depth in ins., I length be¬ 
tween supports in feet, and W destructive weight in lbs. 

Illustration.— What is the destructive weight of a rectangular girder, 35.75 ins. 
in depth by 24 in breadth, metal .75 inch thick, and length between supports 45 feet? 

Assume C or coefficient = 37 000, as per case (17) in preceding table, page 806, 
and area = 87.375 ins. 

Then 87-375 X 35-375 X 37 oo = ” 557 5 23 = 25fi 8 33 . 8 lhs _ 

45 45 

. W l . , l W 

By experiment it was 257 080 lbs. By Inversion — A, and — a. 


Hodgkinson’s 
303 907 lbs. 


formula would give a result of 259373 lbs., and Molesworth’s 

3 Y* 

























8io 


STRENGTH OF MATERIALS.-TRANSVERSE, 


l 2 W 


TJ nexqually Loaded Beams, etc. 


= w. I representing length between supports, and m and n distances from 

points of support, all in like denomination, and W and w destructive and safe weights, 
also in like denomination. 

To Compute Destructive 'Weigh.t and. Area of Bottom 

Blate. 


A d C _ W l W m n 

—= w i 7V7 — A ; and - - = A. 

L C d . 25 C d l 


A representing area of plate in sq. 


ins., d and l depth and length, m and n distances of load at other points than in 
middle, all in feet, and W weight in lbs. 

Note.— Sufficient metal should be provided in sides to resist transverse and 
shearing stress, and in upper flange to resist crushing. 

Illustration. —What area of wrought iron is necessary in bottom plate of a rec¬ 
tangular tubular girder, 3 feet in depth, supported at both ends, and loaded in middle 
with 130000 lbs. ? 

C, ascertained by experiment for destructive stress, 180000 lbs., and area 7.1 sq.ins. 

130000 x 30 

—x-—— = 7.22 sq. ins. 

180000 X 3 

Wrouglit-iron. Cylindrical Beams or Tubes. 

A d C 


l 


: W. Illustration. —W r hat is destructive weight of a cylindrical tube, 

12.4 ins. in diameter, .131 inch in thickness, and 10 feet between its supports? 

Area of metal = 5.05 sq. ins., and C = 2856, as in the 19th case of table, page 806. 
5.05 X 12.4 X 2856 


D. K. Clark. 


Then 

3.14 d 2 t S 
l 


- —17 884.2 lbs. 

1U 

= W. d representing diameter, t thickness of metal, and 
l length, all in ins., S tensile strength of metal per sq. inch, and W weight, both in lbs. 

3.14 X 12.42 X .131 X45°o° 2846250 

S = 45000 lbs. --=-= 23718.7 lbs. 

10 x 12 120 

Molesworth’s formula gives a result of 23 286.1 lbs. 

"Wrought-iron Elliptical Beams or Tubes. 

■ A - ^ ^ = W. Illustration. —Assume diameter of tube 9.75 and 15 ins., metal 

.143 inch 11 thickness, and distance between supports 10 feet. 

A = 5.56 sq. ins. C = 3147, as per case (20) in preceding table, page 806. 

Then 5-56X15X3147 = £ 62 ^ = z69 ^ 


D. K. Clark. 


1.57 (b 2 + d 2 )t S 
l 


: W. b and d representing conjugate and trans¬ 


verse diameter, l length between supports, t thickness of metal, all in ins. , S tensile 
strength of metal per sq. inch, and W destructive weight, both in lbs. 

n. i-57 (9-75 2 +i5 2 ) X .143 X 44000 3161840 , nrn 

S = 44 000/os. -—-— =-= 26348.6 lbs. 

f IO X 12 120 ^ 

Note —B. Baker, in his work on Strength of Beams, etc., London, 1870, page 26, 
shows that ordinary method of computing transverse strength of a hollow shaft by 
difference of diameter alone is erroneous, in consequence of loss of resistance to 
flexure in a hollow beam. 

Grirders and Beams of Unsymmetrical Section. 
— ^ — = W. S representing tensile resistance of metal, and W destructive weight, 

both in lbs., d distance between centres of compression and extension, or crushing and 
tensile resistances, in ins., and l length between supports, in feet. 

Note. —To ascertain d, see Rule, page 8x9, 
















STRENGTH OF MATERIALS.-TRANSVERSE. 8 I I 


Bottom flange.4 X .38 inch. 

Depth.7 ins. 


Illustration. —Dimensions of a rolled wrought-iron girder, n feet in length be¬ 
tween its supports, is as follows : 

Top flange.2.5X1 inch. 

Web. 325 “ 

What is its destructive weight? 

d = 5.22 ins. S assumed at 45000 lbs. Then 4 X _ 45ooo x 5-22 _ g l8 ^ 

11 X 12 

Strength of Riveted Beams or Girders, compared with Solid, is less, and deflec¬ 
tion is greater. 


L W 


"Wrought-iron Inclined. Beams, etc. 

: io. L and l representing lengths or inclination, and horizontal line, in like 

denominations , and W and w destructive and safe weights on horizontal line and in¬ 
clination, also in like denominations. 


AdC 

l 


!Plate Grirders. 

; W. A representing section in sq. ins ., d depth in ins ., and l length be- 
miL tween supports in feet. 

Illustration.— What load will destroy a wrought-iron plate girder or beam of 
following dimensions, 10 feet in length between its supports? 


Top flange. 4.5 X -375 inch. 

Bottom flange. 4-5 X .375 “ 

Angle pieces.2 X.3125 “ 

Avea of Section = 13 sq. ms. 

Assume coefficient of 5180 as per case (14) in preceding Table, page 806. 


Width of web.375 inch. 

Depth of web.13.5 ins. 

Depth of beam.14-25 “ 


Then 


13X14-25X5180 960154 


= 96 015.4 ^s. 


£* = W. 
8 d 


10 10 

Moleswortit. = W. L representing load equally distributed , and W destruc¬ 
tive weight , both in tons, and d effective depth of girder in feet. 

By actual experiment L = 43 tons for 16.5 feet between supports; hence, 

10: i6.5:i48:79.2 tons = 39.6 when supported in middle, and 14.25 ins. = 1.1875 feet. 
39.6X10 _ 396 
9-5 


Then 


D. K. Clark. 


8 X 1-1875 
d (4 a -f 1.155 a') 
.6 l 


41.68, which X 2240 = 93 363.2 lbs. 


— W. d representing depth of girder or beam 


less depth of lower flange in ins., a and a' areas of sections oj bottom flange and of 
web, at its reputed depth, both in sq. ins., and l length between supports in feet. 

d = 14.25 — .375 = 13-875 ins. a — 3, and a' — 5 sq. ins. 

12.875 (4 X 2 - 4 -1.155 X 5) 246.63 ... 

Then —— 5 ^ ^ ———— = ■ ^ = 41.105, which x 2240 = 92 075.2 lbs. 

.0 X 10 6 

Mr. Clark assumes, however, that for girders of like construction the destructive 
stress should be taken at two thirds of that deduced by the formula. 

I Girders or Beams without Upper and Lower Flanges. 

Illustration. —Assume angles 2.125 X -28 above, 2.125 X -3 below, web 
.25, depth 7 ins., and length between supports 7 feet. 

Area of section = 6.35 sq. ins., and C = 3840, as per case (15) in preceding Table, 
page 806. 

6 -35 X 7 X 3840 170688 


Then 


24 384 lbs. 


Approximate. 


- + .25 a'X5<! 

2 

r~ 


W. a representing area of sections of upper 


and lower angles, a' area of section of web for total depth , both in sq. ins., d depth of 
girder in ins., and W load or stress in lbs. 
























812 STRENGTH OF MATERIALS.-TRANSVERSE. 


a = 4.6 sq. ins ., and w 


Then 


4.6 , 1.75 

V+ 4 X5Xj 


7 X .25 = 1.75 sg. ins, 
90.81 


— 12 -973) which X 2240 = 29059.5 lbs. 


IRON AND STEEL RAILS. 
Symmetrical Section.. 

To Compute Transverse Strength. (D. K. Claris.) 

( d'2 . _ X 

4 « -^-+ i-155 t d2 J 


- = W, and 


W l 


( <*' 2 , \ 
V 4 ® 1,155 


= S. S representing ten¬ 


sile strength in lbs. or tons per sq. inch, a a?-ea 0/ one head or Jiang e exclusive of cen¬ 
tral portion composing web, in sq. ins., d' depth or distance between centres of heads 
d depth of rail, t thickness of web, l distance between supports, all in ins and W 
weight in lbs. or tons, alike to S. 

Illustration 1.—What is destructive weight of a wrought-iron double-headed 
rail, 5.4 ins. deep, having a web of .8 ins., an area of head of 1.9 sq. ins. distance 
between centres of its heads 4.2 ins., and between its supports 5 feet ? ’ 

S assumed at 50000 lbs. 

50000 (4 X 1.9 X — + 1.155 X .8 X 5-4 2 ^ 

Then --- 5 G_So000 X (25.23 -f- 26.93) _ 

,, , „ 5X12 15 - 

43460.0 lbs. 

2.—What is destructive weight of a Bessemer steel double-headed rail c 4 ins 
deep, having a web of .75 inch, an area of head of 2 sq. ins., and distance’between 
heads 4.2 ins. ? 

S assumed at 80000 lbs. 


Then 


80000 (4X2X Z - J 55 X -75 X 5 - 4 2 ) 


5 X 12 


80000 x 51.59 
60 


= 68 520 lbs. 


th?proportion'd CartonT'if ° f BeSSemer Rails increa9es ver X g^erally, in direct proportion with 


6.92 S d" A 


TJ11 symmetrical Section. 


— -_ W. d" representing vertical distance between centres of tension 

and compression, h height of neutral axis above base of section, and l length between 
supports, all in ms., and A sum of products, obtained by multiplying areas of strins 
of reduced section under tensile stress, by their mean distances, respectively'that is 
the distances of their centres of gravity, from the neutral axis, in ins. ’ 

Bowstring Girder. 

To Compute Diameter of a Wrought-iron Tie-rod of an 
Arc hed or Bowstring Girder of Cast Iron 
/ W l 

V 4500 X h ®’ '' representing weight distributed over beam in lbs., l length 

between piers or supports in feet, and h height between centre of area of section of 
girder and centre of rod m ms. J J 

Illustration. Required diameter of tie-rod for an arched girder 2^ feet be 
tween its piers, and 30 ins. between centres of its area and of rod, to safely support 
a uniformly distributed load of 25 000 lbs. ? > j 

/25 000 X 25 _ 7625 000 

V 4500X30 “ V = 62 = 2 ‘ ins. 

.62 


4500 X 3 ° 
If two rods are used. Then 


U. 62 

W — = 1-52 ms. = diameter of each rod. 



















STRENGTH OF MATERIALS.-TRANSVERSE. 


813 


CAST IRON. 

Transverse Strength of Grirders and Beams. 

(Deduced from Experiments of Barlow, Hodakinson, Hughes, Bramah, Cubitt, 

Tredgold, and others.) 

Reduced to a Uniform Measure of One Foot in Length. 

Supported at Both Ends. Stress or Weight applied in Middle. 

Section. 


C£3 











Destructive Weight. 

1 w „ 

—r = C. 

Flanges. 

Web. 

Depth. 

Distance. 

Area. 

For Dis- 

Length 





tance. 

of One Foot. 

A d 

Ins. 

Ins. 

Ins. 

Feet. Ins. 

Sq. Ins. 

Lbs. 

Lbs. 


— 

I 

I 

I 


I 

2 24O 

2 24O 

2240 

— 

I 

I 

4 

6 

I 

500 

2 250 

2250 

— 

3 

3 

13 

6 

9 

5080 

68 580 

2540 

— 

I 

3 

4 

6 

3 

5100 

22950 

2550 

— 

I 

4 

4 

6 

4 

IO 3OO 

46350 

2896 

4 X 2 

2 

4 

5 


12 

6 720 

33600 

700 

1.52 x .78 

1.56 

4.07 

4 

6 

2-35 

6666 

30 000 

3136 

i -5 X -5 

•5 

3 

3 

I 

2 

5 208 

16145 

2676 

i -5 X .5 

• 5 

3 

3 

I 

2 

4 536 

14 062 

2331 

i -5 X .5 

• 5 

4 

3 

I 

I 

7 io 4 

22 42O 

5475 

r -5 X .5 

• 5 

4 

3 

I 

I 

3 3 i 2 

10 267 

2553 

1-53 X 1 

•s 

2.04 

4 


2.6 

4 004 

16016 

3019 

2 X - 5 1 

I 

2.02 

4 


2-59 

2569 

10 276 

1963 

— 

— 

2.52 

5 


4.98 

4143 

20715 

1650 

— 

— 

2.83 

5 


4 

2 988 

14940 

1320 

2.28 X -S 3 

f ' 3 

l -425 

} 5-13 

4 

6 

2.28 

9503 

42 763 

3656 

23-9 X 3-12 

3-29 

36.1 

20 


iS 3-5 

403 312 

8 0 66 240 

1220 

1.76 X .4 

.29 

5 -i 3 

4 

6 

2.82 

6678 

30512 

2077 

1.74 X .26 
i-7 8 X -55 

} <3 

5 -i 3 

4 

6 

2.87 

7368 

33200 

2250 

1-07 X -3 
2.1 X .57 

} - 3 2 

5 -i 3 

4 

6 

3.02 

8 270 

37215 

2402 

1.54 X .32 
6.5 X .51 

} ’ 34 

5 -13 

4 

6 

5 - 4 i 

21 OO9 

94 54 ° 

3406 

2.5 X1.5 
3-75 X 1.4 

1 1-25 

8.18 

II 


i 5 

35 620* 

39 1 853 

3 i 93 


* Stirling iron. 

Hence, — y— = W. A representing area of section, d depth in ins., I length in feet, 
and W destructive weight in lbs. 

Note.—W hen lengths are less than those instanced, breaking weight will be in¬ 
creased, in consequence of increased stability of girder. 




















8 14 STRENGTH OF MATERIALS.-TRANSVERSE. 


To Compute Transverse Strength, or Destructive Stress 

of Cast-iron Beams or C3-ird.ers, of various Digores. 

Supported at Both Ends. Weight applied in Middle. 

When Section of Beam or Girder is alike to any of Examples given in 
preceding Table. Rule i.*— Divide product of area of section and depth 
in ins., and Coefficient for girder, etc., from preceding Table, by length be¬ 
tween supports in feet, and quotient will give breaking weight in lbs. 

Example. —Dimensions of a beam, having top and bottom flanges in proportion 
of i to 6, give an area of section of 25.6 sq. ins., a depth of 15.5 ins., and a length 
between its supports of 18 feet; what is its destructive weight? 

Note.— In consequence of increased area of metal over case No. 21 in Table, Coef¬ 
ficient of 3402 is reduced to 3300. 

Dimensions. —Top flange, 3 X -75 ins.; bottom, 18 X .75 a = 13.5 sq. ins. ; web, 
15-5 X .7 a'= 10.8 sq. ins.; and d' — 15.5 — .75= 14.75 ins. 

Theu 25.6X .S. 5 X 33 ~ = » 3 °;^ &6 , bs 

io 18 


D. K. Clark. 


d' (6.5f a-f- 2 a') 
3 1 


= W. a representing area of bottom flange, a' 

of web at depth d' of beam , less depth of bottom flange in sq. ins., I length between 
supports in feet, and W destructive weight in tons. 


Then 


14-75 (7 X i 3'5 + 2 X 10.8) 1712.4 


31.71, which X 2240 = 71030.4 lbs. 


3X18 54 

Hodgkinson’s formula would give a result of 53491.2 lbs., and Molesworth’s 
54 248.3 lbs. 


Rule 2 . — From product of breadth and square of depth in ins. of rec¬ 
tangular solid, the dimensions of which are the depth and greatest breadth of 
beam in its centre, subtract product of breadths and square of depths of 
that part of the beam which is required to make it a rectangular solid, and 
then determine its resistance by rule for the particular case as to its being 
supported or fixed, etc. 

This rule is applicable only in case referred to, viz., when area of section is great 
compared with area of extrepie dimensions. 

Mr. Baker, in case of a hollow cylindrical shaft, where thickness of metal is but 
one eighth of extreme diameter, computes result at but .4 of that of a solid beam. 
This is in consequence of resistance to flexure in hollow beam being more than 
proportionally greater than in solid. 

Example.— Take 7th case from preceding Table, page 813, for length of one foot. 

Coefficient for cold-blast iron — 500. 


Then 1.52 X 4.07 2 —1.52 X 2-5i 2 X 4 X 500 = 25.17 — 9.58 X 2000 = 31180 lbs. 

Result as by experiment, 30000 lbs. 

Note i. —These rules are applicable to all cases where flange of beam is as shown 
in Table, and beam rests upon two supports, or contrariwise, as to position of flange 
when beam is fixed at one end only. 

2.—When case under consideration is alike in its general character to one in 
Table, but differs in some one or more points, an increase or decrease of metal is ob¬ 
tained by an increase or reduction of the Coefficient, according as the differences may 
affect resistance of beam. 


3.—The Coefficients here given are based altogether upon experiments with Eng¬ 
lish iron. 


* Utility of these rules in preference to those of Hodgkinson, Fairbairn, Tredgold, Hughes, and 
Barlow is manifest, as in one case the Coefficient of the metal is considered, and in the other cases the 
metal is assumed to be of a uniform value or strength. 

Only variable element not embraced in this rule is that consequent upon any peculiarity of form of 
section ; as, for instance, in that of a Hodgkinson, or like beam, where area of one flange greatly ex¬ 
ceeds the rest of section, and this flange is other than below, when beam rests upon two supports or is 
fixed at both ends, or than above, when beam is fixed at one or both ends. 

This deficiency is met to some extent by the three cases in table, where proportion of flanges are i to 
2,1 to 3, and x to 6.5. 

t For thick castings put 7, and put Coefficient same as tensile strength of metal in tons per sq. inch. 









STRENGTH OF MATERIALS.-TRANSVERSE. 


815 


Flanged Hollo'W' or Aixmalar Beams of Symmetrical 
Sections. ( D. K. Clark.) 

When Depth is Great Compared with Thickness of Flanges. —Figs, 1,2, and 3. 

x. 2. 3. 


d X S (4 a-\- 1.155 a') 


l 


= W. a representing area of one 

flange, a' area of web or ribs, both in sq. ins. , d depth of 
beam, less depth of one flange, and l distance between sup¬ 
ports, both in ins., S tensile strength of metal, and W 
weight between supports, both in lbs. 

When Depth of Flanges is Great Compared with Depth of Beam. —Figs. 

4 and 5. 

5 - 


S 1-155 1 d2 ) 


= W. a representing area of one flange less 


1 1 

JJL thickness of web, in sq. ins., t thickness of web, d’ reputed depth or 
distance between centres of flanges, and d depth of beam, all in ins. 

When Section of Circular or Elliptic Beam is Small Compared with Diam¬ 
eter. —Figs. 6, 7, and 8. 

6. , ,J 2 / a 7. 8. 



3.14 d 2 t S 


l 


= W. 




1.57 (& 2 + ^)*S 


l 


= w. 


b and d representing mean breadth and depth. 

Illustration i. —Assume Figs. 1, 2, and 3, 20 ins. in depth, width of flanges on 
top and bottom ribs 5 ins., thickness of flanges and webs 1 inch, and of sides of 
Fig. 3 -5 inch; length between supports 10 feet, and S 20000 lbs.; what would be 
breaking weight of each ? 


Then 


20 —1 X 20000 (4 X 5 + i-i 55 X 18) _ 380000 (20 + 20.79) _ i68 _ 4 lbs 


10 X 12 


120 


2._Assume Figs. 4 and 5, 6 ms. in depth, area of flanges 3 ins., widths of webs 1 

inch, and length and S as in preceding case. 


20 000 (4 X 3 X 


1.155 X 1 X 6 5 


Then 


20000 X 91-58 


15263.3 lbs. 


IO x 12 120 

o.—Assume Fig. 6 io ins. in diameter, Fig. 7, 7.5 ins. in depth and 12 ins. in width, 
and Fig. 8,12 ins. in depth and 7.5 ins. in width, and thickness of all metal 1 inch. 

Then Fig 6 3 -*4 X io 2 X i X 20000 = 6280000 ^ g2 33^ ^ w]iic h is .4 of 
10 X 12 120 

that of solid cylinder. 

F s aDd 8 .■ 57 x(..- + 7.iqx,x «.<■■» = » = 52 398.75 m, 

10 X ^2 

m ote _For all ordinary purposes, operation of computing their strength, by first 

computing that of their circumscribing figure, and then deducting from it strength 
due to difference between it and section of beam under computation, will be suf¬ 
ficiently accurate. See Illustration, page 814. 

If greater accuracy is required, see page 810, or D. K. Clark's Manual , pp. 513-17. 

Note._ To compute location of neutral axis of beams of unsymmetrical section, 

see also D. K. Clark, pp. 5 I 4 _I 5 - 

* This result agrees with deduction of Mr. Baker, as given by him in his work on Strength of Beams, 
etc pp. 26-7, foAollow or annular beams of small area of section compared with that of diameter, 
evcn P i!p to a thickness of metal of one eighth of diameter. He assigns their strength so low as .4 of 
that of solid cylinder, in consequence of loss of resistance to flexure. 

































8l6 STRENGTH OF MATERIALS.—TRANSVERSE. 


Greneral Formulas for Destructive Weight of Solid 
Beams of Symmetrical Section. 

Supported at Both Ends. Weight applied in Middle. 

Line of Neutral Ascis runs through centre of gravity of section. 

2 a d r $> , l W _ , 

- - - = W, and -— = S. In square beams for a d put d 3 . a and d rep- 

l ’2 a dr * 

resenting area and depth of section, r radius of gyration (half depth of beam = 1), 
l length of beam between its supports in ins., W destructive weight in tons or lbs., 
and S tensile strength of material in lilce tons or lbs. per sq. inch. 


Illustration.— Assume dimensions of cast-iron beams, Figs. 1, 2, 3, 4, and 5, as 
follows, viz.: 1 and 2, 5 X 5 ins.; 3, 2.5 X 10; 4, 5.64 diameter; and 5, 7.25 X 4.39, 
or equal areas; distance between supports 60 ins., and tensile strength of irou = 
20000 lbs. 



Areas of each 25 sq. ins. Radius of gyration, No. 1, .5775; 2, .4083; 3, .5775; >, 
.5; and 5, 1.43. 

• 2 X 25 X xo X .5775 X 26000 

1 • - — - = 125 125 lbs. 

2 X 25X7-°7*X.4083X26000 
2. - — - =62 545 tfrs. 


2 X 5 3 t X -5775 X 26000 , , „ 

3 - - f -■ ——-— = 62562105. 


60 


4. For formula for square beams substitute 


ad S 
l 


= W 


. 25X5-64X26000 „ „ , „ .7854 bd 2 S „ 

then 4. ---= 57766 lbs.; and for 5. ----- -— — W. 


■ 7854 x 4.39 X 7.25 s X 26000 
60 


= 78 532 lbs. 


These formulas give a result equal to a transverse strength for Cast iron of 550 for 
a tensile strength of 26000 lbs., and of Wrought iron of 600 lbs. for a like strength 
of 50000 lbs. (as per table, page 788). 


6. 


Q 


8 . 

dl 


9 - 

1 ™ 


IO. 


4 C bd 2 


} —W.c representing coefficient of strength of metal in lbs., b and d 
breadth and depth in ins., I length in feet, and W destructive weight in tons. 

R 4 — 7*4 


6 . 


j:> 4-7 — 6 d R and r representing external and internal radius. 


7 - ^ — b d 2 . b' and d' representing interior breadth and depth, 

b d 2 

8. . 38 R 3 — bd 2 . 9. —— — W. d representing depth or height. 


*°: d 2 + 2 b'd ' 2 = W. b and d representing breadth and depth of centre and 
vertical rib, and b and d breadth and depth of horizontal rib, external to central rib. 


\ alues of C 550 for a tensile strength of Cast Iron of 26000 lbs. per sq inch and 
of 600 for a like strength of Wrought Iron of 50000 lbs., and pro rata. ’ 


* Diagonal of square. 


t In square beams <23 = aX d. 
























STRENGTH OF MATERIALS.—TRANSVERSE. 817 

Flanged Beams of Unsymmetrical Section.. (D. K. Clark.) 


0 



4 S d 


l 


: W. S representing total tensile strength of section in lbs. per sq. inch, d 

vertical distance between centres of tension and compression in ins., l length in ins., 
and W weight in lbs. 

Illustration.— If the sectional area of a beam of cast iron is 5.9 sq. ins., the 
depth or distance between centres of tension and compression 5.6 ins., distance be¬ 
tween supports 5.5 feet, and tensile strength of metal 30000 lbs. per sq. inch. 


Then 


4 X 5-9 X 30000 X 5-6 _ 3964 800 


5-5 X 12 


66 


= 60072.7 lbs. 


STEEL. 

'-To Compnte Transverse Strength, of Steel Bars. 
Supported at Both Ends. Weight applied in Middle. 

1.155 S b d s 


l 


W. S representing tensile strength in lbs., l length between supports 
in ins ., and W iveight in lbs. 

Illustration.— What is ultimate destructive stress of a bar of Crucible steel 

2 ins. square, and 2 feet between supports? 8 = 90000 lbs. 

™ 1. iS 5 X 90000 X 2 3 831600 

Then ——— f - = -2 - — 34 650 lbs. 


2 X 12 


24 


To Compote Section of Bower Flange of a Grirder or 
Cylindrical Shaft of Cast Iron to Sustain a Safe Load 
in its Middle. {Baker.) 

I dW 

——— = M. Z representing distance between supports infect, d depth of girder, etc. 
in ins., W weight in tons , C coefficient, and M moment of weight around support. 

Illustration.— What should be section of a girder, 12 ins. deep, to sustain a safe 
load of 10 tons in its middle, between supports 16 feet apart? 


Stress assumed 2 tons per sq. inch, and Factor of safety 4. 
M 


16 X 12 X 10 


=48o=M. 


And 


tZXS 
flange in sq. ins. 


a. S representing stress assumed in tons, and a area of section of 


Then 


480 


20 sq. ms. 


12 X 2 

Bor Rectangolar, Diagonal, or Circular Beam or Shaft. 


d * 2 b 


= M. 


10.2 


M. 


<Z 3 

8 ^ 


= M. 


General Formolas for Compotation of Destroctive 
Weight of a Beam or Girder of any form of Cross 
Section and of any Material. (B. Baker.) 

Load applied at Middle. 

V (1 + Q ) _ ^ g ,. e presenting tensile strength of material per sq. inch in tons, 
4 * 

M moment of resistance of section — product of effective depth of girder or beam, and 
effective area of flange portion of section, in sq. ins., Q resistance due to flexure, l dis¬ 
tance between supports in feet, and Q' = Q X thickness of web of section , both in ins. 


Average Values of S for Various Materials. 


Tons. 

Cast Iron. 7 

Wrought Iron.21 


Tons. 

Steel.40 to 50 

“ plates. 35 

3 Z 


Oak. 

Pine 


Tons. 

2.5 to 4.5 
2 “ 3-5 































818 


STRENGTH OF MATERIALS.-TRANSVERSE. 


Substituting Values of S and Q in a General Equation. 


Section. 

Cast Iron. 

Wrought 

Iron. 

Steel. 

Oak. 

Pine. 


Hi 

IK 


d 2 b 

W =-875 7 T 

d 2 b 

=x -75 — 

d 2 b 

=3 t0 5 — 

d 2 b 

=.14 to.25 — 

d 2 b 

= .II 10,2 —r- 

V 




d 3 

W=. 7 S T 

d 3 

=I ' 5 T 

. . d 3 

=2.625 to 4.25 — 

. d 3 

=. 1 to 16 — 

i 

d 3 

=.08 to. 14 — 

L 


gl 


W=. 5 62 5 y 

d 3 

=i - i25 t 

. d 3 

=2 to 3.25 — 

J J l 

0. d 3 

=.08 to 14 — 

l 

dZ 

=.06 to. II — 

L 


d representing depth of a rectangular bar, side of a square, or diameter of a round , 
b breadth of a vertical bar, all in ins., and l distance between supports in feet. 


Moment of Resistance. 

Moment of Resistance of a cross section is the static force resisting an ex¬ 
ternal force of tension or compression, and it is equal to moment of Inertia, 
divided by distance of centre of effect of the area of fibres which are respec¬ 
tively the most extended or compressed from the neutral axis of the section. 


To Compute Moment of Resistance. 

— = M. I representing moment of inertia, and d distance of centre oj effect of 
area of fibres of extension or compression. 

Work of Resistance. 

Under a Quiescent Load. —Intensity of Elastic resistance increases uni¬ 
formly with total space through which action of stress operates; hence, it 
may be defined by a triangular section. 

Consequently, .5 s L = R. s representing space passed through, L load, and R re¬ 
sistance. 

To Compute Moment of Resistance. 

—and —— R. C a coefficient = one sixth of destructive weight, I moment 

of inertia, h height of neutral axis from base of section, R moment of resistance, and 
M modulus of rupture. 

Note.— Neutral axis, for all practical purposes, is at centre of gravity of any 
section. 

For Radius of Gyration, see Centre of Gyration, page 609. 

For other rule for computation of Moment of Resistance, see Strength of Beams, 
B. Baker, London, 1870. 

Moment of Inertia. 

Moment of Inertia is resistance of a beam to bending, and moment of any 
transverse section is equal to sum of products of each particle of its area into 
square of their distance from neutral axis of section. 



C D 


Illustration.—I f transverse section of a beam. A B C D, Fig. 1, is 
8 X 20 ins., its neutral axis will be at middle of its depth, 0 r; divide 
A B, o r, into an} 1- number of equal spaces, as shown, then each space 
will be 2X2 = 4 sq. ins., and the distances of the centre of each 
square from neutral axis will be as follows : 


1 , x. 2X2X4 Xi 2 = 16 

2, 2. 2 X 2 X 4 X 3 2 = 144 

3, 3. 2 X 2 X 4 X 5 2: = 4 00 


4,4. 2X2X4X7 2 — 784 
5, 5. 2 X 2 X 4 X 9 2 = 1296 

2640 X 2 for low¬ 
er half = 5280 = moment. 


Note.— If the area of the figure in illustration had been more minutely divided, 
the result would have approximated more nearly to the above result. 


For Moment of Inertia of a Revolving Body, see Centre of Gyration, page 609. 





























STRENGTH OF MATERIALS.-TRANSVERSE. 819 

To Compute .Moment of Inertia of a Solid. Beam.-Fig. 2 . 


b d 3 


= M. 


Illustration. —Take elements of preceding case. 

8 x 20 3 64000 

then - = -— 5333.33 moment. 

12 12 3000 03 

Or, .3 t 3 n 3 b = M. t representing breadth of vertical divisions , n number of hori¬ 
zontal divisions from plane of neutral axis , b breadth , and d depth of beam. 

Illustration. —Take elements of preceding case. 

t = 2, n = 5, and b = 8. 

Then .3 X 2 3 X 5 3 X 8 = 2400 X 2 for lower half = 4800 = moment. 


*-~b- 


<r-b—t 



Beams of Various Figures. —Figs. 3, 4, 5. 

bdS — b'd '3 J l)d 3 — 2 b'd'Z 

3- - -- , 4 and 5-- = M. 

12 12 

V and d' representing respectively breadth less 
thickness of web, and depth less thiclcness of flanges. 


-7854 r 2 = M. 
.7854 (r*-r'*) = M. 


.7854 c t 3 = M. 


b d-i 


— M. 


- 6 —> 


12 


nr* 


36 

.11 r~ — M. 


r representing radius , t transverse and c conjugate diameters , and s side. 


To Compute Common Centre of Gravity and 'Vertical 
Distance between Centres of Crushing and Tensile 
Stress of a Girder or Beam. 

Rule. — Multiply surface of section of each part or figure composing 
whole, by distance of its centre from centre of one of the two extreme parts 
or figures, as .; divide sum of their products by sum of surfaces of sec¬ 
tion, and result will give distance of common centre of gravity from centres 
of each extreme part or figure. 

Example. —Take annexed figure. 

2.5 X1X0 =2.5 Xo = .0 


+ .2.5 A 

: 

*T .325 


Above < 


—4- 


l 


/ 5.62 , i\ 

•325 x + —) = -325 x 3-31 — i-o 

.38 X 4 X + 5-62 + = 1-52 X 6.31 =9.5 


76 


4.345 10.667 

Dividing 10.667 by 4.345 = 2.455 — distance of common centre from centre of upper 
part. 

1.52 Xo =1.52 Xo = -o 


Below \ 


.325 X 5-62 X (^ + v) = i -826 X 3 = 5 

2.5 X ^ +5-62 + ‘-| 8 j =2.5 X 6.31 = 15. 


.478 


775 


5.846 21.253 

Dividing 21.225 by 5.846 = 3.631 —distance of common centre from centre of lower 
part. 

Hence, 3.631 + ^? = 3.821 = distance of common centre from bottom , and 3.631 -f 
2 

2.652 = 6.283 = distance between centres of gravity. 



















820 STRENGTH OF MATERIALS.-TRANSVERSE. 


To Compute jN'evitral Axis of a Beam of Unsymmetrical 
Section.—Figs. 3 , 4 , 5 , 6, V, S, and. 9 . ( D. K. Clark.) 

Operation. —Divide section as reduced into its simple elements, and 
assume a datum-line from which moments of elements are .to be computed. 
Multiply area of each element by distance of its own centre of gravity from 
datum-line, to ascertain its moment. Divide sum of these moments by to¬ 
tal reduced area; and quotient is distance of centre of gravity of reduced 
section, or of neutral axis of whole section, from datum-line. 

Illustration.— Fig. 8 annexed is 12 ins. deep, 12 ins. wide, aud 1 inch thick. 


8 . 


e- 


f 


Extend web, c d, to the lower surface at d' and d". leaving 5.5 ins. 
of web, a d' and d" b , on each side. Reduce this width in the ratio 
of 1.73 to 1, or to (5.5-r- 1.73 — ) 3.2 ins., and set off d' a' and d" b' 
each equal to 3.2 ins. Then reduced flange, a' b', is (3.2 X 2 = 6.4 -f- 
1—) 7.4 ins. wide, and reduced section consists of two rectangles, 
a' b' and c d. Assume any datum-line, as ef at upper end of sec¬ 
tion, and bisect depths of rectangles, or take intersections of their 
diagonals at g and 0, for their centres of gravity. Distances of these 
ifr-j 1 from datum-line are 5.5 and 11.5 ins. respectively, and areas of the 


a a'd'd"H b rectangles are nXi = nsq. ins., and 7.4 X 1 = 7.4 sq. ins, 

Then, cd =11 X 5 - 5 = 60.5 
a' V — 7.4 X 11.5= 85.1 

18.4 145.6 = 7.91 ins. 

Showing that centre of gravity of reduced section, being neutral axis of whole 
section, is 7.91 ins. below upper edge, in line i i. Centre of gravity of entire section 
at • , it may be added, is 8.65 ins. below upper edge, or .74 inch lower than that of 
reduced section. 

Neutral axes of other sections, Figs. 3 to 7, found by same process, are marked on 
the figures. Section of a flange rail, No. 7, which is very various in breadth, may be 
treated in two ways: either by preparatorily averaging projections of head and 
flange into rectangular forms; or, by taking it as it is, and dividing it into a con¬ 
siderable number of strips parallel to base, for each of which the moment, with re¬ 
spect to assumed datum-line, is to be ascertained. First mode of treatment is ap¬ 
proximate; second is more nearly exact. 

To Compute Ultimate Strengtla of Homogeneous Beams 
of TJiisymiXLetrical Section. 

Operation. —Resuming section, Fig. 9, for which neutral axis has been 
ascertained, 

To Compute Tensile Resistance , 

Divide portion below neutral axis i i, Fig. 9, with reduced width of 
flange, a 1 b\ into parallel strips, say .5 inch deep, as shown, 
and multiply area of each strip by its mean distance from 
neutral axis for proportional quantity of resistance at 
strip. Divide sum of products, amounting in this case 

_ l to 31.3, by extreme depth below neutral axis = 4.09 ins., 

d > and multiply quotient by 1.73 S (ultimate tensile resist- 
“fS I—| ance at lower surface). The final product is total tensile 

V resistance of section; or, 

3^*3 X 1.73 S 

— 1 * -- — = 13.24 S total tensile resistance. 

4.09 

S representing ultimate tensile strength of material per sq. inch. 

Again, multiply area of each strip by square of its mean distance from neu¬ 

tral axis, and divide sum of these new products, amounting to 104.64, by 

sum of first products. The quotient is distance of resultant centre of tensile 

stress, d‘, from neutral axis. Or, resultant centre is, 

10 - 4 S * * * * 10 '— = 3.34 ins. below neutral axis. 

3 I -3 

This process is that of ascertaining centre of gravity of all the tensile resistances. 


VE 


— 
















STRENGTH OF MATERIALS.-TRANSVERSE. 821 


By a similar process for upper portion in compression, sum of first products is 
ascertained to be same as for lower portion = 31.3. 


But maximum compressive stress at upper portion is greater than maximum 
tensile stress at lower portion, in ratio of their distances from neutral axis, or as 

i-73 S X ^ — 3.34 S, and 313 3 ' 34 -— = 13.24 S total compressive resistance. 

4.09 7-9 1 

which is same as total tensile resistance, in conformity with general law of equal¬ 
ity of tensile and compressive stress in a section. 


Sum of products of areas of stress, divided by squares of their distances respec¬ 
tively from neutral axis, is 164.9, and resultant centre c, Fig. 9, is ■■ 9 = 5.27 

3!-3 

ins. above neutral axis. 


Sum of distances of centres of stress or of resistance from neutral axis, 3.34-f- 
5.27 = 8.61 ins. — distance apart of these centres as represented by central line, c'd'. 

Abbreviated Computation .—As upper part of section is a rectangle, its resultant 
centre -of height, or 7.91 x -§ = 5.27 ins. above neutral axis. Average resist¬ 
ance is half maximum stress, viz., that at upper portion, which is 3.34 S per sq. 
inch. 

Area of rectangle therefore = 7.91 x 1 = 7-91 sq. ins., and 7 ~ 9 X X 3-34 _ ^ 2I g 

compressive resistance , as before determined. 


Moment of tensile resistance = 13.21 X8.61 ins.= 113.76 S, also = —- , or 4 ^ - 

4 t 


W. S representing total resistance of section in lbs ., d vertical distance apart of 
centres of tension and compression , and l length between supports, all in ins. 

Strength of Beam Inverted .—When inverted, maximum tensional resistance of 
beam at its lower surface c, Fig. 8, is 1.73 S. 

7.Q1 X 1.73 s 

Area of rectangle i i 0 = 7.91 sq. ins., and —-- -=6.79 S total tensile re- 

2 

sistance, or about one half of beam in its normal position. 


Note. —For other rule for computation of centre of gravity, see Strength of Beams, etc. B. Baker, 
London, 1870. 


Comparative Qualities of Various IVIetals. Major Wade. 


Metals. 

Density. 

Compres¬ 

sion. 

Tensile. 

Torsion. 

Trans¬ 

verse. 

Tensile 
to Com- 

Hard¬ 

ness. 






pression. 





Sq. Ins. 

Sq. Ins. 

Sq. Ins. 

Sq. Ins. 

I to 9.4 



Least.... 

6.9 

84 529 

9 000 

— 

416 

4-57 

Cast Iron- 

Greatest. 

7-4 

I74 120 

45 97 ° 

— 

958 

I “ 3.8 

33-51 


Mean.... 

7.225 

144 916 

31 829 

8 614 

680 

1 “ 4.6 

22.34 

Wrought Iron 

Least.... 
Greatest. 

7.704 

7.858 

40 OOO 
I27 72O 

38 027 
74 59 2 

2 915 

3 6 43 

542 

I u I 

I u 1.7 

10.45 

12. 14 

Cast Steel- 

Least_ 

Greatest. 

7.729 

8-953 

198944 
39 1 9 8 5 

128 000 

28 280 

1916 

I to 3.1 

— 

Bronze. 

Least.... 

7.978 

17 698 

1 852 

— 

— 

4-57 

Greatest. 

8-953 

— 

56 786 

2656 

— 

— 

5-94 


Factors of Safety. 

Girders, Beams , etc., of cast iron should not be subjected to a greater stress 
than one sixth of their destructive weight, and they should not be subjected 
to an impulsive stress greater than one eighth. 

The following are submitted by English Board of Trade, Commission¬ 


ers, etc. 

Structure. 


Cast Irox. 

Girders. 

Columns. 

Tanks. 

Machinery. 


Stress. 

Factor. 

Structure. 

Dead 

3 to 6 

Wrought Irox. 
Girders. 

U 

6 

U 

u 

4 

Bridges. 

Live 

8 

Steel. 

Shock 

IO 

Bridges. 


0 

z* 


Stress. 


Factor. 


Dead 

Live 

Mixed 


3 
6 

4 


Mixed 


4 













































822 STRENGTH OF MATERIALS.-TRANSVERSE. 


G-irclers, Beams, Bixitels, etc.- 

Transverse or Lateral Strength o f any Girder , Beam ) Breast-summer , 
Lintel , etc., is in proportion to product of its breadth and square of its 
depth, and area of its cross-section. 

Best form of section for Cast-iron girders or beams, etc., is deduced 
from experiments of Mr. E. Hodgkinson, and such as have this form of 
section T are known as Hodgkinson’s. 

Rule deduced from his experiments directs, that area of bottom flange 
should be 6 times that of top flange —flanges connected by a thin ver¬ 
tical web, sufficiently rigid, however, to give the requisite lateral stiff¬ 
ness, tapering both upward and downward from the neutral axis; and 
in order to set aside risk of an imperfect casting, by any great dispro¬ 
portion between web and flanges, it should be tapered so as to connect 
with them, with a thickness corresponding to that of flange. 

As both Cast and Wrought iron resist compression or crushing with a 
greater force than extension, it follow's that the flange of a girder or beam 
of either of these metals, which is subjected to a crushing strain, according 
as the girder or beam is supported at both ends , or fixed at one end , should be 
of less area than the other flange, which is subjected to extension or a ten¬ 
sile stress. 

When girders are subjected to impulses, and sustain vibrating loads, as in 
bridges, etc., best proportion between top and bottom flange is as i to 4; as - 
a general rule, they should be as narrow and deep as practicable, and should 
never be deflected to more than .002 of their length. 

In Public Halls, Churches, and Buildings where weight of people alone 
are to be provided for, an estimate of 175 lbs. per sq. foot of floor surface 
is sufficient to provide for weight of flooring and load upon it. In comput¬ 
ing other weight to be provided for it should be that which may at any time 
bear upon any portion of their floors; usual allowance, however, is for a 
weight of 280 lbs. per sq. foot of floor surface for stores and factories. 

In all uses, such as in buildings and bridges, where the structure is ex¬ 
posed to sudden impulses, the load or stress to be sustained should not ex¬ 
ceed from .2 to .16 of breaking weight of material employed; but when load 
is uniform or stress quiescent, it may be increased to .3 and .25 of breaking 
weight. 

An open-web girder or beam, etc., is to be estimated in its resistance on 
the same principle as if it had a solid web. In cast metals, allowance is to 
be made for loss of strength due to unequal contraction in cooling of w T eb 
and flanges. < 

In Cast Iron, the mean resistances to Crushing and Extension are, for 
American as 4.55 to 1, and for English as 5.6 to 7 to 1; and in Wrought Iron 
are, for American as 1.5 to 1, and for English as 1.2 to 1; hence the mass of 
metal below neutral axis will be greatest in these proportions when stress is 
intermediate between ends or supports of girders, etc. 

Wooden Girders or Beams , when sawed in two or more pieces, and slips 
are set between them, and whole bolted together, are made stiffer by the 
operation, and are rendered less liable to decay. 

Girders cast with a face up are stronger than when cast on a side, in the 
proportion of 1 to .96, and they are strongest also when cast with bottom 
flange up. 

Most economical construction of a Girder or Beam, with reference to at¬ 
taining greatest strength with least material, is as follows: The outline of 


STRENGTH OF MATERIALS.-TRANSVERSE. 


823 


top, bottom, and sides should be a curve of various forms, according as 
breadth or depth throughout is equal, and as girder or beam is loaded only 
at one end, or in middle, or uniformly throughout. 

_ Breaking Weights of Similar Beams are to each other as Squares of their 
like Linear Dimensions. 

By Board of Trade regulations in England, iron may be strained to 5 tons 
per sq. inch in tension and compression, and by regulation of the Ponts et 
Chaussees, France, 3.81 tons. 

Rivets .75 and 1 inch in diameter, and set 3 ins. from centre in top of 
girder, and 4 ins. at bottom. 

Character of fracture, as to whether it is crystalline or fibrous, depends 
upon character of blows; thus, sharp blows will render it crystalline, and 
slow will not disturb its fibrous structure. 

For spans exceeding 40 feet, wrought iron is held to be preferable to 
cast iron. 

Riveting, when well executed, is not liable to be affected by impact or 
velocity of load. 

A Coupled Girder or Beam is one composed of two, fastened together, and 
set one over the other. 

Triassed. Beams or GrircLers. 

Wrought and Cast Iron possess different powers of resistance to tension and com¬ 
pression; and when a beam is so constructed that these two materials act in uni- 
• son with each other at stress due to load required to be borne, their combination will 
effect an essential economy of material. In consequence of the difficulty of adjust¬ 
ing a tension-rod to the stress required to be borne, it is held to be impracticable to 
construct a perfect truss beam. 

Fairbairn declares that it is better for tension of truss-rod to be low than high, 
which position is fully supported by following elements of the two metals : 

Wrought Iron has great tensile strength, and, having great ductility, it undergoes 
much elongation when acted upon by a tensile force. On the contrary, Cast Iron 
has great crushing strength, and, having but little ductility, it undergoes but little 
elongation when acted upon by a tensile stress; and, w r hen these metals are re¬ 
leased from the action of a high tensile stress, the set of one differs widely from 
that of the other, that of wrought iron being the greatest. 

Under same increase of temperature, expansion of wrought is considerably great¬ 
er than that of cast iron; 1.81* tons per sq. inch is required to produce in wrought 
iron same extension as in cast iron by 1 ton. 

Fairbairn, in his experiments upon English metals, deduced that within limits 
of stress of 13440 lbs. per sq. inch for cast iron, and 30240 lbs. per sq. inch for 
wrought iron, tensile force applied to wrought iron must be 2.25 times tensile force 
applied to cast iron, to produce equal elongations. 

Relative tensile strengths of cast and wrought iron being as 1 to 1.35, and their 
resistance to extension as 1 to 2.25, therefore, where no initial tension is applied to 
a truss-rod, cast iron must be ruptured before wrought iron is sensibly extended. 

Resistance of cast iron in a trussed beam or girder is not wholly that of tensile 
strength, but it is a combination of both tensile and crushing strengths, or a trans¬ 
verse strength; hence, in estimating resistance of a trussed beam or girder, trans¬ 
verse strength of it is to be used in connection with tensile strength of truss. 

Mean transverse strength of a cast-iron bar, one inch square and one foot in 
length, supported at both ends, stress applied in the middle, without set, is about 
900 lbs.; and as mean tensile strength of wrought iron, also without set, is about 
20000 lbs. per sq. inch, ratio between sections of beams and of truss should be in 
ratio of transverse strength per sq. inch of beam and of tensile strength of truss. 

Girders under consideration are those alone in which truss is attached to beam 
at its lower flange, in which case it presents following conditions: 


* Elongation of cast and wrought iron being 5300 and 10 000, hence 10 000 -7- 5300 = 1.81. 



824 STRENGTH OF MATERIALS.—TRANSVERSE. 


i. When truss runs parallel to loivcr flange. 2. When truss runs at an inclination 
to lower flange, being depressed below its centre. 3. When beam is arched upward , 
and truss runs as a chord to curve. 

Consequently, in all these cases section of beam is that of an open one with a 
cast-iron upper flange and web, and a wrought-iron lower flange, increased in its re¬ 
sistance over a wholly cast-iron beam in proportion to the increased tensile strength 
of wrought iron over cast iron for equal sections of metals. 

From various experiments made upon trussed beams, it is shown : 

1. That their rigidity far exceeds that of simple beams; in some cases it -was from 
7 to 8 times greater. 2. That when truss resists rupture, upper flange of beam be¬ 
ing broken by compression, there is a great gain in strength. 3. That their strength 
is greatly increased by upper flange being made larger than lower one. 4. That 
their strength is greater than that of a wrouglit-iron tubular beam containing same 
area of metal. 

Comparative Value of Wroviglit-iroii Bars, Hollow 
Grii-ders, or Tubes of "Various Figures (English). 


Circular tubes, riveted.1 

Flanged beams.1.2 

Elliptic tubes, riveted.1.3 

Rectangular tubes, riveted.1.5 


Circular, uniform thickness ..1.7 

Plate beams.1.7 

Elliptic, uniform thickness.1.8 

Rectangular, uniform thickness.2 


General Deductions from Experiments of Stephenson , Fairbairn, Cubitt, 

Hughes , etc. 


Fairbairn shows in his experiments that with a stress of about 12320 lbs. per sq. 
inch on cast iron, and 28000 lbs. on wrought iron, the sets and elongations are 
nearly equal to each other. 

A cast-iron beam may be bent to .3 of its breaking weight if load is laid on grad- • 
ually; and .16 of it, if laid on at once, will produce same effect, if weight of beam 
is small compared with weight laid on. Hence, beams of cast iron should be made 
capable of bearing more than 6 times greatest weight which will be laid upon them. 


In beams of cast or w'rought iron, if fixed or supported at both ends, flanges 
should be in proportion to relative resistances of material to crushing or extension. 

Breaking weights in similar beams are to each other as squares of their like linear 
dimensions; that is, breaking weights of beams are computed by multiplying to¬ 
gether area of their section, depth, and a Constant , determined from experiments on 
beams of the particular form under investigation, and dividing product by distance 
between supports. 

Cast and wrought-iron beams, having similar resistances, have weights nearly as 
2.44 to I. 

A box beam or girder, constructed of plates of wrought-iron, compared to a single 
rib and flanged beam X, of equal weights, has a resistance as 100 to 93. 

Resistance of beams or girders, where depth is greater than their breadth, when 
supported at top, is much increased. In some cases the difference is fully one third. 

When abeam is of equal thickness throughout its length, its curve of equilibrium, 
to enable it to support a uniform stress with equal resistance in every part, 
should he an Ellipse , and if beam is an open one, its curve of equilibrium, for a uni¬ 
form load, should be that of a Parabola. Hence, when middle portion is not wdiolly 
removed, its curve should be a compound of an ellipse and a parabola, approaching 
nearer to the latter as the middle part is decreased. 

Girders of cast iron, up to a span of 40 feet, involve a less cost than of wrought 
iron. 


Cast-iron beams and girders should not be loaded to exceed .2, or subjected to a 
greater stress than .166 of their destructive weight; and when the stress is attended 
with concussion and vibration, this proportion must be increased. 

Simple cast-iron girders may be made 50 feet in length, and best form is that of 
Hodgkinson; when subjected to a fixed load, flanges should be as 1 to 6 and when 
to a concussion, etc., as 1 to 4. 

Forms of girders for spaces exceeding limit of those of simple cast iron are vari¬ 
ous; principal ones adopted are those of straight or arched cast-iron girders in 
separate pieces, and bolted together — Trussed, Bowstring, and wrought-iron Box 
and Tubular. 











STRENGTH OP MATERIALS.-TRANSVERSE. 


825 


Straight or Arched Girder , formed of separate castings, is entirely dependent 
upon bolts of connection for its strength. 

Trussed or Bowstring Girder is made of one or more castings to a single piece, 
and its strength depends, other than upon the depth or area of it, upon the proper 
adjustment of the tension, or the initial strain, upon the wrought-iron truss. 

Box or Tubular Girder is made of wrought iron, and is best constructed with 
cast-iron tops, in order to resist compression: this form of girder is best adapted to 
afford lateral stiffness. 

When a girder has four or more supports, its condition as regards a stress 
upon its middle is essentially that of a beam fixed at both ends. 

The following results of the resistances of materials will show how they 
should be distributed in order to obtain maximum of strength with minimum 
of dimensions: 



To Tension. 

To Crushing. 


To Tension. 

To Crush’g. 

Cast iron. 

(21 OOO 

90 300 

Oak, w 7 hite, mean. 
“ English “ . 

Wrought iron. 

“ English 
Yellow pine. 

11 OOO 

7 5oo 

“ English.. 

Granite. 

( 32 OOO 
(13 OOO 
(23 OOO 
578 
f 6 7 ° 

140 500 

58 OOO 

116 OOO 

15 OOO 

4 000 

6500 
( 45 000 

159 000 
(31OOO 
(53 °°o 

3 mo 

4 7 OOO 

83 OOO 

40 OOO 

65 OOO 

Limestone. 


\ 2 800 

9 000 

16 OOO 

4000 


The best iron has greatest tensile strength, and least compressive or crushing. 


Conditions of Forms and Dimensions of a Symmetrical 
Beam or Girder. 

When Fixed at One End , and Loaded at the Other. 

1. When Depth is uniform throughout entire Length , section at every point 
must be in proportion to product of length, breadth, and square of depth, and 
as square of depth is in every point the same, breadth must vary directly as 
length; consequently, each side of beam must be a vertical plane, tapering 
gradually to end. 

2. When Breadth is uniform throughout entire Length , depth must vary 
as square root of length; hence upper or lower sides, or both, must be deter¬ 
mined by a parabolic curve. 

3. When Section at every point is similar , that is , a Circle, an Ellipse , a 
Square, or a Rectangle, Sides of which hear a fixed Proportion to each other , 
the section at every point being a regular figure, for a circle, the diameter 
at every point must be as cube root of length; and for an ellipse or a rec¬ 
tangle, breadth and depth must vary as cube root of length. 

Illustration.—A rectangular beam as above, 6 ins. wide and 1 foot in depth at 
its extreme end, and 4 feet in length, is capable of bearing 6480 lbs. ; what should 
be its dimension at 3 feet? 3/4 = 1.587, and 3/3 = 1.442. 

Then 1.587 : 1.442 ” 1 : .9086, and 6 and 12 X .9086 = 5.452 and 10.9. 

5.452 X 10.9 2 6 X 12 2 . 

Hence r-AA - z_ — 2 i 6. and —--- = 216. 

3 4 

When Fixed at One End , and Loaded uniformly throughout its Length. 

1. When Depth is uniform throughout its entire Length, breadth must in¬ 
crease as the square of length. 

2. When Breadth is uniform throughout its entire Length , depth will vary 
directly as length. 

3. When Section at every point is similar , as a Circle , Ellipse , Square , and 
Rectangle , section at every point being a regular figure, cube of depth must 
be in ratio of square of length. 

















826 STRENGTH OF MATERIALS.—TRANSVERSE. 


Illustration.—T ake preceding case. 

Then 4 2 : 3 2 12 3 : 972, and V 97 2 — 9-9 i n depth. 

When Supported at Both Ends. 

1. When Loaded in the Middle, Coefficient or Factor of Safety of the beam, 
or product of breadth and square of depth, must be in proportion to distance 
from nearest support; consequently, whether the lines forming the beam are 
straight or curved, they meet in the centre, and of course the two halves are 
alike. 

2. When Depth is Uniform throughout, , breadth must be in ratio of length. 

3. When Breadth is Uniform throughout , depth will vary as square root 
of length. 

4. When Section at every point is similar , as a Circle , Ellipse , Square , and 
Rectangle , section at every point being a regular figure, cube of depth will 
be as square of distance from supported end. 

When Supported at Both Ends , and Loaded uniformly throughout its 

Length. 

1. When Depth is Uniform , breadth will be as product of length of beam 
and length of it on one side of given point, less square of length on one side 
of given point. 

2. When Breadth is Uniform , depth will be as square root of product of 
length of beam and length of it on one side of given point, less square of 
length on one side of given point. 

3. When Section at every point is similar , as a Circle , Ellipse , Square , and 
Rectangle , section at every point being a regular figure, cube of depth will 
be as product of length of beam and length of it on one side of given point, 
less square of length on one side of given point. 

JGlli.ptioal-sid.ecL Beams. 


To Determine Side or Curve of air Elliptical-sided Beam. 

/= d. L representing load in lbs ., l length in feet , C coefficient, and b 

Y 2 C o 
breadth in ins. 

Illustration.— What should be depth in centre of a beam of white pine, 10 feet 
in length between its supports, and 5 ins. in breadth, to support a load of 10000 lbs.? 

_ „ /10000 X 10 /100000 

Assume C = 100. Then / - == /- = 10 ms. 

V 2 X 100 X 5 V 1000 

Hence, outline of beam is that of a semi-ellipse, having 10 feet for its transverse 
diameter, and 9 ins. for its semi-conjugate. 

Note. —Weight of Girder, Beam, etc., should in all cases be added to stress or load. 


Ntiscellaneons Illustrations. 

1.—What should be side of a rectangular white oak beam, 2 ins. in width, and 6 
feet between its supports, to sustain a load of 360 lbs. ? 


Assume stress at .2 of breaking weight of 150 lbs. = 30. 

6 X 360 /2160 

240 




3 ms. 


4 X 2 X 30 

2.—What should be breadth and depth of such a beam if square? 
6 x 360 


3 J 6 X 360 = 3 /2160 = 2 
V 4 X 3° V 120 


4 X 30 

-What should be diameter of a cylinder? 
360 X 6 


62 ins. 


£ v „ =120, and 3 /— = 3.1 ins. 
6 X 30 V 4 







STRENGTH OF MATERIALS.—TRANSVERSE. 


STEEL. 


To Compute Transverse Strength, of Steel Bars. 
Supported at Both Ends. Weight applied in Middle. 

15 5 S b d _ yr g re p resen ting tensile strength in lbs., I length between supports 


in ins., and W weight in lbs. 

Illustration. —What is ultimate destructive stress of a bar of Crucible steel, 
2 ins. square, and 2 feet between supports ? S = 90000 lbs. 


Then 


i-i 55 X 90000 X23 
2 X 12 


831600 

24 


= 34 650 lbs. 


Elastic Transverse Strength is 50 per cent, of its ultimate strength. 

Hardening in oil increases its strength from 12 to 56 per cent. Thus, 

Soft steel, 121 520 lbs.; soft steel, cooled in water, 90160 lbs.; soft steel, 
cooled in oil, 215 120 lbs. 

Krupp's is about .45 of its tensile breaking weight, .24 of its compressive 
or crushing strength, .38 of its transverse, and .39 of its torsional. 

Friction of a steel shaft compared to one of wrought iron is as .625 to 1. 

Capacity of steel to resist a transverse stress is much less than to resist 
torsion. 


Relative diameters of steel and wrought-iron shafts, to resist equal trans¬ 
verse stress, are as .98 to 1, and weight of such a proportion of steel shaft 
compared with one of wrought iron will be about 4 per cent, less, and friction 
of bearing will be 6 per cent. less. 


CYLINDERS, FLUES, AND TUBES. 

Hollow Cylinders. Cast Iron. 

To Compute Elements of* Hollow Cylinders within 
Limits of - Elastic Strength. {D. K. Clark.) 


S X hyp. log. R = P. 


S. 


— = hyp. log. R. S representing 

O 


hyp. log. R 

elastic tensile strength of metal in lbs. per sq. inch, R ratio of external diameter to in- 
v / 

ternal, — — — —, and P internal pressure in lbs. per sq. inch, d and d’ representing 

internal and external diameter, and r and r’ internal and external radii, all in ins. 
Notk.-— Hyperbolic Logarithm of a number is equal to product of its common logarithm and 2.3026. 

Illustration i.—D iameters of a hydrostatic cylinder 5.3 by 13.125 ins.; what 
pressure within its elastic strength will it sustain per sq. inch? 

I 3. 125 

Assume S = 10000 lbs. Hyp. log. R = -- X 2.3026 = log. 2.5 X 2.3026 - .92. 

5-3 

Then 10000 X -92 = 9200 lbs. per sq. inch. 

Note. —For Bursting Strength take maximum strength of metal. 

2.—A water-pipe .75 inch thick has an internal diameter of 10 ins., what is its 
bursting pressure ? 


10 + -75 X 2 


.1398. 


S = 30 000 lbs. Hyp. log. 

AC 

Then 30000 X -1398 = 4194 lbs. 

3.—If it were required of a hydrostatic press to sustain a pressure of 589050 lbs. 
upon a ram of 5 ins. in diameter, what would be pressure on ram, and what should 
be thickness of metal, assuming it equal to an elastic tensile stress of 15000 lbs. 
per sq. inch ? 

Area of 5 ins. = 19.635. -^——- = 3000o —pressure per sq. inch on ram. 

Then 3 °° — = 2, which = hyp. log. R — 7.39, and 7.39 X 5 = 36-95 = external di- 
15000 _ . 

ameter. 36.95 — 5 = 31.95, which -f- 2 — 15-975 ins. thickness of metal. 









828 STRENGTH OF MATERIALS.-TRANSVERSE. 


d' 

R + hyp. log. — ■ 


Wroiaglit Iron. and. Steel 
2 P „ 2 P 


P. 


d' 

R + hyp- 1 °9--^- 


: S. 


S 


-f i = (P + byp. log. R). 


Illustration i. —If diameters of a wrougbt-iron cylinder are 5 and 15 ins., and 
ultimate or destructive strength of metal is 40 000 lbs. per sq. inch, what is its break¬ 
ing pressure? is 

— = 3. Hyp. log. 3 = .477 12 X 2.3026 = 1.0986. 

Then ^ ~t~ I- ° 9 ^ -* x 40000 = 61 972 lbs. per sq. inch = 61 972 X 5 "F 15 — 5 = 

2 

30986.2 lbs. per sq. inch of section of metal. 

2.—A steam-boiler 6 feet in internal diameter, of wrought-iron plates .375 inch 
thick and double riveted longitudinally, burst at a joint by a pressure of 300 lbs. per 
sq. inch; what was resistance of joint per sq. inch of its section? 


72 + - 37 S X 2 


72 


Then 


1.0104. Hyp. log. 1.0104 .010345. 

2 X 300 600 


1.0104 4-.010 345 —1 


.020745 


,= 29 405 lbs. per sq. inch of section of joint. 


BOILER AND SHIP PLATES. 

(See pages 753-757 for Boiler Riveting.) 

Ultimate Tensile Strength, of Riveted, and NUelded 
Joints of Wrought-iron [Plates. (D. K. Clark.) 

Entire Plate — 100. 


Joints. 

•5 

Plate. 

•4375 

•375 

Aver¬ 

age. 

Joints. 

•5 

Plate. 

•4375 

• 37 S 

Aver¬ 

age. 

Scarf, welded. 

_ 

106 

102 

104 

Double riv’d,snap-) 




67 

Lap, welded. 

5 ° 

6o 

66 

62 

headed. j 

59 

7° 

7 2 

Single riveted. 

40 

5 ° 

60 

50 

“ “ counter-) 





“ “ snap-) 



rfs 


sunk and snap-1 

53 

72 

69 

65 

headed.J 

5 ° 

5 2 

5 ° 

53 

headed.) 





“ “ by machine 

40 

54 

5 2 

49 

“ “with single) 





“ “ counter-) 





welt, counters'lc [ 

5 2 

60 

65 

59 

sunk head. j 

44 

5 ° 

5 2 

49 

and snap-headed.) 






Strength of Riveted Joints per Sq. Inch of Single Plate. ( Wm. Fairbairn.) 
Single Lapped. —Machine riveted. Pitch 3 times, 25 000 lbs. 

Hand riveted. Pitch 3 times, 24 000 lbs. 

Eivets “ staggered,” and equidistant from centres, 30 500 lbs. 

Abut Joints. —Hand riveted. Rivets not u staggered,” and equidistant 
from centres, single cover or strip, 30 000 lbs. 

Rivets “ square,” single cover or strip, 42 000 lbs. double covers or strips, 
55 000 lbs. 


Comparative Strength, of Riveted Joints. 
Entire Plate .375 ins. thick = 100. 


Double riveted, double welt, or fish-) 

plated joint.j 

Double riveted lap joint. 


80 

7 2 


Double riveted, single welt, or fish-) , 

plated joint..} 

Single riveted lap joint.60 


For all joints of plates over .5 inch, other than double welded, these proportions 
are too high. 

A closer pitch of rivets should be adopted in single than in double riveted abuts, etc. 







































STRENGTH OF MATERIALS.-TRANSVERSE. 


829 


Results of Experiments on Doifble Riveted, and Domble 
welded 3 ?late Joints. {Mr. Brunei.) 


Plates 20 ins. in Width, .5 inch Thick, Abut Jointed with a Welt or Fish-plate on 
each side, 10 ins. deep. Holes Punched. 



. 


Area of Section of 


Tensile Strength 

Description. 

1 ? 

0 


Plate. 


Shearing Strength 


cr*Q a 
m o 



Ph 

En- 

Net trans- 

of Rivets per Net 

Net. 

, -+-< 



tire. 

versely. 

Section. 


® - V 

Plate as above de-) 

Inch. 

Ins. 

S. Ins. 

Sq.Ins. 

Per c’t. 

Sq. Ins. 

Per c’t. 

Per c’t. 

scribed with 20 [ 
rivets “sq’re”) 

.6875 

4 

IO 

8.28 

82.8 

( 7.42 or 90 per) 

1 cent. j 

77 

93 

U 6C U 

•75 

4 

IO 

8.125 

81.25 

( 8.84 or no | 

( per cent. ) 

76 

93-5 

“ “ but) 








with 18 rivets > 
set “staggered”) 

•75 

4 

IO 

8-5 

85 

S 7-95 or 93.5 1 
( per cent. ) 

78.6 

9 2 -5 

“ 24) 

“square” or > 
“chain” rivets) 

•75 

5 

IO 

8.5 

85 

j 10.61 or 125 1 
\ per cent, j 

84 

99 


Proportions of Rivet Joints. 


Elements. 

Double 
staggered. 

Single. 

Elements. 

Double 
staggered. 

Single. 

Plate . 

Diameter of Rivet. 

Breadth of Lap. 

Distance of rivet fro 

I 

!-7 

8-3 

m edge oi 

1 

!-7 

5-4 

plate. 

Pitch of rivets in line ... 

Pitch lines apart. 

Rivets from edge of plate 

7- 1 

2.8 

2.7 

ter of riv 

4.6 

i-7 

3 t. 


Maximum pitch of rivets... =4 “ 

Minimum “ “ .. —2.75103 “ 

“ overlap, single riveting. =3-25 “ 

“ “ double “ . — c. t “ 


Proportions of Single Rivet Wrought-iron Joints. 

{French.) 


Thickness 
of Plate. 

Diameter 
of Rivets. 

Pitch of 
Rivets. 

Width of 
Lap. 

Thickness 
of Plate. 

Diameter 
of Rivets. 

Pitch of 
Rivets. 

Width of 
Lap. 

Mil’s 

Inch. 

Mil’s 

Inch. 

Mil’s 

Ins. 

Mil’s 

Ins. 

Mil’s 

Inch. 

Mil’s 

Ins. 

Mil’s 

Ins. 

Mil’s 

Ins. 

3 

.118 

8 

•315 

27 

1.06 

30 

H 

H 

CO 

IO 

•394 

20 

.787 

56 

2.2 

58 

2.28 

4 

.158 

IO 

•394 

3 2 

1.26 

34 

i-34 

II 

•433 

21 

.827 

57 

2 . 24 

60 

2.36 

5 

.197 

12 

.472 

37 

1.46 

40 

1.58 

12 

.472 

22 

,866 

58 

2.28 

60 

2.36 

6 

.236 

14 

•55i 

43 

1.69 

44 

i-73 

13 

.512 

2 3 

.906 

60 

2.36 

62 

2.44 

7 

.276 

16 

•63 

48 

1.89 

5o 

1.97 

14 

•55i 

24 

•945 

62 

2.44 

64 

2.52 

8 

•315 

17 

.669 

5i 

2.01 

54 

2.13 

15 

•59 1 

25 

•984 

63 

2.48 

66 

2.6 

9 

*354 

J 9 

•748 

54 

2.13 

56 

2.2 

l6 

•63 

26 

1.024 

65 

2.56 

68 

2.68 


Jo Compute Diameter of Rivet. 

T 1.25 + • 1875 — d. T representing thickness of plate, and d diameter of rivet. 

Steam Boilers. 

Assuming shearing strength of wrought-iron rivets at 42000 lbs. per sq. 
inch, and its crushing strength at 66000 lbs. 

Diameter in single shear, up to .375 inch thickness of plate = 2 thickness 
of plate, and for thicker plates it decreases to 1.5 thickness. 

Diameter in double shear = 1.05 thickness of plate, and when more than 
two plates are jointed put diameter + .125. 

Lap of Plate 1.5 diameter of rivet. 

Pitch of Rivets .—111 single shear, when d—2t, 2.75 d, and when d= 1.5 t, 
2.3 d. In double shear, when d = 2 t, 4.5 d, and when cl — 1.5 t, 4.5 d. 

4 A 































































830 


STRENGTH OF MATERIALS.-TRANSVERSE. 


Hulls of Vessels. 


Diameter of Rivets. 


Plate. 

U. S. and 
British 
Lloyds. 

Liverpool 

Reg’y. 

Admiralty, 

Eng. 

Mill wall, 
Eng. 

Pitch 
of Rivets. 

Length 

Counter¬ 

sunk. 

)f Rivets. 
Snap¬ 
headed. 

Inch. 

Inch. 

Ins. 

Ins. 

Inch. 

Ins. 

Ins. 

Ins. 

•3125 

.625 

•5 

•5 

.625 

1-75 

1-125 

1-5 

•375 

.625 

.625 

.625 

.625 

2 

1.25 

1.625 

•4375 

.625 

.625 

•75 

.625 

■2.125 

i -375 

i -75 

•5 

•75 

•75 

•75 

•75 

2.25 

i -5 

2 

.5625 

•75 

•75 

•875 

•75 

2-437 

1.6875 

2.1875 

.625 

•75 

.8125 

•875 

•875 

2.56 

x -9375 

2-375 

.6875 

•875 

•875 

•875 

•875 

2.812 

2.1875 

2.625 

•75 

•875 

•875 

I 

■875 

3-125 

2-375 

2-75 

.8125 

•875 

•9375 

I 

•875 

3-375 

2-5 

2.875 

•875 

I 

I 

1-125 

I 

3.625 

2.625 

3 

•9375 

I 

1.0625 

1-125 

I 

3-875 

2-75 

3-125 

I 

I 

1-125 

1-125 

I 

4.125 

2.875 

3-25 


Lap of Joint or Course should be .5 pitch of rivets added to .3 diam. of rivet. 

Note.— Lloyd’s requires a spacing of 4.5 diameter. Liverpool Registry, 4. Ad¬ 
miralty, 4.5 to 5 in edges and abuts of bottom and bulkhead plates, and 5 to 6 in 
other water-tight work. Bureau Veritas , 4 diameters for single riveting, and 4.5 
for double. 

STEEL PLATES. 

Steel Plates, according to M. Barba, .354 inch thick are equal to wrought 
iron .472 inch thick, or as 3 to 4; consequently, when iron rivets are used, 
their diameter should be in proportion to an iron plate. 

It is ascertained also that they are best united by iron rivets. 

A steel plate .3x25 inch thick requires an iron rivet .5625 inch in diam¬ 
eter, and 1.375 ins. apart. 


Bridge Ulates and Rivets. 


Plates .25 to .5 inch thick. Rivets .75 to 1 inch diameter, and 3 ins. apart 
from centres in upper flange or girder, and 4 ins. in lower. 

Rivet Heads. 

x./Sl>, Ellipsoidal , Fig. 1. — D diameter , R radius of head = D, r radius of 


flange = . 4 D, c depth at centre = .5 D, 

Segmental , Fig. 2. —D diameter , c depth at centre = .625 


D, R radius of head =. 75 D, 0 depth below head —. 125 D. 


.:rrJA.E 


Countersunlc .—Head 1.52 D, angle 6o°. Countersink .45 diam. of plate. 

Cheesehead or heads, section of which is a parallelogram. Head .45 D, 
diameter 1.5 D. 

Rivets. 


Shearing strength of a Lowmoor rivet = 40 320 d 2 or 18 d 2 in tons. 

d representing diameter of rivet in ins. 

Memoranda. 

Punching holes for riveting weakens plates, varying from 10 to 20 per cent., ac¬ 
cording to their temper, hardest losing most. 

Countersunk riveting does not impair strength of joint, as compared with ex¬ 
ternal head. 

Diagonal abut joints are stronger than square. 

Shearing strength of rivets should not exceed that of plates. 

Maximum strength of joint is at tained at 90 to 100 per cent, of net section of plate. 

Shearing strength of English wrought iron is taken at 80 per cent, of its tensile 
strength. 























STRENGTH OF MATERIALS.—TRANSVERSE. 83 1 


LEAD PIPE. 


Resistance of Lead. Ripe to Internal Pressure. 
( Kirkaldy , Jardine, and Fairbairn.) 


Diam. 

Thick¬ 

ness. 

Weight 

per 

Foot. 

Bursting 
Pressure. 

Diam. 

Thick¬ 

ness. 

Weight 

per 

Foot. 

Bursting' 
Pressure.J 

Diam. 

Thick¬ 

ness. 

Weight 

per 

Foot. 

Bursting 

Pressure. 

Inch. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Lbs. 

• 5 

.2 

2-3 

1579 

1.25 

.21 

5-3 

683 

2 

.21 

9.2 

498 

.625 

.2 

2.6 

1349 

i-5 

.24 

7 - 1 

734 

2 

.2 

— 

448 

•75 

.22 

3-8 

II9I 

i-5 

.2 

— 

528 

3 

• 25 

— 

364 

I 

.2 

4 - 1 

9 11 

i-5 

.2 

— 

626 

3 

•25 

-- 

374 


Tensile strength of metal = 2240 lbs. per sq. inch. 

To Compute Thickness of a Lead Ripe when Diameter 
and Pressure in Lbs. per Sep Inch, is given. 

RupE.—Multiply pressure in lbs. per sq. inch by internal diameter of pipe 
in ins., and divide product by twice tensile resistance of metal in lbs. per sq. 
inch. 

Illustration.— Diameter of a lead pipe is 3 ins., and pressure to which it is to 
be submitted is 370 lbs. per sq. inch; what should be thickness of metal? 


370X3 1110 . 


.248 ins. 


2240 x 2 4480 

Difference in Weight between Pipes of “Common,” “Middling,” and “Strong” 
is 12 per cent. 

To Compute Weight of Lead Ripe. 

p2 __^2 3.86 = W. D and d representing external and internal diameters in ins., 
and W weight of a lineal foot in lbs. 


To Compute Maximum or Bursting Pressure that may 
he home by a Lead Ripe. 

p> UL E.— Multiply tensile resistance of metal in lbs. per sq. inch by twice 
thickness of pipe, and divide product by internal diameter, both in ins. 
Illustration. —What is bursting pressure of a lead pipe 3 ins. in diameter and 

.5 inch thick? _ 

2240 X - 5 X 2 = 2240 = > 6 lbg 

3 3 

Assume a column of water 34 feet in height to weigh 15 lbs. per sq. inch; what 
head of water would such a pipe sustain at point of rupture? 

15 : 34 746-6 : 1692.3 feet. 

Resistance of G-lass GJ-lohes and Cylinders to Internal 
Pressure and Collapse. (Flint Glass.) 


Per Sq. Inch. 




Bursting Pressure. 



GLOBES. 



CYLINDER. 

Diameter. 

Thickness. 

Per Sq. Inch. 

Diameter. 

Length. 

Thickness. 

Ins. 

Inch. 

Lbs. 

Ins. 

Ins. 

Inch. 

4 

.024 

84 

4 

7 

.079 

q 

.022 

90 

Elliptical (Crown Glass). 

6 

•059 

152 

4 - 1 

7 

.019 


Collapsing Pressure. 


.014 

.025 

•059 


292 

ioocd 

900 s * 


3 

14 

.014 

4 

7 

•034 

4 


.064 


Lba. 

282 


109 

85 

202 

297 


* Unbroken. 













































832 STRENGTH OF MATERIALS.—TRANSVERSE. 


Manganese Bronze. 

Manganese Bronze, No. 2, has a Tensile strength of 72000 to 78600 lbs. 
per sq. inch, its elastic limit is from 35000 to 50000 lbs., its ultimate elon¬ 
gation 12 to 22 per cent., and its hardness alike to that of mild steel. 

Transverse Strength. —Destructive stress of a bar 1 inch square, supported 
at both ends at a distance of 1 foot = 4200 lbs., bending to a right angle be¬ 
fore breaking, and requiring 1700 lbs. to give it a permanent set. 

MEMORANDA. 

Cast Iron. 

Beams cast horizontally are stronger than when cast vertically. 

Relative strength of columns of like material and of equal weights is: 
Cylindrical, 100; Square, 93; Cruciform, 98; Triangular, no. ( Hodgkinson .) 

If strength of a cylindrical column is 100, one of a square, a side of which 
is equal to diameter of the cylinder, is as 150. 

Repetition of Stress. — A piece submitted to transverse stress broke at 
1956th strain, with a stress .75 of that of its original ultimate resistance. 

Resistance to Bursting of Thick Cylinders. —Mean resistance to bursting, 
of chambers of cast-iron guns is as follows ( Major Rodman , U.S.A.) : 

Thickness of metal = 1 calibre, length = 3 calibres, 52 217 lbs. per sq. inch. 

Thickness of metal = .5 calibre, length = 3 calibres, 49 100 lbs. per sq. inch. 

The tensile strength of the iron being 18 820 lbs. 

Diam. of cylinder 2 ins., length 12 ins., metal 2 ins., 80229 ^s. P er S T inch. 

Diam. of cylinder 3 ins., length 12 ins., metal 3 ins., 93702 lbs. per sq. inch. 

Tensile strength of iron being 26 866 lbs. 

Sudden Applications of Stress. —Loss of strength by sudden application 
of load was, by experiment, 18.6 per cent, in excess of load applied gradually, 
and its elongation 20 per cent, greater. 

Low Temperature. —Tensile strength at 23 0 under sudden application of 
load, was reduced 3.6 per cent., and elongation 18 per cent. 

Wrought Iron. 

Increased Hammering gives 20 per cent, greater strength with decreased 
elongation. 

Hardening. —Water increases strength more than oil or tar. A bar .87 
inch in diameter, forged and hardened in water, attained a tensile strength 
of 73448 lbs. (Mr. Kirlcaldy.) 

Case Hardening. —Loss of tensile strength 4950 lbs. per sq. inch. 

Cold Rolling added 18.5 per cent, to tensile strength, and when plates 
were reduced .33 in thickness, strength was nearly doubled, with but .1 per 
cent, elongation. Specific gravity was reduced. 

Fibre. —Plates are about 12 per cent, stronger with fibre than across it. 

Angles , Tees , etc., have from 2200 to 4500 lbs. less tensile strength than 
rectangular bars. 

Galvanizing does not perceptibly affect strength. 

Welding. —Strength as affected by welding varies by experiment from 2.6 
to 43.8 per cent, less, average being 19.4. 

Elastic Strength is about .45 of its tensile breaking weight, .15 of its com¬ 
pressive or crushing strength, and .5 of its transverse strength. 

Effect of Screw Threads. —1 inch bolts lose by dies 6.11 per cent., and by 
chasing 28 per cent. 

Steel. 

Steel can be hardened in water at a temperature of 310°. 


STRENGTH OF MATERIALS.-TRANSVERSE. 833 


WOODS. 

To Compute Transverse Strength, of Large Timber. 

Destructive Stress. 

.3 S b d 2 


Fixed at One End, and Loaded at the Other. 
Fixed at Both Ends, and Loaded in Middle. 


I 

x.8 S 6 d 2 


W. 


I 


:W. 


* Supported at Both Ends, and Loaded in Middle. 


1.2 S bd 2 


W. 


:W. 


Fixed at Both Ends, and Loaded at any other point than) -45 Sb d 2 
the Middle. J l 

Supported at Both Ends, and Loaded at any other point) -3 Sb d 2 l _ 
than the Middle. f in n ~ 

W l w l 

* Hence,-— — = S, and ——- = 1.2 S. 

’ 1.2 bd 2 ’ bd 2 

b, d, and l representing breadth , depth , and length to or between supports , all in 
ins.. S mean of tensile and crushing strengths of material at two thirds of its Value, 
as determined by experiments , W ultimate weight or stress in lbs., and m and n dis¬ 
tances of load from nearest supports in ins. 

When a beam is uniformly loaded, the stress is twice that if applied in its middle 
or at one end. 

Values of 1.3 S. 


Hence, for other coefficients, as .3, 1.8, etc., the values will be proportional. 


Woods. 


Ash, white. 

“ Canadian. 

“ English. 

Beech . 

Birch. 

Cedar. 

“ Cuban. 

Chestnut. 

Cypress. 

Elm, English. 

‘ 1 Roclc, Canada. 

Fir, Dantzic. 

Greenheart. 

Gum, blue. 

Hackmatack. 

Iron wood. 

Larch. 


1.2 S 


2.38 

2.4 

2.46 

2-55 

2-5 

1.6 

1.6 

1- 53 

•85 

x. 12 
2.63 

2- 5 
3.81 
2 

I. 36 

3- 64 

1.77 


Woods. 


Locust . 

Mahogany, Honduras. 

Oak, Pa. 

“ Va. 

“ white. 

“ English. 

“ Dantzic. 

“ French. 

Pine, Va. 

“ pitch. 

“ white. 

“ yellow. 

“ “ Canada.. 

Redwood, Cal. 

Spruce...'. 

Teak. 

Walnut, black. 


1.2 S 

3-7 

2-3 

2 

2-3 

2- 5 
i-7 
i-35 
2.44 

3 

2.2 
2.71 

3- 87 

1.8 

1.1 

1.2 
3 -i 7 
1.25 


Illustration i. —What is destructive stress of a beam of English oak, 2 ins. 
square, and 6 feet between its supports? 

1.2 from table = i.7, and S=66 of 5700 (mean of tensile and crushing strength) 
= 3762 lbs. 

x. 7 x 2 X 2 2 X_37_62 = 51163 = Ws 

6 X 12 72 

By experiment of Mr. Laslett it was 688 lbs. 

2.—What is destructive stress of a beam of yellow pine, 3 ins. by 12, and 14 feet 
between its supports? 

1.2 from table = 3.87, and S = .66 of 10200 (mean of tensile and crushing strength) 
= 6732 lbs. 

3.87 x 3 X 12 2 X 67_3g = 11654827 == 6 374 lbs _ 

14 X 12 168 

If the beam was fixed at both ends then 3.87 would be 5.8. 

Or, as 1.2 : 1.8 3.87 : 5.8. 

4 A* , 





















































Safe Statical Loads for Rectangular Beams of Various ^Materials, One Incli in Breadtli and. 

One Foot in Length. 

Supported at Both Ends and Loaded in Middle. 

A American, Af African, B Baltic, B’k Black, C Canadian, D Dantzic, E English, G Georgia, M Memel, P Pitch, R Riga, W White, 

Y Yellow.—Figures at Head of Columns denote Destructive Weight of Material in Lbs . 


834 


STRENGTH OF MATERIALS.—TRANSVERSE. 


■s 0 

3 CO 

0 — 

0 _ 

ft 

vo 'vt* t}-vo 0 vo th 'vfvo 0 vo Thvo 0 vo 
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Hickory. 

Maple. 

Af Oak. 

G Pine. 
800 

0 0 0 0 c 
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300000000000 

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Spruce. 

Sycamore. 

Elm. 

W Pine. 
500 

Lbs. 

100 

400 

900 

1 600 

2 500 

3 600 

4 900 

6 400 

8 100 

10000 

12 100 

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19 600 

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Birch. 

Hack¬ 

matack. 

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400 

Lbs. 

80 

320 

720 

1 280 

2 000 

2 880 
3920 
5.120 

6 480 

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to 


03 

H 

t>> 

ft 


^ a 

^ a 

T- 


d 

0 

02 

Es 

-*-> 

<D 

ft 


o 

2 


d 

d 

o 

ft 

02 02 

3 "3 

ft'd 

1 *r-» 

5 a 

• *—• 

ft d 
d 

'd 

d 

<22 

Tj «f 
d tj 
2 o 

ft 
ft 
a 

01 


d 

O 


d rn 

cc ' r_l 


a 

c 3 


ft 

o 

"d . 

co 
G x? 

CO o 

.a s- 

N ? 

M 

r- O 

5| 
2 -2 
k 

^ 02 
d ^ 

CO 

ft * 
0 h 

53 s 

ft 


05 

rO 

r*o 

o 

00 

co 

II 


vn 

H. 


X 

o 

o 


o 

o 

d 

02 

a 


-■§ 

cti 


II 

o CO 

ft 


03 

ft 


o 

ft 

ft 

S3 

02 

02 


i 
































STRENGTH OF MATERIALS.-TRANSVERSE. 835 


Following Coefficients or Factors of Safety are for .125 of average de¬ 
structive weight: 


Coefficients for 

Ash. 85 

“ English. 80 

“ Canada. 60 

Beech.,. 58 

Chestnut.. 65 

Elm, Canada. 80 

“ English. 42 

Fir, Riga. 47 

Hemlock.;. 60 


"Varions Woods. 
Hickory. j ^ 


Larch. 40 

Locust. 150 

Maple. 105 

Oak, white_ .... 80 

“ English... 60 

“ Dantzic .. 62 

“ Adriatic.. 55 

Spruce. 65 

Illustration. —What safe weight will a beam of white pine sustain, 4 ins. in 
breadth, 12 in depth, and 15 feet between its supports, when loaded in its middle? 
and what when uniformly loaded? Coefficient as above, 62. 


(.Hatfield and others.) 

Oak, Canada. 70 

“ French. 85 

Pine, pitch .. 68 

“ yellow..... 65 

“ red. 55 

“ Georgia.{ J°° 

“ white.... 62 

“ Canada red. 60 


Then 


4 X 12 2 X 62 
15 


= 2380.8 lbs. loaded in its middle , and 2380.8 X 2 = 4761.6 lbs. 


if uniformly loaded. 

•Floor ZB earns of Wood. 

Condition of stress borne by a Floor beam is that of a beam supported 
at both ends and uniformly loaded; but from irregularity in its loading 
and unloading, and from necessity of its possessing great rigidity, it is 
proper to estimate its capacity as a beam loaded at middle of its length. 

To Compute Capacity- of Floor Bea\ns, Grirders, etc. 

Supported at Both Ends. 

Rule. — Divide product of breadth and square of depth, in ins., and Coef- 
cient for material, by length in feet, and result will give weight in lbs. 

Or, ^ d : - - = W. Fixed at Both Ends. = W. 

L i 

Example. —The dimensions of a white-pine floor timber are 4 by 12 ins., and its 
length between supports 15 feet; what weight will it sustain in its centre? 

4 X 12^ X 62 

C as per preceding table = 62. Then - ---= 2380.8 lbs. 


When Uniformly Loaded. Multiply the results by 2. 


To Compute Deptli of a Floor 13 earn. 

Supported at Both Ends. 

When Lenrjth between Supports , Breadth and Distance between Supports , 
for One Foot , between Centres of Beams are Given. Rule. —Divide product 
of length in feet, and weight to be borne in lbs., by product of breadth in 
ins., and Coefficient for material, and square root of quotient will give depth 
in ins., for distance between centres of one foot. 

Or, = d. Fixed at Both Ends. ~ 

When Uniformly Loaded , W represents but half required or given weight. 

Example.— Take elements of preceding case, distance between centres of beam 


15 ins. C = 62. 


Then 


/ 


5 X 2380.8 


/35712 


4 x 62 v 248 

When Distance between Centres of Beams is greater or less than one Foot. 
Rule— Divide product of square of depth for a beam, When distance between 
centres is one foot, by distance given, by 12, and square root of quotient will 
give depth of beam. 





































836 


STRENGTH OF MATERIALS.—TRANSVERSE. 


Example.— Assume beam in preceding case to be set 15 ins. from centres of ad¬ 
joining beams; what should be its depth? 

/12 2 X 15 ,- 

' — ' —. 13.42 ins. 


Then 


/ 12 2 X 15 _ / 2160 _ 

\ 12 V 12 


To Compute Breadth, of a Floor Beam or Grirder. 
Supported at Both Ends. 

When Length and Depth are given. Rule.—D ivide product of length in 
feet, and weight to be borne in lbs., by product of square of depth in ins., 
and Coefficient for material, and quotient will give breadth in ins. 

I \y t l W 

Or, — b. Fixed at Both Ends. - — b. 

’ d 2 C 1.5 a 2 C 

When Uniformly Loaded , W represents but half required or given weight. 


Example. —Take elements of preceding cases. 


Then 


15 X 2380.8 
12 2 X 62 


35 7^2 
8928 


— 4 ins. 


When Distance between Centres of Beams is greater or less than One Foot. 
Rule.—D ivide product of breadth for a beam, When distance between centres 
is one foot , and distance given, by 12, and result will give breadth. 

Example. —Assume beam, as in preceding case, to be set 15 ins. from centre of 
adjoining beams; what should be its breadth? 

4X15 60 

Then ——— = — = 5 ms. 

12 12 

When Weight is Suspended or Stress borne at any other point than the 
Middle , See Formulas, page 801. 

Header and Trimmer or Carriage Beams. 

Conditions of stress borne or to be provided for by them are as follows: 

1 leader supports .5 of weight of and upon tail beams inserted into or at¬ 
tached to it, and stress upon it is due directly to its length and weight of 
and upon tail beams it supports, alike to a girder loaded at different points. 

Trimmer or Carriage beams support, in addition to that borne by them 
directly as floor beams, each .5 weight on headers. 


Note. —In consequence of effect of mortising (when bridles or stirrups are not 
used), a reduction of fully one inch should be made in computing the capacity of 
depth of headers and trimmers. 

To Compute BreacLtH. of a Header Beam. 

When Uniformly | Loaded. Rule.—C ompute weight to be borne in lbs. by 
tail beams, divide it by two (one half only being supported by header), mul¬ 
tiply result by length of beam in feet, and divide product by product of 
twice Coefficient of material and square of depth, and result will give breadth 

in ins. _ 

W -r- 2 l 

Or, ■ ^ = 0. W representing iveight per sq. foot. 


Example. —What should be breadth of a Georgia pine header, 13 ins. in depth, 
10 feet in length, supporting tail beams 12 feet in length, bearing 200 lbs. per sq. 
foot of area supported ? 

C, as per preceding table, 100, and depth = 13 —1 = 12 ins. 


Then 


12 X 10 X 2004-2X10 
2 


120000 
28 800 


= 4.17 ins. 


2 X IOO X 12' 











STRENGTH OF MATERIALS.-TRANSVERSE. 


837 


To Compute Depth, of a Header Beam. 

Rule.— See rule for depth of a floor beam, page 835, with the exception 
that a header is assumed to be always uniformly loaded. 

. ! l -5 W 
’V b C 


Or, 


— d. 


b. on and n representing distances of the weight or load from each 


To Compute Breadth of a Trimmer Beam. 

With One Header and One Set of Tail Beams. Rule.— Proceed as for 
computation of dimension of a beam loaded at any other point than middle. 
m n W 

' 5 Id* C : 
end in feet. 

Illustration. —What should be breadth of a trimmer or carriage beam of Georgia 
pine, 23 feet in length, 15 ins. in depth, sustaining a header 10 feet in length, with 
tail beams 19 feet, and designed for a load of 540 lbs. per sq. foot of floor? 

Assume C = 100; d = 15— 1 = 14; m and n — 19 and 4 feet. 


19X4X19X10-^2X54° 3898800 

•5 X —— !L — f --- 5 — — = -5X — = 4-32 ins. 


23 X 100 X 14" 


450 800 


Note r.—Depth of trimmer beams is usually determined by depth of floor beams; 
when not, proceed to determine it as for a header. 

2.—When a trimmer beam is mortised to receive headers, it is proper to deduct 
1 inch from its depth, as in preceding illustrations. When bridle or stirrup irons 
are used to suspend headers, a deduction of the thickness of the iron only is neces¬ 
sary, usually .5 inch. 

With Tico Headers and One Set of Tail Beams. —Fig. 1. 

Operation. —Proceed for each weight or load as for a beam, when weights 
are sustained or stress borne at other point than the middle. 

a k = W and w. a representing area of floor in sq. feet, L load per sq. foot, 

• 5 X .5 

and W and w weights or loads at points of rest on trimmers. 

Note. —Hatfield and some other authors give 
4_->f- complex and extended formulas, to deduce the di¬ 

mensions of a Girder or Beam, under a like stress. 

<■— m.-~- i xl-— n--i Upon consideration, however, it will readily be 

recognized that a beam loaded at more than one 
point is simply two or more beams, as the case 
may be, loaded at different points, and connected 
together. 

Illustration. —What should be breadth of a 
-vy w trimmer beam of Yellow or Georgia pine, 25 feet 

in length, 12 ins. in depth, sustaining two headers 
12 feet in length, set at 15 feet from one wall and 5 feet from the other, to support 
with safety 300 lbs. per sq. foot of floor? 

I — 25, m = 15, n—10, s — 5, o'—20, C = 100, and 4 = 12 — 1 = 11 for 

loss by mortising. 


Fig. 1. 






--S-- > 


12 X 5 X 300 _ 18000 

•5X-5 ~ -25 

15 X 10 X 4500 


Then 


= 4500 lbs. at W, and 
675 000 


12 x 5 X 300 18000 


= 4500 lbs. at w. 


.5X.5 - 2 5 

= 2.23 ins. breadth for load on header at 15 feet, 


25 X 11 X 100 302 500 

and 5 X 20 x 45 °° _ 4joo°o __ £ nS ' i re adth for load on header at 5 feet, and 
25 X11 2 X100 302500 

2.23 -f-1.48 = 3.71 ins. combined bo'eadth. 




































838 


STRENGTH OP MATERIALS.-TRANSVERSE. 


With Two Headers and Two Sets of Tail Beams. —Fig. 2. 


Fig. 2. 

k--- V 


<— -Wt-T-f 

1 1 

1 

1 

| 

i 

1 

1 









- 1 

1 - 


W W 


Operation. —Piocead as directed for Fig.i. 

Illustration. — What should be breadth of a 
trimmer beam of yellow pine 25 feet in length, 15 
ins. in depth, sustaining two headers 12 feet in 
length, set at 15 feet from one wall and 5 feet from 
the other, to support with safety 300 lbs. per sq. 
foot of floor? 

I = 25, th — 15, n — 10, s — 5, r= 20, C = ioo, 
and d —is — 1 = 14 for loss by mortising. 


12 X 15 X 300 
•5 X .5 


54000 

.25 


= 13 500 lbs. at W, and 


12 X 5 X 3 °° 

■5 X -5 


18000 

•25 


4500 lbs. at w. 


Then 


15 X 10 X 13 500_2 025 000 

25 X 14 2 X 100 490000 


14.14 ins., and 


ins., and 4.14 -{- .92 = 5.06 ins. combined oreadth. 


5 X 20 X 4500 
25 X 14 2 X 100 


450000 

-=1.92 

490 000 


With Three Headers and Two Sets of Tail Beams. —Fig. 3. 


Fig- 3 - 







f- 

1 


1 

-n— 

*--m— 



| | 

| 






to' to w 


Operation. —Proceed as directed for Fig. 1. 


Illustration. — What should be breadth of a 
trimmer beam of yellow pine, 20 feet in length, 13 
ins. in depth, sustaining 3 headers 15 feet in length, 
set at 3, 7, and 13 feet from one wall, to sustain a 
load of 200 lbs. per sq. foot of floor? 

1 = 20, m — 7 , n = 13, s = j, o — 3, d = i 3 —1 
= 12 ins., and C = 100. 


15 X 7 X 200 
•5 X .5 


21000 

•25 


= 5250 lbs. at W; 


1 5 X 7 — 3 X 200 12 000 


•5 X -5 


•25 


: 3000 lbs. at w ; and 


15 X 7 — 3 X 20 

•5X.5 


— 3000 lbs. at w'. 


Then 


7 X 13 X 5250 _ 477 750 . 
20 X 12 2 X 100 288000 


.66 ins. 


7 X13 X 3000 273000 


.95 ms. 


and 


3X17X3000 153000 


20 X 12 2 X 100 
bined breadth. 


288 000 


= .53 ms. 


20X I2 2 X IOO 288 000 

Hence, 1.66 + .95 + .53 = 3.14 ins. com- 


Stirrups or Bridles. 

Stirrups are resorted to in flooring designed for heavy loads, in order to 
avoid the weakening of the trimmers by mortising. 

Average wrought iron will sustain from 40000 to 50000 lbs. per sq. inch. 

Hence 45 000 lbs. as a mean, which -4- 5 for a factor of safety, = 9000 lbs. 

A stirrup supports one half weight of header, and being doubled (looped), 
the stress on it is but .5 - 5 - 2 = .25 of load on header. 


To Compute Dimensions of Stirrups or Bridles. 

W-t- 2 „ area 

—-= area. Hence --— width. 

2 X 9000 thickness 

Illustration. —What should be area and width of .75 inch wrought-iron stirrup 
irons for a weight on a header beam of 240 000 lbs. ? 


240 000 -r- 2 
2 X 9000 


120000 . .. , 6.66__ 

— -— 6.66 sq. ms., and-= 8.8 ms. = width. 

18000 .75 






































































STRENGTH OF MATERIALS.-TRANSVERSE. 


839 


Grirder. 


Condition of stress borne by a Girder is that of a beam fixed or supported 
at both ends, as the case may be, supporting weight borne by all beams 
resting thereon, at the points at which they rest. 

To Compute Dimensions of a. Grirder. 

Rule. —Multiply length in feet by weight to be borne in lbs., divide 
product by twice* the Coefficient , and quotient will give product of breadth 
and square of depth in ins. 

1 w / z w 

Or, —- = b and d 2 , and . / — ■ = d. 

2 C V 2 b C 

Example.— It is required to determine dimensions of a yellow-pine girder, 15 feet 
between its supports, to sustain ends of two lengths of beams, each resting upon it 
and adjoining walls, 15 feet in length, having a superincumbent weight, including 
that of beams, of 200 lbs. per sq. foot. 

Condition of stress upon such a girder is that of a number of beams, 30 feet in 
length (15 X 2), supported at their ends, and sustaining a uniform stress along their 
length, of 200 lbs. upon every superficial foot of their area. 

Coefficient .2 of 500= 100. 

30 X 15 X 200 - 4 - 2, for half support on their walls = 45 000 lbs. 

Then — = 3375 = b and d 2 . Assuming & = 12 ins., then = 16.77 

2 X 100 V 12 

ins. Or, if 15 ins., then — I 5 ins - 


To Compute Gfreatest Load upon a Grirder, and Dimen¬ 
sion s thereof.—Fig. 1. 

When a Beam is Loaded at Two Points. 



^22 = effect of weight at 1, 

— — effect of weight at 2, 
y (W X n -f- w s) = the two effects 


at 1 , and y (w r -f W m) == two effects at 2 . 

Then, for weight and dimensions, same formulas will apply. 


Illustration.— Assume weight of 8000 lbs. at 3 feet from one end of a white-pine 
beam 12 feet in length between its bearings, and another weight of 3000 lbs. at 5 
feet from other end. C .2 of 500 — 100. 


8000 X3X12 — 3 = 216 000 effect of weight at location 1, and 3000 x 5 X 12 — 5 
— Io5 000 effect of weight at location 2. Hence 1, being greatest, = W, and 2 = w. 

Then, 5219 x 8ooo — ^000 at W, and x 8750 = 3750 at w ; and 
’12 12 


- 5 - (8000 X 9 + 3000 x 5) = 21750 = total effect at W, and j- (3000 X 7 + 8000 X 3) 

— 18 750 total effect at w. 

Hence, to ascertain dimensions at greatest stress, 

2175° X 3 X 9 __ ^ anc j assum e d=io, then 1 ^ — 4-89 ins. breadth; or, 

12 X 3 00 10 


2 


489-33 

4.89 


10 ins. depth. 


* For being uniformly loaded. 




















84O STRENGTH OF MATERIALS.-TRANSVERSE. 


Verification .—Assume a beam as above loaded with 21 750 lbs. at 3 feet from end. 


Then, by formula for 801, 


3X9X21750 587250 


4.89 ins. 


12 X io* X 100 120000 

Equivalent Weight at Middle. —Fig. 2. 

A: 



Z = 2 


W S 


W n 

1 - 7-2 

LZ 


B; 


_E; and- '-2*=.D — 

l -r- S l 

equivalent load at middle. 

Illustration. —What should be 
breadth of a beam of Georgia pine, 
E 20 feet in length, 15 ins. in depth, 

uniformly loaded with 4000 lbs., and sustaining 3 headers or concentrated loads of 
6000 lbs., at respective distances of 4 and 9 feet from one end and 7000 lbs. at 6 feet 
from other end ? 

o — 4, r=i6, m — 9, n = n, s = 6, d — 15 — 1 = 14, L = 4000, and 

6000 X 4 6000 X 9 7000 X 6 

0 = 050 X -2 = 170. - = 24.00; 1 —- = 5400; - = 4200; 

20 -r- 2 ' 20= 2 20 - 7-2 

4000 X 20 _ . . . 7 , 

and — -— 2 = 2000. 2400 -f- 5400 -j- 4200 -j- 2000 = 14 000 tbs. 

14000 X 10 X 10 ,, _ , . . 

Then -- ■ = 70 000 lbs., effect at middle. 


Hence, 


20 x 70 000 


20 

1400 000 


= 10.5-}- ins. 


4 X 14 2 X 170 133 280 

Operation deduced by Graphic Delineation of Greatest Stress without uni¬ 
form Load. 

Fig. 3. <--7-> Moments of weights = 

A r . r~ - 2--- 1 R w' o r W m n 



i ■ —— > 1 - 

19200, 29700, and 29400, and 
let fall perpendiculars 1, 2, and 3 
proportionate thereto. 

yj- Connect w', W, and w with 

A B, and sum of distances of in¬ 
tersections of these lines upon perpendiculars, from 1, 2, and 3, respectively, will 
give stress upon A B at these points. 

Whence, greatest stress at greatest load will be ascertained to be 61800 lbs. 

When Loaded at Three Points, ™ . , ,,,r no 

as in Fig. 2. 1 (W» + «>«) + «> — = Greatest &tress - 

Illustration. —Take elements of above case, omitting uniformly distributed load. 

— (6000 X 11 X 7000 X 6) 4 - 6000 11 ^ = — X ic8 oco 4 -13 200 = 61 800 lbs. 

20 20 20 

Deflection of Girders and. Beams. 

WZ 3 Cb(Z 3 /W Z 3 . _/CZ>(Z 3 D . , 

3 = D; ta = W; 3 / 7 rrT. = d i and ^/-sr-= & l represent- 


w s u 


Gbd 3 ’ Z 3 ’ VC&D ’ V W 

ing length in feet , b and d breadth and depth , and D d flection in ins. 

Values of C for Various Woods. {Hatfield.) 


Ash.. 


Chestnut.. 

.2550 

Hemlock.. 


Hickory.. 

.3850 


Pine, Georgia.5900 

“ pitch.2836 

white.2900 


red. 


4259 


Larch.2093 

Oak, white.3100 

“ English, mean.. 2686 

Spruce,.3500 

Illustration.— What would be deflection of a floor beam of white pine, 10 feet 

in length, 4 ins. in breadth, and 8 in depth, with 4000 lbs. loaded in its middle? 

_ 4000 X 10 3 4000000 „ . , 

C = 2900. - - 7= =-= .674 inch. 

2900 X 4 X 8 3 5 939 200 


* Load uniformly distributed. 














































STRENGTH OF MATERIALS.-TRANSVERSE. 84 1 


.625 W l 3 


D; 


When Weight is Uniformly Distributed, 
Cbd 3 __ „ /C b d 3 D 


W; 


V- 

V '< 


: l: and 3 


W .625 l 3 


Cbd 3 ’ .625 £3 ‘ V -625 W ’ V C b 

Hence, Deflection in preceding illustration would be .674 X -625 = .421 ins. 
Illustration.— What should be length of a white-pine beam 3 by 10 ins., to sup¬ 
port 6000 lbs. uniformly distributed, with a deflection of 2 ins. ? C = 2900. 

/2900 X 3 X io 3 x 2 , /17400000 


f 


: = 3 lYL 
V : 


.625 x 6000 V 3750 
A fair allowance for deflection of floor beams, etc. 
.04 inch may be safely resorted to. 


16.68 feet. 
is ,03 inch per foot of length; 


Weights of Floors and. of Loads. 

Dwellings .—Weight of ordinary floor plank of white pine or spruce, 3 lbs. 
per sq. foot, and of Georgia pine, 4.5 lbs. 

Plastering, Lathing, and Furring will average 9 lbs. per sq. foot. 

Clay Blocks (Flat Arch ) 5.25 x 7.25 ins. in depth and 1 foot in length, 
21 lbs. — 80 lbs. per cube foot of vo^ime. 

Floors of dwellings will average 5 lbs. per sq. foot for white pine or spruce, 
and on iron girders will average from 17 to 20 lbs. per sq. foot. 

Weight of men, women, and children over 5 years of age, 105.5 lbs., and 
one third of each will occupy an average area of 12 x 16 ins. = 192 sq. ins. 
= 78.5 lbs. per sq. foot. 

Of men alone 15 X 20 ins. = 300 sq. ins. =48 in 100 sq. feet. 

Bridges , etc .—Weight of a body of men, as of infantry closely packed, = 
138 lbs. each, and they will occupy an area of 20 X 15 ins. = 300 sq. ins. = 
66.24 lbs. per sq. foot of floor of bridge, and as a live or walking load, 80 lbs. 
per sq. foot. 

Weight of a dense and stationary crowd of men, 120 lbs. per sq. foot. 

Bridging of Floor Beams increases their resistance to deflection in a very 
essential degree, depending upon the rigidity and frequency of the bridges. 


Weiglit oil Floors, etc., in addition to Weight of Struct¬ 
ure, per Sq. Foot. 


Ball rooms. 85 lbs. 

Brick or stone walls..... 115 to 150 “ 

Churches and Theatres... 80 “ 

Dwellings. 40 “ 

Factories.200 to 400 “ 

Grain. 100 “ 


Roofs, wind and snow.... 

Slate roofs.... 

Snow, per inch. 

Street bridges. 

Warehouses...... 

Wind. 


30 to 35 lbs. 

45 “ 

• 5 lb. 
80 lbs. 

250 to 500 “ 
50 “ 


Scarfs. 

Relative resistance of scarfs in Oak and Pine, 2 ins. square, and 4 feet in 
length, by experiments of Col. Beaufoy. 

Scarf 12 ins. in Length and 13 ins. from End , or 1 inch from Fulcrum. 

Vertical .—no lbs. gave away in scarf. 

Horizontal , large end uppermost and toivards fulcrum .—101 lbs. fastenings 
drew through small end of scarf; small end uppermost , etc., 87 lbs. gave 
away in thick part of scarf. 

Factors of Safety. 

Statical or Dead Load at .2 of destructive stress, but for ordinary pur¬ 
poses it may be increased to .25, and in some cases with good materials to .3. 

IJve Load at .1 to .125 of destructive stress. 

See also page 802. 



















842 


SUSPENSION BRIDGE. 


SUSPENSION BRIDGE. 


To Compute Elements. 



= stress at •. C representing chord or span, a half chord, and v versed sine of 
chord or curve of deflect ion, in feet, L distributed load, inclusive of suspended struct¬ 
ure , Q load per lineal foot, and S stress at centre, all in tons, x distance of any point 
from centre of curve, and h height of chain at x above centre of it, both in feet, s 
stress on chain at any point, as x, from centre of span, s stress on any tension-rod, 
and t stress at abutments, all in tons, n number of tension-rods, 0 angle of tangent 
of chain with horizon at any point, as x, r angle of chain with vertical at abutments, 
l length of chain, in feet, and z angle of direction of chain. 

Assume C = 300 feet, L = 1000 tons, v = 25 feet, x = 100 feet, n = 30, r — 71 0 34', 
and 0 = 12 0 32'. 


Then, 


300 X 1000 
8 X 25 


1500 tons = S; 


25 X 100 2 
(.5 X 300) 2 


11 feet = h ; 


1500 


2X1 


O / 


i = 1536.56 tons — s ; 


4 X 11.11 
2 X 100 


.2222 = I2 U 32 — : 


tan. 0; 2 (.5 X 300) 2 + -^-25 2 = 305.5/^:^; 3 °° X IOO ° = 25 — v; 

v 3 o 1 .^00 


4 x 25 

300 

1500 


: -3333 = 71° 34'— = cot. angle r\ 


./ ^- 2 S N ) -f- 1 = 1428.6 j tons = t: and — 

V \ 300 / v 


30—1 
2 X 25 


V(2 X 25)^+ (300-^2) 2 
For a deflection of .125 of span, horizontal stress is equal to total load 
To Construct curve, see Geometry, page 230. 


X 1500 
34.48 tons = s'; 

.3162 = 18 0 26'. 


To Compute Batio which Stress on Chains or Cables at 
either Point of Suspension Bears to whole Suspended 
Nyeight of Structure and Load. 


-= R. R representing ratio. 

2 X sin. 2 mu 

Illustration. —Assume elements of preceding case. 

2 x \ i62 = 1-58 ratio. By a preceding formula it would be 1.536. 

Stress on Back Stays. —The cables being led over rollers, having free mo¬ 
tion. tension upon them is same, whether angle i is same as that of r or not. 

Stress on Piers— When angles r and i are alike, stress on piers will be 
vertical, but when angle of i is greater or less than r, stress will be oblique. 

To Compute Horizontal Stress and Vertical Pressure 

on Biers. 

S cos. 2 = St, S cos. n— S 0, S sin. z = Vi, and S sin. n — Vo. S i and S 0 
representing stress, and P i and P 0 pressure, inward and outward. 

Note.— Span of New York and Brooklyn Bridge 1595.5 feet, deflection 128 feet 
angle of deflection at piers from horizontal 15 0 10'. 































TRACTION. 


843 


TRACTION. 

Res nits of Experiments on Traction of Roads 
and Pavements. (M. Norm.) 

1st. Traction is directly proportional to load, and inversely proportional 
to diameter of wheel. 

2d. Upon a paved or Macadamized road resistance is independent of 
width of tire, when it exceeds from 3 to 4 ins. 

3d. At a walking pace traction is same, under same circumstances, for 
carriages with or without springs. 

4th. Upon hard Macadamized, and upon paved roads, traction increases 
with velocity: increments of traction being directly proportional to incre- 
ments of velocity above velocity of 3.28 feet per second, or about 2.25 miles 
per hour. The equal increment of traction thus due to each equal increment 
of velocity is less as road is more smooth, and carriage less rigid or better 
hung. 

5th. Upon soft roads of earth, sand, or turf, or roads thickly gravelled, 
traction is independent of velocity. 

6th. Upon a well-made and compact pavement of dressed stones, traction 
at a walking pace is not more than .75 of that upon best Macadamized 
roads under similar circumstances; at a trotting pace it is equal to it. 

7th. Destruction of a road is in all cases greater as diameters of wheels 
are less, and it is greater in carriages without springs than with them. 

Experiments made with the carriage of a siege train on a solid gravel 
road and on a good sand road gave following deductions : 

1. That at a walk traction on a good sand road is less than that on a good 
firm gravel road. 

2. That at high speeds traction on a good sand road increases very rapidly 
with velocity. 

Thus, a vehicle without springs, on a good sand road, gave a traction 2.64! 
times greater than with a similar vehicle on same road with springs. 

Resnlts with. a. Dynamometer. 

Wagon and Load 2240 lbs. * 


Roadway. 

Relat’e num¬ 
ber of horses 
for like effect. 

Roadway. 

Relat’e num¬ 
ber of horses 
for like effect. 

On railway, 8 lbs. 

On best stone tracks, 12.5 lbs. 
Good plank road, 32 to 50 lbs. 
Stone block pavement, 32.5 “ 
Macadamized road, 65 lbs.... 

i 1 . 56 

4 to 6.25 
' 4.06 

8.12 

Telford road, 46 lbs. 

Broken stone or con’te, 46 lbs. 

Gravel or earth, 140-147 lbs. | 
Common earth road, 200 lbs.. 

5-75 

5-75 

17 - 5 

1 8 - 37 

25 


Note.—B y recent experiments of M. Dupuit, he deduced that traction is inversely 
proportional to square root of diameter of wheel. 

Relation of force or draught to weight of vehicle and load over 6 different con¬ 
structions of road, gave for different speeds as follows: 

Walk. Trot. Walk. Trot. 

Stage coach, 5 tons.. X. 3 1 | Carriage, seats only, on springs.. 1.29 1 

Resistance to Traction 01a Common Roads. 

On Macadamized or Uniform Surfaces. (M. Dupuit.) 

1. Resistance is directly proportional to pressure. 

2. It is independent of width of tire. 

3. It is inversely as square root of diameter of wheel. 

4. It is independent of speed. 


* See Treatise on Roads, Streets, and Pavements, by Brev. Maj.-Gen’l Q. A. Gilhnore, U. S. A. 

t Telford estimated it at 3.5. 














844 


TRACTION, 


On Paved and Rough Roads. 

Resistance increases with speed, and is diminished by an enlargement of 
tire up to a moderate limit. 

Traction on Various Roads .—Traction of a wheeled vehicle is to its weight 
upon various roads as follows : 


Per Ton. 

Stone track, best 12.5 to 15 
“ “ .... 28 to 39 

“ pavement. 14 

Asphalted.22 

Plank.22 

Block stone 1 
pavement.... J 32 


to 36 
to 28 
to 45 

to 35 


Per 100 lbs. 
.55 to .58 

Telford road_ 

Per Ton. 

46 to 78 

1.25 to 1.3 

Macadamized... 

46 to 90 

.5 to 1.5 

“ loose 

67 to 112 

1 to 1.25 

Gravel. 

134 to 180 

.98 to 2 

Sandy. 

140 to 313 

1.4 to 1.6 

Earth. 

200 to 290 


Per 100 lbs. 
2.1 to 3.5 

2 to 4 

3 to 5 


to 
to 

6 to 8 
6.3 to 14 
9 to 13 


Hence, a horse that can draw 140 lbs. at a walk, can draw upon a gravel road 

6 + 8 
140 - 4 - - 


X 100 — 2000 lbs. 


Resistance on Common Roads or Fields. 
{Bedford Experiments , 1874.*) 


Gkavelled Road. 

(Hard and dry, rising 1 in 430.) 


2 horse wagon without springs. 

u " " “ 

4 


with 

without 


Maxi¬ 

mum 

Draft. 

Average 

Draft. 

Average 

Speed 

per 

Hour. 

H * 1 2 3 * * * * * * * de¬ 
veloped 
per 

Minute. 

Draft per Ton 
on Level. 

Work 
per H' 
per 
Horse. 

Lbs. 

Lbs. 

Miles. 

LP. 

Lbs. 

IP. 

320 

!59 

2-5 

1.06 

43.5 or.0192 

•53 

400 

251 

2.6 

1.74 

44.5 “ .02 

.87 

300 

133 

2.47 

.88 

34-7 “ -oi 5 

•44 

180 

49.4 

2.65 

•35 

28 “ .0125 

•35 

IOOO 

700 

2-35 

4-36 

210 or.099 

to 

w 

CO 

1200 

997 

2.52 

6.7 

194 “ .083 

3-35 

IOOO 

710 

2-35 

4-45 

210 £t .099 

1.22 

4OO 

212 

2.61 

1.48 

140 “ .0625 

1.48 


u 

1 “ cart 

Arable Field. 

{Hard and dry , rising 1 in 1000.) 

2 horse wagon without springs. 

U ll (( K 

2 “ “ with “ 

1 “ cart without “ 

Fore wheels of wagons were 39 ins., and hind 57 ins. in diam.; tires varying from 
2.25 to 4 ins.; and wheels of cart were 54 ins. in diam., and tires 3.5 and 4 ins. 
Springs reduced resistance on road 20 per cent., but did not lessen it in the field. 

From these data it appears, that on a hard road, resistance is only from .25 to .16 
of resistance in field. Lowest resistance is that of cart on road = 28 lbs. per ton; 
due, no doubt, to absence of small wheels alike to those of the wagons. 

Assuming average power without springs to be .6 EP on road, as average for a 
day’s work, it represents .6 X 33000 = 19800 foot-lbs. per minute for power of a 
horse on such a road. 

Resistance of a smooth and well-laid granite track (tramway), alike to those in 
London and on Commercial Road, is from 12.5 to 13 lbs. per ton. 


01 X 11111 ) 118.1 (Weight 5758 lbs.) 


Average Speed per Hour. 

Granite pavement (courses 3 to 4 ins.). 2.87 miles. 

Asphalt roadway. 3.56 “ 

Wood pavement. 3.34 “ 

Macadam road, gravelly. 3.45 “ 

“ “ granite, new. 3.51 “ 


Total. 
44.75 lbs. 

69-75 “ 

106.88 “ 
114.32 “ 
259.8 “ 

Note.— The resistance noted for an asphalt roadway is apparently inconsistent 
with that for a granite pavement, for when it is properly constructed it is least 
resistant of all pavements. 


Per Ton. 
17.41 lbs. 
27.14 “ 
41.6 11 

44.48 “ 

101.09 U 


* See report in Engineering, July 10, 1874, page 23. 


t Report Soc. Arts, London, 1875. 





























TRACTION, 


845 


Wagon. 
Weight 2342 lbs. 


(Sir John Macneil.) 
Speed 2.5 Miles per Hour. 


Resistance. 


Per Ton. 

Total. 

31.2 

lbs. 

33 

lbs. 

44 

U 

46 

U 

62 

a 

65 

U 

140 

u 

H 7 

u 


Well-made stone pavement. 31.2 lbs. 

Road made with 6 ins. of broken hard stone, on a foundation) 

of stones in pavement, or upon a bottom of concrete. ] 

Old hint road, or a road made with a thick coating of broken ) , 

stone, on earth.) 02 

Road made with a thick coating of gravel, on earth. 140 

Stage Coach. (Sir John Macneil.) 

Weight 3192 lbs. Gradients 1 to 20 to 600. 

Speed. Metalled Road. 

At 6 miles per hour. 62 lbs. per ton. 

“ 8 “ “ . 73 u “ 

11 xo u u . 79 “ “ 

Note.—I t was found that, from some unexplained cause, the net frictional resistance at equal speeds 
varied considerably, according to gradient, resistances being a maximum for steepest gradient, and a 
minimum for gradients of i in 30 to 1 in 40; for these they are less than 1 in 600. Mode of action of 
the horses on the carriage may have been an influential element. ( D. K. Clark.) 

r 

To Compute Resistance to Traction on Various Roads. 

(Sir John Macneil.) 

ON A LEVEL. 

Rule.—D ivide weight of vehicle and load in lbs. by its unit in following 
table, and to quotient add .025 of load; add sum to product of velocity of 
vehicle in feet per second, and Coefficient in following table for the particular 
road, and result will give power required in lbs. 

W + w 


Or, 


unit 


-j -10 .025 -}* C v — T. W and w representing weights of vehicle and load 

Coefficients for Traction of Various Vehicles. 

2 horse w T agon without springs. 54 

2 “ “ with “ 42 

1 “ cart without “ 36 


Stage coach. 100 

Heavy wagon. 93 

4 horse wagon without springs. 55 


Coefficients f or Roads of Various Construction. 


Pavement. 2 

Broken stone, dry and clean. 5 

“ “ covered with dust- 8 

“ “ muddy. 10 


Macadamized road. 4.3 

Gravel, clean. 13 

“ muddy. 32 

Stone tramway. 1.2 

Sand and Gravel. 12.1 

Illustration. —What is the traction or resistance of a stage coach weighing 2200 
lbs., with a load of 1600 lbs., when driven at a velocity of 9 feet per second over a 
dry and clean broken stone road? 

2200-I-1600 , —— - .- ,, 

-b 1600 x .025 -(-5X9 = 123 lbs. 


To Compute I J ower necessary to Sustain, a 'Vehicle upon 
an Inclined. Road, and also its Pressure thereon, omit¬ 
ting Effect of Eriction. 

AT AN INCLINATION. 

W : A C :: 0 : B C, and W : A C :: p : A B. 

Or, r e : e 0 :: A B : BC; W : e 0 :: l: h; whence, 
„ T h 

W — = eo. 

t 

Assume A B of such a length that vertical rise, 
W A B _ WAB _ cos a =p. 



W _ W 
A C VA B 2 -b 1 


; = W sin. A = 0, and 


A C 


4 b 


Va B 2 +i 

































TRACTION. 


846' 


Or,T = P; 


W V 


—Pi 


or, 


W 


Vr 2 + 


: P, and 


W l' 2 

•— = y. W representing 

Vl' 2 +x 


weights of vehicle and load 0, and P power or force necessary to sustain load on road , 
p pressure of load on surface, all in lbs., h height of plane , l inclined length of road 
or plane , and l' horizontal length , all in feet. 

Illustration.— What is power required to sustain a carriage and its load, weigh¬ 
ing 3800 lbs., upon a road, inclination of which is 1 in 35, and what is its pressure 
upon road? 


Sin. A = .028 56. Cos. A = .99994. £ = 35.014. 


Then 3800X -028 56 = 108.53 lbs. = power, and 3800 x -999 94-3799-77 ^hs. pressure. 


To Compute Resistance of a Load, on an Inclined Road. 

Rule. —Ascertain the tractive power required, and add to it the power 
necessary to sustain load upon inclination, if load is to ascend, and subtract 
it if to descend. 

Example i.— In preceding example tractive power required is 123 lbs., and sus¬ 
taining power for that inclination 108.53; hence 1234-108.53 = 231.53 lbs. 

2.—If this load was to be drawn down a like elevation. 

Then 123 — 108.53 = 14.47 lbs. 


To Compute Rower necessary to Move and Sustain a 
Vehicle either Ascending or Descending an Elevation, 
and at a given Velocity, omitting Effect of Friction. 



cos. L 4= (W 4- w) sin. L_ 4- v c = R. 


L_ representing angle of 


elevation for a stage wagon and a stage coach , and t units as preceding; upper sign 
taken when vehicle descends the plane , and lower when it ascends. 

Illustration.— Assume a stage coach to weigh 2060 lbs., added to which is a 
load of noo lbs., running at a speed of 9 feet per second over a broken stone road 
covered with dust, and having an inclination of 1 in 30; what is power necessary 
to move and sustain it up the inclination, and what down it? 


Then 


u = 9, c = 8, sin. of L_ = sin. of i° 54'-}-= .0333, and cos. L_ =.9995. 

(2.060 1100 


\ 100 40 

105.234-72 = 236.3 lbs. up inclination, 
f 2060 4- noo 


+ iioo\ x >99g5 _|_ ( 2o6o 4— IIOO ) x .0333 4-8X9 = 59-°7 + 

40 / 


And 


—105.23 = 25.84 lbs. down inclination. 


—— ) x .9995 + 8x9 — (20604- iio °) x .0333 = 59.07 4- 72 
4 ° / 


Tractive and Statical Resistance of Elevations. ( Gillmore.) 
T 

- -g'- T representing traction in lbs. per ton, W weight of load in lbs., 

V'V 2 — T 2 
and g’ grade of road. 


Illustration. —Assume traction as per preceding table, page 844, 200, and weight 
of vehicle 2 tons; what should be least grade of road? 


200 X 2 

\/4480 2 — 200 X 2 



Showing that, for a road upon which traction is 200 lbs. per ton, the grade should 
not exceed one in height to one eleventh fall of base; hence, generally, the proper 
grade of any description of road will be equal to force necessary to draw load upon 
like road when level. 


Practically, greatest grade of a Telford or Macadamized road in good condition 
= .05, and a horse can attain at a walk a required height upon this grade, without 
more fatigue and in nearly same time that he would require to attain a like height 
over a longer road with a grade of .033, that he could ascend at a trot. 

For passenger traffic, grades should not exceed .033. 

















TRACTION. 


Resistance of Gravity at Different Inclinations o 
Grade. For a Load of 100 Lbs. 


Grade. 

R 

Grade. 

R 

Grade. 

R 

Grade. 

R 

i in 5 

Lbs. 

19.61 

1 in 25 

Lbs. 

4 

1 in 45 

Lbs. 

2.22 

1 in 70 

Lbs. 

i -43 

1 in 10 

9-95 

1 in 30 

3-33 

i in 50 

2 

1 in 80 

1.25 

1 in 15 

6.65 

1 in 35 

2.85 

1 in 55 

1.82 

i in 90 

I .11 

1 in 20 

4.99 

1 in 40 

2-5 

i in 60 

1.67 

1 in 100 

I 


Inclination of Roads .—Power of draught at different inclinations and velocities 
is as follows (Sir John Macneil): 





Traction at Speeds of per 

Frictional Resistance per 

Inclination. 

Angle. 

Feet 


Hour of 


Ton at Speeds of per Hour of 



per Mile. 

6 Miles. 

8 Miles. 

10 Miles. 

6 Miles. 

8 Miles. 

10 Miles. 

1 in 20 

2 0 52' 

264 

268 

296 

318 

76 

96 

112 

1 in 26 

2° 12 ' 

203.4 

213 

219 

225 

63 

68 

72 

i in 30 

1 ° 55 ' 

176 

165 

196 

200 

4 i 

63 

66 

1 in 40 

1° 26' 

132 

160 

166 

I72 

56 

6l 

65 

1 in 60 

57 - 5 ' 

88 

III 

120 

128 

7 2 

78 

81 


Grade. 

Grade of a road should be reduced to least of practicable attainment, and 
as a general rule should be as low as i in 33, and steepest grade that is ad¬ 
missible on a broken stone road is 1 in 20. 


The condition of traction is /-f-sin. a L, which should not exceed P, and sin. a 
p 

should not exceed - f or f f representing coefficient of friction, a angle of in- 

J_i 

clination , L load , and P power in lbs. 

Illustration. — In case, page 846, weight or load = 2060+ 1100 = 3x60 lbs., Co¬ 
efficient of friction for such a road = .042 per 100 lbs., and sin. i° 54' = .033 16. 

Then .042 -j- .033 16 X 3160 = 237.5 lbs. 

Traction of a Vehicle compared to its Weight on Different Roads. 

(F. Robertson , F. R. A. S.) 

Stone pavement. 1 in 68 I Flint foundation. 1 in 34 

Macadamized road.... x “ 49 | Gravel road. 1 “ 15 

Sandy road. 1 in 7. 

Assuming a horse to have a tractive force of 140 lbs. continuously and steadily at 
a walk, he can draw at a walk on a gravel road 15 X 140 = 2100 lbs. 

Friction of Roads. 

Friction of Roads. —According to Babbage and others, a wagon and load 
weighing 1000 lbs. requires a traction as follows: 

Of Load. 


Loose sand.25 

Fresh earth.125 

Common side roads.1 

Gravelled road. j 


Of Load. 

Macadamized.033 

Dry high road.025 

Well paved road.014 

Railroad. j ,00 ^ 

1 .0059 


Gravel road, new.083 

Common road, bad order.. .07 

Sand road.063 

Broken stone, rutted.052 

“ “ fair order... .028 

“ “ perfect order .015 

Macadamized, new.045 

“ 033 

gravelly.02 

Earth, good order.025 


Sled , hard snow, iron shod.033 of load. 

Coefficients of Friction in Proportion to Load. 

Per 100 lbs. 

Wood pavement.019 

Asphalt roadway.012 

Stone pavement.015 

Granite “ 008 

Stone “ very smooth .006 
Plank road.01 

Railwa y. {zii 

Stone track. .05 


Per 100 lbs. Per Ton. 

186 
157 
141 
117 

63 

34 

101 

74 
44 
56 


Per Ton. 
42 
27 

34 

18 

13 

22 

8 

*3 

112 





































































848 


TRACTION. 


To Compute Frictional Resistance to Traction of a 
Stage Coach on a Metalled. Road in, Good Condition. 

30 -J- 4 v -f- -f 10 v — R. v representing speed in miles per hour , and R frictional 
resistance to traction per ton. 

Note. —Formula is applicable to wagons at low speeds. 

V 

Canal, Slacltwatex*, and River. 

On a canal and water, resistance to traction varies as square of velocity, 
from that of 2 feet per second to that of 11.5 feet. 

When velocity is less than .33 miles per hour, resistance varies in a less 
degree. 

In towing, velocity is ordinarily 1 to 2.5 miles per hour. 

Resistance of a boat in a canal depends very much upon the comparative 
areas of transverse sections of it and boat, it being reduced as difference 
increases. 

In a mixed navigation of canal and slack-water, 3 horses or strong mules 
will tow a full-built, rough-bottomed canal boat, with an immersed sectional 
area of 94.5 sq. feet, and a displacement of 240 tons, 1.75 to 2 miles per hour 
for periods of 12 hours. 

With a section of but 24.5 sq. feet, or a displacement of 65 tons, an aver¬ 
age speed of 2.5 miles is attained for a like period. 

By the observations of Mr. J. F. Smith, Engineer of the Schuylkill Navigation 
Co., a canal boat, with an immersed section alike to that above given, can be towed 
for 10 hours per day as follows: 

Per Hour. 


By x horse or 

By 2 horses or 

By 3 horses or 

By 4 horses or 

By 8 horses or 

mule. 

mules. 

mules. 

mules. 

mules. 

1 mile. 

1.5 miles. 

1.75 miles. 

1.875 miles. 

2.5 miles. 


Assuming then, the tractive power of a horse as given in table,page 437, the above 
elements determine results as follows: 


Horses. 

Miles. 

Tractive Power 
divided by Load. 

in Lbs. per Ton. 

Fraction 

in Lbs. per Sq. Foot of 
immersed Section. 

I . 

I 

250- 

- 240 

1.04 

2.65 

2. 

i -5 

165 X 2- 

- 240 

1.38 

3-49 

3. 

i -75 

140 X 3 “ 

- 240 

i -75 

4.44 

3. 

1-875 

132 x 3- 

- 240 

1.65 

4.19 

3. 

2 

125 x 3- 

~ 240 

1.56 

3 - 9 8 

3 (bght). 

2-5 

100 X 3 _ 

- 65 

4,61 

12.24 


Upon a canal of less section and depth, a displacement of 105 tons, with an im¬ 
mersed section of 43 sq. feet, a speed of 2 miles with 2 horses was readily obtained, 
which would give a traction of 2.38 lbs. per ton, and of =5.71 lbs. per sq. foot of im¬ 
mersed section. 

Maximum Rower of a Horse on a Canal. ( Molesworth.) 


Miles per hour. 2.5 

Duration of work in 1 

hours.j IX '5 

Load drawn in tons .. 520 


3 3-5 4 5 6 

8 5.9 4.5 2.9 2 

243 153 102 52 30 


78 9 10 

1.5 1.125 .9 .75 

19 13 9 6.5 


Street Railroads or Tramways. ( GenH Gillmore.*) 

Upon a level road, and at a speed of 5 miles per hour, the power required to draw 
a car and its load is from gA^ to gl^ of total weight, varying with condition of 
rails and dryness or moisture of their surface. 


* Treatise on Roads, Streets, and Pavements. D. Van Nostrand, 1876, N. Y. 



























TRACTION.-WATER. 


849 


To Compute Resistance of” a Car. 


T X 6 —f ; = 

3 . __ 400 


= r ; and /-}- c r = R. T representing weight 


in tons, f friction in lbs., v speed in miles per hour , a area of front or section of car 
in sq. feet , c concussion, r resistance of atmosphere, and R total resistance, all in lbs. 

Illustration. —Assume a car and load of 8960 lbs., with an area of section of 56 
sq. feet, and a speed of 5 miles per hour. 

4X5 


8960 

Then —— = 4 tons; 
2240 

5 2 X 56 


4 X 6 = 24 lbs. friction ; 


= 6.66 lbs. concussion ; 


400 


= 3.5 lbs. resistance of air ; and 24 + 6.66 -f- 3.5 = 34.16 lbs. 


In average condition of a road, the resistance of a car may be taken at which, 
in preceding case, would be 74.66 lbs. On a descending grade, therefore, of 1 in 
74.66, the application of a brake would not be required. 


WATE R. 

Fresh Water. Constitution of it by weight and measure is 

By Weight. By Measure. I By Weight. By Measure. 

Oxygen... 88.9 1 | Hydrogen.. 11.1 2 

Cube inch of distilled water at its maximum density of 39. i°, barom¬ 
eter at 30 ins., weighs 252.879 grains, and it is 772.708 times heavier 
than atmospheric air. 

Cube foot (at 39. i°) weighs 998.8 ounces, or 62.425 lbs. 

Note. —For facility of computation, weight of a cube foot of water is 
usually taken at 1000 ounces and 62.5 lbs. 

At a temperature of 32 0 it weighs 62.418 lbs., at 62° (standard tem¬ 
perature) 62.355 lbs., and at 212 0 59.64 lbs. Below 39.i° its density 
decreases, at first very slow, but progressing rapidly to point of conge¬ 
lation, weight of a cube foot of ice being but 57.5 lbs. 

Its weight as compared with sea-water is nearly as 39 to 40. 

It expands .085 53 its volume in freezing. From 40° to 12 0 it ex¬ 
pands .00236 its volume, and from 40° to 212° it expands .0467 — 
times = .000 271 5 for each degree, giving an increase in volume of 1 
cube foot in 21.41 feet. 

~V olnmes of IPnre "Water. 


At 32 0 27.684 cube ins. = 1 cube foot. 

« !q.i° 27.68 “ “ =1 “ “ 

“ 62° 27.712 “ “ =1 “ “ 

“ 212° 28.978 “ “ =1 U ££ 


At 62° i Ton =35-923 cube feet. 

“ “ 1 Lb. —27.71 “ ins. 

“ 39.1 0 1 Tonneau = 35.3156 “ feet. 

“ “ 1 Kilogr. = .0353 “ “ 


Height of a Column of Water at 62° or 62.355 lbs. 

1 lb. per sq. inch = 2.3093 feet, and at pressure of atmosphere = 33.947 feet = 
10.347 meters. 


Ice and. Snow. 


Cube foot of Ice at 32 0 weighs 57.5 lbs., and 1 lb. has a volume of 
30.067 cube ins. 

Volume of water at 32 0 , compared with ice at 32 0 , is as 1 to 1.085 S 3 ; ex¬ 
pansion being 8.553 per cent. 

Cube foot of new fallen snow weighs 5.2 lbs., and it has 12 times bulk of 
water. 







850 


WATER. 


Rainfall. 


A nnual Fall at different Places. 


Location. 


Ins. 


Location. 


Ins. 


Alabama. 

3 °-i 7 

Albany. 

4 i -35 

Algiers . 

7-75 

Alleghany. 

46.66 

Antigua. 

45 

Archangel. 

14.52 

Auburn... 

30.17 

Bahamas. 

42.19 

Baltimore. 

39-9 

Barbadoes. 

55-87 

Bath, Me. 

34-58 

Belfast. 

39-46 

Biskra. 

.2 

Bombay. 

no 

Bordeaux. 

29.7 

Boston. 

39-23 

Brussels. 

29 

Buffalo. 

27.27 

Burlington, Yt. 

32 

Calcutta. 

81 

Cape St. Franpois.. 

150 

Cape Town. 

23.31 

Charleston. 

54 

Cherbourg . 

39-7 

Cologne. 

24 

Copenhagen. 

23 

Cracow. 

13-33 

Demerara. 

91.2 

“ 1849 . 

132.21 

Dover (Engl.). 

37-52 

Dublin. 

30.87 

Dumfries. 

36.92 

East Hampton .... 

38.52 

Edinburgh. j 

24-5 

29 

Fairfield. 

32-93 


Ft. Crawford, Wis.. 

Ft. Gibson, Ark- 

Ft. Snelling, Iowa.. 
Fortr. Monroe, Va.. 

Florence. 

Frankfort, Oder... 
“ Main... 

Geneva. 

Gibraltar. 

Glasgow.| 

Gordon Castle, Sc’d 
Granada. j 

Great Britain_ 

Greenock. 

Halifax. 

Hanover. 

Havana. 

Hong-kong. 

Hudson. 

India. j 

Jamaica. 

Jerusalem. 

Key West. 

Khassaya, India... 

Lewiston. 

Liverpool. 

London. 

Louisiana. 

Madeira. j 

Manchester. j 

Marseilles. 


29-54 

30.64 
3°- 3 2 
52-53 
35-9 

21.3 

16.4 
32.6 
47.29 

21.3 

3 1 

2 9-3 

105 

126 

32 
61.8 

33 

22.4 

52 

8 i -35 

39-3 2 

60 

130 

34-3i 

65 

3i-39 

610 

23-15 

34.12 

25.2 

51-85 

22 

49 

36.14 

43 

18.2 


Location. * 

Ins. 

Michigan. 

33-5 

45 

54-94 

41.8 

40-5 

36 

Mississippi. 

Mobile, 1842. 

Naples. 

Newburg. 

New York. 

Ohio. 

Palermo. 

22.8 

Paris..... 

23.1 

49 

44 

29.16 

Philadelphia. 

Plymouth (Engl.).. 
Port Philip. 

Poughkeepsie. 

Providence. 

32.06 

36.74 

29 

on 

Rochester. 

Rome. 

Santa Fe. 

jy 

74.8 

55 

47-77 

7-75 

84 

85-79 

48 

120 

Savannah. 

Sclienectadv. 

Siberia. 

Sierra Leone. 

Sitka.. 

St. Bernard. 

St. Domingo. 

St. Petersburg. 

State of N.Y. 

17.6 

33-79 

Sydney. 

Tasmania. 

‘iy 

35 

46.4 

263.21 

Trieste. 

Ultra Mullay, India 
Utica. 

Venice. 


Vera Cruz. 

62 

Vienna. 


Washington. 

34.62 

48.7 

West Point. 


Average rainfall in England for a number of years was, South and East, 34 ins.; 
West and hilly, 43.02 to 50 ins., and percolation of it was estimated at 30 per cent. 

Mean volume of water in a cube foot of air in England is 3.789 grains. 

Globe, mean depth. 36 ins. 

Cape of Good Hope in 1846.. in 3 days, 6.2 “ 

At Khassaya, in 6 rainy months.. 550 ins.; in 1 day, 25.5 “ 

Evaporation .—Mean daily evaporation, in India .22 inch; greatest .56; in Eng¬ 
land .08. At Dijon, when mean depth of rainfall was 26.9 ins. in 7 years, evapora¬ 
tion was for a like period 26.1 ins., and in Lancashire, Eng., when fall was 45.96 
ins., evaporation was 25.65. 


Volume of Rainfall. 

Rainfall, depth in ins., X 2 323 200 = cube feet per sq. mile. 

“ “ “ X 17.378 74 = millions of gallons per sq. mile. 

“ “ “ X 3630 = cube feet per acre. 

“ “ “ X 27 154.3 = gallons per acre. 

Mineral Waters are divided into 5 groups, viz.: 

1. Carbonated, containing pure Carbonic acid — as, Seltzer, Germany; Spa, Bel¬ 
gium; Pyrmont, Westphalia; Seidlitz, Bohemia; and Sweet Springs, Virginia. 

2. Sulphurous, containing Sulphuretted hydrogen—as, Harrowgate and Chelten¬ 
ham, England; Aix-ia-Chapelle, Prussia; Blue Lick, Ky.; Sulphur Springs, Va., etc. 

3. Alkaline, containing Carbonate of soda—these are rare, as, Vichy, Ems. 











































































































WATER. 


851 


4. Chalybeate, containing Carbonate of iron—as, Hampstead, Tunbridge, Chelten¬ 
ham, and Brighton, England; Spa, Belgium; Ballston and Saratoga, N. Y.; and 
Bedford, Penn. 

5. Saline, containing salts—as, Epsom, Cheltenham, and Bath, England; Baden- 
Baden and Seltzer, Germany; Kissingen, Bavaria; Plombieres, France; Seidlitz, 
Bohemia ; Lucca, Italy ; Yellow Springs, Ohio; Warm Springs, N. C.; Congress 
Springs, N. Y.; and Grenville, Ky. 

Brief Rules for Qualitative Analysis of Mineral Waters. 

First point to bo determined, in examination of a mineral water, is to which of 
above classes does water in question belong. 

1. If water reddens blue litmus paper before boiling, but not afterwards, and blue 
color of reddened paper is restored upon warming, it is Carbonated. 

2. If it possesses a nauseous odor, and gives a black precipitate, with acetate of 
lead, it is Sulphurous. 


3. If, after addition of a few drops of hydrochloric acid, it gives a blue precipitate, 
with yellow or red prussiate of potash, water is a Chalybeate. 

4. If it restores blue color to litmus paper after boiling, it is Alkaline. 

5. If it possesses neither of above properties in a marked degree, and leaves a 
large residue upon evaporation, it is a Saline water. 


River or canal water contains 
Spring or well water “ 


07 ] v0 ^ ume gaseous matter. 


Tie-agents. 


When water is pure it will not become turbid, or produce a precipitate 
with any of following Re-agents: 

Baryta Water , if a precipitate or opaqueness appear, Carbonic Acid is present. 

Chloride of Barium indicates Sulphates, Nitrate of Silver, Chlorides, and Oxalate 
of Ammonia, Lime salts. Sulphide of Hydrogen, slightly acid, Antimony, Arsenic, 
Tin, Copper, Gold, Platinum, Mercury, Silver, Lead, Bismuth, and Cadmium; Sul¬ 
phide of Ammonium , solution alkaloid by ammonia, Nickel, Cobalt, Manganese, 
Iron, Zinc, Alumina, and Chromium. Chloride of Mercury or Gold and Sulphate of 
Zinc, organic matter. 

Filter Deds. 


Fine sand, 2 feet 6 ins.; Coarse sand, 6 ins.; Clean shells, 6 ins., and Clean gravel 
2 feet, will (liter 700 gallons water in 24 hours by gravitation. 


Ska Water. Composition of it per volume: 


Chloride of Sodium (common salt).. 

Sulphuret of Magnesium. 

Chloride of “ . 


2.51 

•53 

•33 


Carbonate of Lime 

“ of Magnesia 

Sulphate of Lime. 

Water... 


.02 


.01 

96.6 


By analysis of Dr. Murray, at specific gravity of 1.029, it contains 

Muriate of Soda.220.01 I Muriate of Magnesia. 42.08 

Sulphate of Soda. 33.16 | Muriate of Lime. 7.84 


3°3- 09 

Or, 1 part contains .030309 parts of salt = 3V part of its weight. 


Mean volume of solid matter in solution is 3.4 per cent., .75 of which is 
common salt. 


Boilin'” ^Points at Different Degrees of Saturation. 


Salt, by Weight, 

Boiling 

in too Parts. 

Point. 

3*°3 — 3 T 3 

213.2° 

6.06 = in* 

214 - 4 ° 

9-°9 = TTs 

2 i 5 - 5 ° 

12.12 —Z 

216.7° 


Salt, by Weight, 
in 100 Parts. 

Boiling 

Point. 

15.15 {= -/g- 

217.9 0 

Jpt. 

II 

00 

M 

00 

H 

219 0 

21-22 — 3% 

220.2° 

2 4-25 =^3 

221.4° 


* Saturn te l. 


Salt, by Weight, 

Boiling 

in 100 Parts. 

Point. 

27.28 = -^ 

222.5 0 

3 °- 3 i=H 

223.7° 

33-34 — ■§■§■ 

224.9° 

* 36.37 = 

226° 
























852 WATER.-WAVES OF THE SEA. 


Deposits at Different Degrees of Saturation and. Tem¬ 
perature. 

When 1000 Paris are reduced by Evaporation. 


Volume of Water. 

Boiling Point. 

Salt in 100 Parts. 

Nature of Deposit. 

IOOO 

214 0 

3 

None. 

299 

217 0 

IO 

Sulphate of Lime. 

102 

228° 

29-5 

Common Salt. 


It contains from 4 to 5.3 ounces of salt in a gallon of water. 


Saline Contents of Water from several Localities. 


Baltic.... 
Black Sea 
Arctic ... 


6.6 

21.6 

28.3 


British Channel.... 

Mediterranean. 

Equator. 


35-5 

39-4 

39-42 


South Atlantic. 
North Atlantic 
Dead Sea. 


There are 62 volumes of carbonic acid in 1000 of sea-water. 


41.2 

42.6 

385 


Cube foot at 62° weighs 64 lbs. Its weight compared with fresh water 
being very nearly as 40 to 39. 

Height of a Column of Water at 6o° or 64.3125 lbs. 

At 62°, 1 Ton = 35 cube feet. 1 Lb. per sq. inch = 2.239 f ee b ail 4 a t pressure of 
atmosphere = 32.966 feet = 10.048 meters. 

"Weights. 

A ton of fresh water is taken at 36, and one of salt at 35 cube feet. 


WAVES OF THE SEA. 

Arnott estimated extreme height of the waves of an ocean, at a distance 
from land sufficiently great to be freed from any influence of it upon their 
culmination, to be 20 feet. 

French Exploring Expedition computed waves of the Pacific to be 22 feet 
in height. 

By observations of Mr. Douglass in 1853, he deduced that when waves had 
heights of 

8 feet, there were 35 in number in one mile, and 8 per minute. 

15 “ “ s and 6 “ “ 5 “ 

20 “ “ 3 “ “ 4 “ 

J. Scott Russell divides waves into 2 classes—viz.: 

Waves of Translation, or of 1st order; of Oscillation, or of 2d order. 

Waves of* the First Order. 

1. Velocity not affected by intensity of the generating impulse. 

2. Motion of the particles always forward in same direction as wave, and 
same at bottom as at surface. 

3. Character of wave, a prolate cycloid, in long waves, approaching a true 
cycloid. When height is more than one third of length, the wave breaks. 

Waves of the Second Order. 

1. Ordinary sea waves are waves of second order, but become waves of the 
first order as they enter shallow water. 

2. Power of destruction directly proportional to height of wave, and great¬ 
est when crest breaks. 

3. A wave of 10 feet in height and 32 feet in length would only agitate 
the water 6 ins. at 10 feet below surface; a wave of like height and 100 feet 
in length would only disturb the water 18 ins. at same depth. 

Average force of waves of Atlantic Ocean during summer months, as de¬ 
termined by Thomas Stevenson , was 611 lbs. per sq. foot; and for winter 
months 2086 lbs. During a heavy gale a force of 6983 lbs. was observed. 



















WAVES OF THE SEA. 


853 

J. Scott Russell deduced that a wave 30 feet in height exerts a force of 1 
ton per sq. foot, and that, in an exposed position in deep water, 1.75 tons 
may be exerted upon a vertical surface. 

At Cassis, France, when the Avater is deep outside, blocks of 15 cube me¬ 
ters were found insufficient to resist the action of waves. 

Breakwaters with vertical walls, or faces of an angle less than 1 to 1, will 
reflect waves without breaking them. Waves of oscillation have no effect 
on small stones at 22 feet below the surface, or on stones from 1.5 to 2 feet, 
12 feet below surface. 

A roller 20 feet high will exert a force of about 1 ton per sq. foot. 

Greatest force observed at Skerryvore, 3 tons per sq. foot; at Bell Rock, 
1.5 tons per sq. foot. 

Waves of the second order, when reflected, will produce no effect at a depth 
of i2 feet below surface. 

Action of waves is most destructive at low-water line. 

Waves of first order are nearly as powerful at a great depth as at surface. 

To Compute Velocity. 

When l is less than d. .55 fl or 1.818 fl = V. 

When l exceeds 1000 d. V 32.17 d = V, and When Height of Wave becomes a sen¬ 
sible Proportion to Depth, 32.17 -f- 3 = V. 

To Compute Height of Waves in Reservoirs, etc. 

1. S\/L + (2.5 — v'L) = height in feet. L representing length of Reservoir, Pond, 
etc., exposed to direction of wind, in miles. 

Tidal Waves. 

Wave produced by action of sun and moon is termed Free Tide Wave. 
Semi-diurnal tide wave is this, and has a period of 12 hours 24-j- minutes. 

Professor Airy declared that when length of a wave was not greater than 
deptli of the water, its velocity depended only upon its length, and was pro¬ 
portionate to square root of its length. 

When length of a wave is not less than 1000 times depth of water, velocity of it 
depends only upon depth, and is proportionate to square root of it; velocity being 
same that a body falling free would acquire by falling through a height equal to half 
depth of water. 

For intermediate proportions, velocity can be obtained by a general equation. 

Under no circumstances does an unbroken wave exceed 30 or 40 feet in height. 

A wave breaks when its height above general level of water is equal to general 
depth of it. 

Diurnal and other tidal waves, so far as they are free, may be all considered as 
running with the same velocity, but the column of the length of wave must be 
doubled for diurnal wave. 


Depth of Water. 

Length of Wave. 

Feet. 

1 

Feet. 

10 

Feet. I Feet. 

100 1 1000 

Velocity per Second. 

Feet. 

lOOOO 

Feet. 

ICO OOO 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

I 

2.26 

5-34 

5-67 

— 

— 

— 

IO 

2.26 

7- 1 5 

16.88 

17.92 

17-93 

— 

IOO 

— 

7 -i 5 

22.62 

53-19 

56.67 

56 - 7 I 

I OOO 

— 

— 

22.62 

71-54 

168.83 

179.21 

IOOOO 

— 

— 

— 

7 I -54 

226. 24 

533-9 


4 C 













854 


WHEEL GEARING. 


WHEEL GEARING. 

Pitch Line of a wheel is circle upon which pitch is measured, and it 
is circumference by which diameter, or velocity of wheel, is measured. 

Pitch is arc of circle of pitch line, is determined by number of tee<th 
in wheel, and necessarily an aliquot part of pitch line. 

True or Chordial Pitch , or that by which dimensions of tooth of a 
wheel are alone determined, is a straight line drawn from centres of 
two contiguous teeth upon pitch line. 

Line of Centres is line between centres of two wheels. 

Radius of a wheel is semi - diameter bounded by periphery of the 
teeth. Pitch Radius is semi-diameter bounded by pitch line. 

Length of a Tooth is distance from its base to its extremity. 

Breadth of a Tooth is length of face of wheel. 

Depth of a Tooth is thickness from face to face at pitch line. 

Pace of a Tooth , or Addendum , is that part of its side which extends 
from its pitch line to its top or Addendum line. 

Flank of a Tooth is that part of its side which extends from pitch 
line to line of space at base of and between adjacent teeth; its length, 
as well as that of face of tooth, is measured in direction of radius of 
wheel, and is a little greater than face of tooth, to admit of clearance 
between end of tooth and periphery of rim of wheel or rack. 

Cog Wheel is general term for a wheel having a number of cogs or teeth set in or 
upon, or radiating from, its circumference. 

Mortice Wheel is a wheel constructed for reception of teeth or cogs, which are 
fitted into recesses or sockets upon face of the wheel. 

Plate Wheels are wheels without arms. 

Rack is a series of teeth set in a plane. 

Sector is a wheel which reciprocates without forming a full revolution. 

Spur Wheel is a wheel having its teeth perpendicular to its axis. 

Bevel Wheel is a wheel having its teeth at an angle with its axis. 

Crown Wheel is a wheel having its teeth at a right angle with its axis. 

Mitre Wheel is a wheel having its teeth at an angle of 45° with its axis. 

Face Wheel is a wheel having its teeth set upon one of its sides. 

Annular or Internal Wheel is a wheel having its teeth convergent to its centre. 

Spur Gear .—Wheels which act upon each other in same plane. 

Bevel Gear .—Wheels which act upon each other at an angle. 

Inside Gear or Pin Gearing.— Form of acting surfaces of teeth for a pitch-circle 
in inside gearing is exactly same with those suited for same pitch-circle in outside 
gearing, but relative position of teeth, spaces, and flanks are reversed, and adden¬ 
dum-circle is of less radius than pitch-circle. 

A Train is a series of wheels in connection with each other, and consists of a 
series of axles, each having on it two wheels, one is driven by a wheel on a preced¬ 
ing axis and other drives a wheel on following axis. 

Idle Wheel.—A wheel revolving upon an axis, which receives motion from a pre¬ 
ceding wheel and gives motion to a following wheel, used only to affect direction of 
motion. 

Trundle , Lantern , or Wallower is when teeth of a pinion are constructed of round 
bars or solid cylinders set into two disks. Trundle with less than eight staves can¬ 
not be operated uniformly by a wheel with any number of teeth. 

Spur , Driver , or Leader is term for a wheel that impels another; one impelled is 
Pinion , Driven , or Follower. 


WHEEL GEARING. 855 

Teeth of wheels should be as small and numerous as is consistent with 
strength. 

When a Pinion is driven by a wheel , number of teeth in pinion should not 
be less than 8. 

When a Wheel is driven by a pinion , number of teeth in pinion should not 
be less than 10. 

When 2 wheels act upon one another, greater is termed Wheel and lesser Pinion. 

When the tooth of a wheel is made of a material different from that of wheel it is 
termed a Cog ; in a pinion it is termed a Leaf J in a trundle a Stave, and on a disk 
a Pin. 

Material of which cogs are made is about one fourth strength of cast iron. 
Hence, product of their b d 2 should be 4 times that of iron teeth. 

Number of teeth in a wheel should always be prime to number of pinion; 
that is, number of teeth in -wheel should not be divisible by number of teeth 
in pinion without a remainder. This is in order to prevent the same teeth 
coming together so often and uniformly as to cause an irregular -wear of their 
faces. An odd tooth introduced into a -wheel is termed a Hunting tooth or Cog. 

The least number of teeth that it is practicable to give to a wheel is regu¬ 
lated by necessity of having at least one pair always in action, in order to 
provide for the contingency of a tooth breaking; and least number that can 
be employed in pinions having teeth of following classes is : Involute, 25 ; 
Epicycloidal, 12 ; Staves or Pins, 6. 

Velocity Ratio in a Train of Wheels. —To attain it -with least number of 
teeth, it should, in each elementary combination, approximate as near as 
practicable to 3.59. A convenient practical rule is a range from 3 to 6. 

Illustration. 1 6 36 216 1296 velocity ratio. 

123 4 elementary combination. 

To increase or diminish velocity in a given proportion, and with least 
quantity of wheel-work, number of teeth in each pinion should be to number 
of teeth in its wheel as 1: 3.59. Even to save space and expense, ratio 
should never exceed 1 : 6. {Buchanan.) 

To Compute TPiteli.. 

Rule.—D ivide circumference at pitch-line by number of teeth. 

Example.— A wheel 40 ins. in diameter requires 75 teeth; what is its pitch? 

3.1416 X 4°-=“ 75 — 1-6755 ins. 

To Compute True or Clrordial IPitclr. 

Rule.—D ivide 180 0 by number of teeth, ascertain sine of quotient, and 
multiply it by diameter of wheel. 

Example.— Number of teeth is 75, and diameter 40 ins. ; wEat is true pitch? 

180-4-75 = 2 0 24', and sin. of 2 0 24' = .041 88, tvhich X 40 = 1.6752 ins. 

To Compute Diameter. 

Rule.— Multiply number of teeth by pitch, and divide product by 3.1416. 

Example.— Number of teeth in a wheel is 75, and pitch 1.6755 ins.; w-hat is di¬ 
ameter of it? 75 X 1-6755 -f- 3.1416 = 40 ins. 

When the True Pitch is given. Rule.—M ultiply number of teeth in wheel 
by true pitch, and again by .3184. 

Example.— Take elements of preceding case. 

75 X 1-6752 X - 3 i8 4 = 4 ° ins - 

Or, Divide 180° by number of teeth, and multiply cosecant of quotient by 
pitch. 

180-4-75 = 2 0 24', and cos. 2 0 24'= 23.88, which X 1.6752 = 40 ins. 


856 


WHEEL GEARING. 


To Compute IN"nmIber of Teeth. 

Rule. —Divide circumference by pitch. 

To Compute Number of Teeth in. a Pinion or Follower 
to have a given Velocity. 

Rule. —Multiply velocity of driver by its number of teeth, and divide 
product by velocity of driven. 

Example i. —Velocity of a driver is 16 revolutions, number of its teeth 54, and 
velocity of pinion is 48; what is number of its teeth? 

16 X 54 - 4 - 48 = 18 teeth. 

2.—A wheel having 75 teeth is making 16 revolutions per minute; what is num¬ 
ber of teeth required in pinion to make 24 revolutions in same time? 

16 X 75 - 4 - 24 = 50 teeth. 

To Compute Proportional Piadins of a "Wheel or IPinion. 

Rule. —Multiply length of line of centres by number of teeth in wheel, 
for wheel, and in pinion, for pinion, and divide by number of teeth in both 
wheel and pinion. 

Example. —Line of centres of a wheel and pinion is 36 ins., and number of teeth 
in wheel is 60, and in pinion 18; what are their radii? 

36 x 60 , 36 x 18 

---= 27.60 ms. wheel. ——--= 8.3 ms. pinion. 

60+18 ' y 60 + 18 

To Compute Diameter of a IPinion. 

When Diameter of Wheel and Number of Teeth in Wheel and Pinion are 
given. Rule. —Multiply diameter of wheel by number of teeth in pinion, 
and divide product by number of teeth in wheel. 

Example. —Diameter of a wheel is 25 ins., number of its teeth 210, and number 
of teeth in pinion 30; what is diameter of pinion? 

25 X 30-r- 210 = 3.57 ins. 

To Compute UNTnmber of Teeth required in a Train o± 
Wheels to produce a given Velocity. 

Rule.—M ultiply number of teeth in driver by its number of revolutions, 
and divide product by number of revolutions of each pinion, for each driver 
and pinion. 

Example.—I f a driver in a train of three wheels has 90 teeth, and makes 2 revo¬ 
lutions, and velocities required are 2, 10, and 18, what are number of teeth in each 
of other two ? 


10 : 90 :; 2 : 18 = teeth in 2 d wheel. 18 : 90 ;: 2 : 10 ==. teeth in 3c? wheel. 

To Compute Velocity of a IPinion. 

Rule. —Divide diameter, circumference, or number of teeth in driver, as 
case may be, by diameter, etc., of pinion. 

When there are a Series or Train of Wheels and Pinions. Rule. —Divide 
continued product of diameter, circumference, or number of teeth in wheels 
by continued product of diameter, etc., of pinions. 

Example i.—I f a wheel of 32 teeth drives a pinion of 10, upon axis of which there 
is one of 30 teeth, driving a pinion of 8, what are revolutions of last? 

32 30 960 , 

— X -v = — 12 revolutions. 

10 8 80 


2.—Diameters of a train of wheels are 6, 9, 9, 10, and 12 ins.; of pinions, 6, 6, 6,6, 
and 6 ins.; and number of revolutions of driving shaft or prime mover is 10 • what 
are revolutions of last pinion? 


6 


X 9 X 9 X 10 X 12 X 10 
. 6X6X6X6X6 


= —= 75 revolutions. 
7776 




WHEEL GEARING, 


857 


To Compute Proportion, that Velocities of "Wlieels in 
a Train should Lear to one another. 


Rule.—S ubtract less velocity from greater, and divide remainder by one 
less than number of wheels in train ; quotient is number, rising in arithmet¬ 
ical progression from less to greater velocity. 

Example.— What should be velocities of 3 wheels to produce 18 revolutions, the 
driver making 3 ? 


^ ^_— = 7.5 — number to be added to velocity of driver = 7.5 -f-3 —10.5, and 

3 * —— 2 

10.5 4 - 7 -5 = 18 revolutions. Hence 3, 10.5, and 18 are velocities of three wheels. 


IPitcH of “Wlieels. 


To Compute Diameter of a Wheel for a given Pitch, 
or Pitch for a given Diameter. 

From 8 to 192 Teeth. 


No. of 
Teeth. 

Diame¬ 

ter. 

No. of | 
Teeth, j 

Diame¬ 

ter. 

No. of 
Teeth. 

Diame¬ 

ter. 

No. of 
Teeth. 

Diame¬ 

ter. 

No. of 
Teeth. 

Diame¬ 

ter. 

8 

2.6l 

45 

14-33 

82 

26.II 

119 

37-88 

156 

49.66 

9 

2-93 

46 

14.65 

83 

26.43 

120 

38.2 

157 

49.98 

10 

3-24 

47 

14.97 

84 

26.74 

121 

38-52 

158 

50.3 

11 

3-55 

48 

15.29 

85 

27.06 

122 

38.84 

159 

50.61 

12 

3.86 

49 

15.61 

86 

27.38 

123 

39.16 

160 

50.93 

13 

4.18 

50 

15-93 

87 

27.7 

124 

39-47 

l6l 

51-25 

14 

4.49 

5i 

16.24 

88 

28.02 

125 

39-79 

162 

51-57 

IS 

4.81 

52 

16.56 

89 

28.33 

126 

40.11 

163 

51-89 

16 

5.12 

53 

16.88 

90 

28.65 

127 

40.43 

164 

52.21 

17 

5-44 

54 

17.2 

9i 

28.97 

128 

40.75 

165 

52.52 

18 

5-76 

55 

I7-52 

92 

29.29 

129 

41.07 

166 

52.84 

19 

6.07 

56 

17.8 

93 

29.61 

130 

41.38 

167 

53-16 

20 

6 -39 

57 

18.15 

94 

29-93 

131 

41.7 

168 

5348 

21 

6.71 

58 

18.47 

95 

30.24 

132 

42.02 

169 

53-8 

22 

7-03 

59 

18.79 

96 

3 0 -56 

133 

42.34 

170 

54.12 

23 

7-34 

60 

19.11 

97 

30.88 

134 

42.66 

I 7 I 

54-43 

24 

7.66 

61 

19.42 

98 

31.2 

135 

42.98 

172 

54-75 

25 

7.98 

62 

19.74 

99 

3I-52 

136 

43-29 

173 

55-07 

26 

8-3 

63 

20.06 

100 

31.84 

137 

43.61 

174 

55-39 

27 

8.61 

64 

20.38 

101 

32-15 

138 

43-93 

175 

55-7i 

28 

8-93 

65 

20.7 

102 

32-47 

139 

44-25 

176 

56.02 

29 

9-25 

66 

21.02 

103 

3 2 -79 

140 

44-57 

1 177 

56-34 

3° 

9 57 

67 

2i-33 

104 

33-ii 

141 

44-88 

178 

56.66 

31 

9.88 

68 

21.65 

105 

33-43 

142 

45-2 

179 

56.98 

32 

10.2 

69 

21.97 

106 

33-74 

143 

45-52 

180 

57-23 

33 

10 52 

70 

22.29 

107 

34.06 

144 

45-84 

l8l 

57.62 

34 

10.84 

7i 

22.61 

108 

34-38 

145 

46.16 

182 

57-93 

35 

11.16 

72 

22.92 

109 

34-7 

146 

46.48 

183 

58.25 

36 

11.47 

73 

23.24 

no 

35-02 

147 

46.79 

184 

58-57 

37 

11.79 

74 

23-56 

III 

35-34 

148 

47.11 

I§5 

58.89 

38 

12.11 

75 

23.88 

112 

35-65 

149 

47-43 

186 

59.21 

39 

12.43 

76 

24.2 

113 

35-97 

150 

47-75 

187 

59-53 

40 

12.74 

77 

24.52 

114 

36.29 

151 

48.07 

188 

59-84 

4 1 

13.06 

78 

24.83 

115 

36.61 

152 

48-39 

189 

60.16 

42 

13-38 

79 

25-15 

Il6 

36.93 

i53 

48.7 

190 

60.48 

43 

13-7 

80 

25-47 

117 

37-25 

154 

49.02 

I 9 I 

60.81 

44 

14.02 

81 

25-79 

! 118 

37-56 

l 155 

49-34 

192 

61.13 


Pitch in this table is true pitch, as before described. 

To Compute Circumference of a Wheel. 

Rule.—M ultiply number of teeth by their pitch. 

4 C* 



































858 


WHEEL GEARING. 


To Compute IR-evolntions of a 'Wlaeel or ^Pinion. 
Rule.— Multiply diameter or circumference of wheel or number of its 
teeth in ins., as case may be, by number of its revolutions, and divide prod¬ 
uct by diameter, circumference, or number of teeth in pinion. 

Example. —A pinion io ins. in diameter is driven by a wheel 2 feet in diameter, 
making 46 revolutions per minute; what is number of revolutions of pinion? 

2X12X464-10=110.4 revolutions. 

To Compute INTvim'ber of Teeth, of a Wheel for a given. 
Diameter and. Pitch.. 

Rule.—D ivide diameter by pitch, and opposite to quotient in preceding 
table is given number of teeth. 

Example.— Diam. of wheel is 40 ins., and pitch 1.675; what is number of its teeth ? 
404- 1.675 =' 23.88, and opposite thereto in table is 75 = number of teeth. 

To Compute Diameter of a Wheel for a given Pitch and 

Number of Teeth. 

Rule.—M ultiply diameter in preceding table for number of teeth by 
pitch, and product will give diameter at pitch circle. 

Example. —What is diameter of a wheel to contain 48 teeth of 2.5 ins. pitch? 

15.29 X 2.5 = 38.225 ins. 


To Compute Pitch of a Wheel for a given Diameter and 

Number of Teeth. 

Rule.—D ivide diameter of wheel by diameter in table for number of 
teeth, and quotient will give pitch. 

Example. —What is pitch of a wheel when diameter of it is 50.94 ins., and num¬ 
ber of its teeth 80? 5 a 94 = 25 . 47 2 ins. 

General Illustrations. 


1. —A wheel 96 ins. in diameter, making 42 revolutions per minute, is to drive a 
shaft 75 revolutions per minute; what should be diameter of pinion? 

96 X 4 2 = 75 — 53 - 76 ins. 

2. —If a pinion is to make 20 revolutions per minute, required diameter of an¬ 
other to make 58 revolutions in same time. 

58 - 4 - 20 = 2.9 = ratio of their diameters. Hence, if one to make 20 revolutions is 
given a diameter of 30 ins., other will be 304-2.9 = 10.345 ins. 

3. —Required diameter of a pinion to make 12.5 revolutions in same time as one 
of 32 ins. diameter making 26. 


32 X 26 - 4 -12.5 = 66.56 ins. 

4. —A shaft, having 22 revolutions per minute, is to drive another shaft at rate 
of 15, distance between two shafts upon line of centres is 45 ins.; what should be 
diameter of wheels? 

Then, 1st. 22 -f 15 : 22 :: 45 : 26.75 = ins. in radius of pinion. 

2d. 22-j- 15 : 15 45 ; 18.24 — ins. in radius of spur. 

5. —A driving shaft, having 16 revolutions per minute, is to drive a shaft 81 revo¬ 
lutions per minute, motion to be communicated by two geared wheels and two pul- 
leys, with an intermediate shaft; driving wheel is to contain 54 teeth and driving 
pulley upon driven shaft is to be 25 ins. in diameter; required number of teeth in 
driven wheel, and diameter of driven pulley. 

Let driven wheel have a velocity of V16 x 81 =36, a mean proportional between 
extreme velocities 16 and 81. 

Then, 1st. 36 : 16 :: 54 : 24 = teeth in driven wheel. 

2d. 81 : 36 :: 25 : 11.11 = ins. diameter of driven pulley. 

6. —If, as in preceding case, whole number of revolutions of driving shaft num- 
ber of teeth in its wheel, and diameters of pulleys are given, what are revolutions 
ot shafts? 

Then, 1st. 18 : 16 :: 54 : 48 = revolutions of intermediate shaft. 

2d. 15 : 48 :: 25 : 80 = revolutions of driven shaft. 



WHEEL GEARING.-TEETH OF WHEELS. 


Teeth, of NV'heels. 

Epicycloida,l. —In order that teeth of wheels and pinions should work 
evenly and without unnecessary rubbing friction, the face (from pitch line 
to top) of the outline should be determined by an epicycloidal curve (see 
page 228), and that of the flank (frompitch line to base ) by an hypocycloidal 
(see also page 228). 

When generating circle is equal to half diameter of pitch circle, hypocy¬ 
cloidal described by it is a straight diametrical line, and consequently out¬ 
line of a flank is a right line, and radial to centre of wheel. 

If a like generating circle is used to describe face of a tooth of other wheel 
or pinion respectively, the wheel and pinion will operate evenly. 

Illustration.— Determine all elements of wheel 
—viz., Pitch circle, Number of teeth, Pitch, Length, 
Face, and Flank. 

Cut a template A to pitch circle c c of wheel, and 
secure it temporarily to a board. 

Having determined depth of tooth, set it off on 
pitch line, as a 0, Fig. 1, and above it apply a sec¬ 
ond template, a\ radius of wheel is equal to half 




radius of pinion; insert into, or attach exactly at its edge, a tracer ., roll template 
a along A, and tracer will describe an epicycloidal curve, a r, and by inverting a 
describe 0 r, and faces of a tooth are delineated. 

To describe flanks, define pitch line c c, Fig. 2, and arc n n, 
drawn at base of teeth or board A (as in Fig. 1), secure a strip 
of wood, w, equal in length to radius of wheel, and locate 
centre of it, x , draw radii x a and x 0, and they will define 
flanks, which should be filleted, as shown at s s. Define arc 
zz, and length of tooth is determined. 

Proceed in like manner conversely for teeth of pinion, and 
wheel and pinion thus constructed will operate truly. 

In construction of the teeth of a wheel or pinion in 
the pattern-shop, it is customary to construct the wheel 
or pinion complete, out to face of wheel at base of teeth, 
and then to insert the teeth in rough, approximately 
shaped blocks, by a dovetail at their base, fitting into face of wheel, and then 
the outline of a tooth is described thereon; the block is then removed, fin¬ 
ished as a tooth, replaced, fastened, and filleted. 

Involute. 

Teeth of two wheels will work truly together when their face is that of an 
involute (see page 229), and that two such wheels should work truly, the 
circles from which the involute lines for each wheel are generated must be 
concentric with the wheels, with diameters in same ratio as those of the wheels. 

Assume A c, B c, Fig. 3, pitch radii of two wheels designed 
to work together, through c, draw a right line, e i, and with 
perpendiculars e c, i c, describe arcs n 0, r s, and involutes 
n c o and res define a face of each of the teeth. 

To describe teeth of a pair of 
wheels of which Ac, Be, Fig. 4, 
are pitch radii, draw c i, c e, per¬ 
pendicular to radials B i and A e, 
and they are to be taken as the 
radials of the involute arcs from 
which the faces of the teeth are 
to be defined; then fillet flanks at 
base, as before described, Fig. 2. \* 

Involute teeth will work with truth, even at varying 
distances apart of the centres of the wheels, and any wheels of a like pitch will work 
in union, however varied their diameters. 













86o wheel gearing.—TEETH of wheels. 


Circular teeth are defined as follows: 


—4 



Assume A A, Fig. 5, pitch-line, b b line of base 
of teeth, and t t face-line. Set off on pitch-line 
divisions both of pitch and depth of teeth, then 
with a radius of 1.25 pitch describe arcs as 0 s 
upon pitch line for faces of teeth, then draw ra¬ 
dial lines 0 v, r u, to centre of wheel for flanks, 
strike arc 11 to define length of tooth, and fillet 
flanks at base as before described. 

Proportions of Teetli. 

In computing dimensions of a tooth, it is to 
be considered as a beam fixed at one end, 
weight suspended from other, or face of beam ; 
and it is essential to consider the element of velocity, as its stress in opera¬ 
tion, at high velocity with irregular action, is increased thereby. 

Dimensions of a tooth should be much greater than is necessary to resist 
direct stress upon it, as but one tooth is proportioned to bear whole stress 
upon wheel, although two or more are actually in contact at all times; but 
this requirement is in consequence of the great wear to which a tooth is sub¬ 
jected, shocks it is liable to from lost motion, when so worn as to reduce its 
depth and uniformity of bearing, and risk of the loss of a tooth from a defect. 

A tooth running at a low velocity may be materially reduced in its dimen¬ 
sions, compared with one running at a high velocity and with a like stress. 

Result of operations with toothed wheels, for a long period of time, has 
determined that a cast-iron (Eng.) tooth with a pitch of 3 ins. and a breadth 
of 7.5 ins. will transmit, at a velocity of 6.66 feet per second, power of 59.16 
horses. 


To Compvite Dimensions of a Tooth to Tbesist a given. 

Stress. 


Rule.—M ultiply extreme pressure at pitch-line of wheel by length of 
tooth in decimal of a foot, divide product by Coefficient of material of tooth, 
and quotient will give product of breadth and square of depth. 


Or ,~=bd*. 


S representing stress in lbs., and l length in feet. 


The Coefficient of cast iron for this or like purposes may be taken at from 50 to 70. 


Pitch A B = 1. Depth rs = .45. 

Length co = . 75. Space s v = . 55. 

Working length c e = . 7. Play s v — r s = . 1. 

Clearance e to 0 — .05. Face B c = .35. 

Note. — It is necessary first to determine pitch, in 
order to obtain either length or depth of a tooth. 

Example.— Pressure at pitch-line of a cast- 
iron wheel (at a velocity of 6.66 feet per sec¬ 
ond) is 4886 lbs.; what should be dimensions 

of teeth, pitch being 3 ins. ? 

3 X .75 = 2.25 length of tooth, which- 4-12 —. 1875 = length in decimal of a foot. 
Coefficient of material is taken at 60. 

4886 x -1875 _ . if length _ 2 .25, pitch = 3, and depth = 1.35 ins. 

60 

Pitches of Equivalent Strength for Cast Iron and Wood. —Iron 1. Hard wood 1.26. 

Then Util. — 8.30 ins. breadth. 

I - 35 2 

When Product ofbd 2 is obtained , and it is required to ascertain either 
dimension. ~ depth, and = breadth. 








WHEEL GEARING.-TEETH OF WHEELS, 


861 


To Compute Depth of a Tooth. 

1. When Stress is given. Rule. —Extract square root of stress, and mul¬ 
tiply it by .02 for cast iron, and .027 for hard wood. 

2. When EP is given. Rule. —Extract square root of quotient of IP di¬ 
vided by velocity in feet per second, and multiply it by .466 for cast iron, 
and .637 for hard wood. 

Example.— IP to be transmitted by a tooth of cast iron is 60, and velocity of it 
at its pitch-line is 6.66 feet per second; what should be depth of tooth ? 

/ 60 

\/6456 X ’ 4 ^ 6 = I- * ns ' 


To Compute IP of a Tooth. 

Rule.—M ultiply pressure at pitch-line by its velocity in feet per minute, 
and divide product by 33000. 

Example.— What is IP of a tooth of dimensions and at velocity given in preced¬ 
ing example. 

4886 X 6.66 X 60" - 4 - 33 000 = 59.16 horses. 

To Compute Stress that may he Lome by a Tooth. 

Rule. —Multiply Coefficient of material of tooth to resist a transverse 
strain, as estimated for this character of stress, by breadth and square of its 
depth, and divide product by extreme length of it in decimal of a foot. 

Example. —Dimensions of a cast-iron tooth in a wheel are 1.38 ins. in depth, 2.1 
ins. in length, and 7.5 ins. in breadth; what is the stress it will bear? 

60 7 ^ I. "28^ 

Coefficient assumed at 60. -—-—— = 4886 lbs. 

2.1 -r- 12 

Following deductions by the rules of different authors for like elements are sub¬ 
mitted for a cast-iron tooth: 

Pitch.3 ins. | Depth_1.38 ins. | Breadth... 7.5 ins. | Length..'.. 2.1 ins. 


Actual Power in Stress Exerted 
at a velocity of 400 feet per min., 4886 lbs. 


/H 

By Above Rule /— X .446. 

“ Fairbairn .025 y/W . 

/ W 

“ Imperial Journal /-- .. 

V 1576 


Depth of 
Tooth. 


Ins. 

1-398* 

i-75 

1.76 


I Actual Power in Stress Exerted 
at a velocity of 400 feet per min ., 4886 lbs. 


By Rankine - 
“ Tredgold — ^/— 


500 

H 


Buchanan 


•556 H 


Depth of 
Tooth. 


Ins. 

1.8 


2.25 


2.24 


H representing horse-power (60), W stress in lbs., and v velocity in feet per second. 


Depth, Ditch, and. Breadth. (M. Morin.) 


Cast iron.028 -/W = d. .057 yfW — P. 

Hard wood.. .038 y/W = d. .079 y/W = P. 


W representing weight or stress upon tooth in lbs., d depth of tooth, and P pitch 
in ins. 

When velocity of pitch-circle does not exceed 5 feet per second h =. 4 cl, 
when it exceeds 5 feet b=.$ cl, and if wheels are exposed to wet b = 6 cl. 

b representing breadth. 

Illustration. —Assume pressure at pitch-line of a cast-iron wheel upon a tooth 
equal 6000 lbs., and velocity 5 feet per second. 

Then .028 1/6000 = 2.17 ins. Depth, and .057 1/6000 = 4.46 ins. Pitch. 

Note. — For further Illustrations of Formation of Teeth, Bevel Gearing;, Willis’s Odontograph, 
Staves, Trundles, etc., see Mosely’s Engineering, Shelton’s Mechanic’s Guide, Fairbairn’s Mechanism 
and Machinery of Construction, etc. 

* This depth, with a breadth of 7.5 ins., is .1 of ultimate strength of average strength of American 
Cast Iron. 




















862 TEETH OF WHEELS.—WINDING ENGINES. 


PROPORTIONS OF WHEELS. 

With six flat A rms and Ribs upon one side of them , as mxmfy ; or a Web 
in centre , as aa. 

Rim .—Depth, measured from base of teeth, .45 to .5 of pitch of teeth, hav¬ 
ing a web upon its inner surface .4 of pitch in depth and .25 to .3 of it in 
width. 

Note.—W hen face of wheel is mortised, depth of rim should be 1.5 times pitch, 
and breadth of it 1.5 times breadth of tooth or cog. 

Hub .—When eye is proportionate to stress upon wheel, hub should he 
twice diameter of eye. In other cases depth around eye should be .75 to .8 
of pitch. 

Arm .—Depth .4 to .45 of pitch. Breadth at rim 1.5 times pitch, increas¬ 
ing .5 inch per foot of length toward hub. 

Rib upon one edge of arm, or Web in its centre, should be from .25 to .3 
pitch in width, and .4 to .45 of it in depth. 

When section of an arm differs from those above given, as with one with 
a plane section, as raaa , or with a double rib, as , its dimensions 

should be proportioned to form of section. 

In a wheel of greater relative diameter, length of hub and breadth of arms, 
or of the rib or web, according as plane of arm is in that of wheel, or con¬ 
trariwise, should be made to exceed breadth of face of wheel (at the hub) 
in order to give it resistance to lateral strain. 

Number of arms in wheels should be as follows •. 

1.15 to 3.25 feet in diameter. 4 I 5 to 8.5 feet in diameter.6 

3.25 “ 5 “ “ . 5 I 8.5 “16 “ “ . 8 

16 to 24 feet in diameter. 10 

With light wheels, number of arms should be increased, in order better to 
sustain rigidity of rim. 

Mortise Wheels .—Their rim or face should he .9 pitch of tooth, and twice 
depth of rim of a solid wheel. 


WINDING ENGINES. 


With Winding Engines, for drawing coals, etc., out of a Pit, where it 
is required to give a certain number of revolutions, it is necessary to 
have given diameter of Drum and thickness of rope, which is flat made, 
and contrariwise. 


To Compute Diameter of a Dram. 


Where Flat Ropes are used , and are wound one part over the other. Rule. 
—Divide depth of pit in ins. by product of number of revolutions and 3.1416, 
and from quotient subtract product of thickness of rope and number of rev¬ 
olutions ; remainder is diameter in ins. 

Example.— If an engine makes 20 revolutions, depth of pit being 600 feet and 
rope 1 inch, what should be diameter of drum ? 


600 X 12 
20 X 3-1416 


— 1 X 20 = 


7200 

62.832 


— 20 = 94.59 


ins. 


To Compate Diameter of Roll. 


Rule. —To area of drum add area or edge surface of rope; then ascertain 
by inspection in table of areas, or by calculation, diameter that gives this 
area, and it is the diameter of Roll. 













WINDING ENGINES.-WINDMILLS. 


863 


Example. —What is diameter of roll in preceding example? 

Area of 94.59 = 7027.2 + (area of 7200 X 1) = 7200= 14227.2, and +14227.2-4- 
.7854 = 134.59 ins. 

Or, Radius of drum is increased number of revolutions multiplied by thickness 
of rope; as 2±_59 _j_ 20 x 1 = 67.295 ins. 

To Compute Number of Revolutions. 

Rule.—T o area of drum add area of edge surface of rope; from diameter 
of the circle having that area subtract diameter of drum, and divide re¬ 
mainder by twice thickness of rope; quotient will give number of revolutions. 

Example.—L ength of a rope is 2600 ins., its thickness 1 inch, and diameter of 
drum 20 ins.; what is number of revolutions? 

Area of 20 +area of rope = 314.16-!-2600 = 2914.16, diameter of which is 60.91, 

and —— -— = 20.45 revolutions. 

1X2 

Or, subtract diameter of drum from diameter of roll, and divide remainder by 
twice thickness of rope; as 60.91 — 20 = 40.91, and 40.91 - 4 - 1 x 2 = 20.45 revolutions. 

To Compute Point of ^Meeting of .A-scending and De¬ 
scending Bnckets when two or more are used. 

To Compute Point of Meeting of Buckets. Rule.—D ivide sum of length 
of turns of rope by 2, and to quotient add length of last turn; divide sum 
by 2, multiply quotient by half number of revolutions, and product will 
give distance from centre of drum at which buckets will meet. 

Note i.— Meetings will always be below half depth of pit. 

2.—At half number of revolutions buckets will meet. 

Example.—D iameter of a drum is 9 feet, thickness of rope 1 inch, and revolu¬ 
tions 20; what is depth of pit, and at what distance from top will buckets meet? 

- b 38.48 -7- 2 X — =-= 35-995 X 10 = 359.95/ee#. 

2 2 2 

To Compute this Depth. Rule.—T o diameter of drum add thickness of 
rope in feet, and ascertain its circumference; to diameter of drum add quo¬ 
tient of product of twice thickness of rope and number of revolutions less 1, 
divided by 12 for a diameter, and circumference of this diameter is length 
of last turn, also in feet; add these two lengths together, multiply their sum 
by half number of revolutions, and product will give depth of pit. 

9 + thickness of rope = 9 + -i. of 1 = 9.083, which X 3-1416 = 28.54 feet = length 
I X 2 X 20 ■ I 

of first turn. 9.0833-)- X 3.1416 = 38.48 feet = length of last turn. 

Then 28.54 + 38.48 X — = 67.02 X 10 = 670.2 feet, depth of pit. 

2 


WINDMILLS. 

Driving Shaft of a vertical windmill should be set at an elevating angle 
with horizon when set upon low ground, and at a depressing angle when set 
upon elevated ground. Range of these angles is from 3 0 to 15°.. A velocity 
of wind of 10 feet per second is not generally sufficient to drive a loaded 
mill, and if velocity exceeds 35 feet per second the force is generally too 
great for ordinary structures. 

Angle of Sails should be from 18 0 to 30° at their least radius, and from 
7 0 to 17 0 at their greatest radius, mean angle being from 15 0 to 17 0 to plane 
of motion of sails. Length of a whip (arm) is divided into 7 parts, sails ex¬ 
tending over 6 parts. 










WIND-MILLS. 


864 


Whip in parts of its length: Breadth .033, at top .016; Depth .025, at top 
.0125; Width of sail .33, at axis .2. Distance of sail from axis .014 of 
length of whip, and cross-bars 16 to 18 ins. from centres. 


t 8 (P 


2 3 


To Compute Angles of Sails. 

— angle of sail with plane of its motion at any part of it. d repre¬ 


senting distance of part of sail from its axis, and r extreme radius of sail, both in feet. 


Illustration. —Assume r = 14, and length of sail 12 feet, d — .5 of 12 or three 
sixths of sail = . 5 X 12 -J~ (14 —12) = 2 = 8 feet. 

18 X 8 2 

Then 23 0 -= 23 — 5.88° = 17.12 0 . 

14 2 

Hence, angle of sail with axis = 90 0 — 17.12 0 — 72.88°. 

If radius of sails is divided into 6 equal parts, angles at each of these parts will 
be as follows: 

Distance from Axis. 


1 2 3.4 S 6 

Angle of sail with axis.67-5° 69° y 1 ^ 0 75 ° 79 - 5 ° 85° 

“ “ with plane of motion. 22.5° 21° 18.5° 15° 10.5 0 5 0 


To Compute Elements of Windmills. 


3.16 v 


r sin. x 
TP X 1080 000 


n: 


11.5 v 


.1047 n — av, 


= A; 


y 


A a 3 4 5 6 7 8 

0-=EP; 

1080 000 


R 2 + ?- 2 

_ _ _ _ — r'. v representing velocity of wind per sec- 

v 3 V 2 

ond, r' radius of centre of percussion of sails, and R and r outer and inner radii of 
sails, all in feet, x mean angle of sail to plane of motion, n number of revolutions of 
arms per minute, a v angular velocity, A area of sails in sq. feet, and IP horse-power. 

Illustration.— If a windmill has 4 arms of 28 feet, with a mean angle (x) of 16°, 
with an area of sail of 150 sq. feet each, having an inner radius of 4 feet, and is op¬ 
erated by wind at a velocity of 40 feet per second; what are its elements? 

11.5X4° /28 s — f— 4 2 , r 3.16X4° 

~ _ n = 22 _^. 


Then 


4 X 150 X 64 000 
1080 000 


' / 2 
= n = 23; /- 

W = 35-55] 


sr' — 20 feet; _ 

2 20 X. 275 64 

35-55 X 1080000 

- = A = 599.9 sq. feet. 

64 000 


DecLmctioiis from "Velocities varying from A to 9 Feet per 

Second. (Mr. Smeaton.) 


1. Velocity of windmill sails, so as to produce a maximum effect, is near¬ 
ly as velocity of wind, their shape and position being same. 

2. Load at maximum is nearly, but somewhat less than, as square of ve¬ 
locity of wind, shape and position of sails being same. 

3. Effects of same sails, at a maximum, are nearly, but somewhat less 
than, as cubes of velocity of wind. 

4. Load of same sails, at maximum, is nearly as squares, and their effect 
as cubes of their number of turns in a given time. 

5. In sails where figure and position are similar, and velocity of wind the 
same, number of revolutions in a given time will be reciprocally as radius or 
length of sail. 

6. Load, at a maximum, which sails of a similar figure and position will 
overcome at a given distance from centre of motion, will be as cube of radius. 

7. Effects of sails of similar figure and position are as square of radius. 

8. Velocity of extremities of Dutch sails, as well as of enlarged sails, in 
all their usual positions when unloaded, or even loaded to a maximum, is 
considerably greater than that of wind. 












WINDMILLS.-WOOD AND TIMBER. 


865 

Results of Experiments 011 Effect of Windmill Sails. 

When a vertical windmill is employed to grind corn, the millstone usu¬ 
ally makes 5 revolutions to 1 of the sail. 

1. When velocity of wind is 19 feet per second, sails make from 11 to 12 
revolutions in a minute, and a mill will grind from 880 to 990 lbs. in an 
hour, or about 22 440 lbs. in 24 hours. 

2. When velocity of wind is 30 feet per second, a mill will carry all sail, 
and make 22 revolutions in a minute, grinding 1984 lbs. of flour in an hour, 
or 47 616 lbs. in 24 hours. 


Results of Operation of Windmills. (A. R. Woolf M. E.) 
Velocity of Wind 15 to 20 Miles per Hour. 

Revolutions of Wheel and Gallons of Water raised per Minute. 


Desig¬ 

nation 

Revolutions 

of 

Water raised to an Elevation of 

Power 

Cost per Hour. 

of Mill. 

Wheel. 

25 Feet. 

j 50 Feet. 

ioo Feet. 

200 Feet. 

developed. 

Actual.* 

Per IP. 

Feet. 

No. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

HP - 

Cents. 

Cents. 

8-5 

70 to 75 

6.16 

3 .C2 

— 

— 

.04 

.60 

15 

10 

60 to 65 

19.18 

9-56 

4-75 

— 

.12 

.70 

5-8 

14 

50 to 55 

45-14 

22.57 

11.25 

5 

.28 

1.63 

5-8 

18 

40 to 45 

97.68 

52.16 

24.42 

12.21 

.61 

2.83 

4.6 

20 

35 to 4 ° 

124.95 

63-75 

31-25 

15-94 

.78 

3-56 

4-5 

25 

30 to 35 

212.38 

106.96 

49-73 

26.74 

i -34 

4.26 

3-2 


* Including interests 5 per cent, per annum. 


WOOD AND TIMBER. 

Selection of Standing Trees .— Wood grown in a moist soil is lighter, 
and decays sooner, than that grown in dry, sandy soil. 

Best Timber is that grown in a dark soil, intermixed with gravel. 
Poplar, Cypress, Willow, and all others which grow best in a wet soil, 
are exceptions. 

Hardest and densest woods, and least subject to decay, grow in warm 
climates; but they are more liable to split and warp in seasoning. 

Trees grown upon plains or in centre of forests are less dense than 
those from edge of a forest, from side of a hill, or from open ground. 

Trees (in U. S.) should be selected in latter part of July or first part 
of August; for at this season leaves of sound, healthy trees are fresh 
and green, while those of unsound are beginning to turn yellow. A 
sound, healthy tree is recognized by its top branches being weil leayed, 
bark even and of a uniform color. A rounded top, few leaves, some of 
them turned yellow, a rougher bark than common, covered with parasitic 
plants, and with streaks or spots upon it, indicate a tree upon the de¬ 
cline. Decay of branches, and separation of bark from the wood, are 
infallible indications that the wood is impaired. 

Green timber contains 37 to 48 per cent, of liquids. By exposure to 
air in seasoning one year, it loses from 17 to 25 per cent., and when 
seasoned it retains from 10 to 15 per cent. 

According to M. Leplay, green wood contains about 45 per cent, of its 
weight of moisture. In Central Europe, wood cut in winter holds, at end of 
following summer, fully 40 per cent, of water, and when kept dry for sev¬ 
eral years retains from 15 to 20 per cent, of water. 

Felling Timber .—Most suitable time for felling timber is in midwinter and 
in midsummer. Recent experiments indicate latter season and month of July. 

4 D 



















866 


WOOD AND TIMBER. 


A tree should be allowed to attain full maturity before being felled. Oak 
matures at 75 to 100 years and upward, according to circumstances; Ash, 
Larch, and Elm at 75; and Spruce and Fir at 80. Age and rate of growth 
of a tree are indicated by number and width of the rings of annual increase 
which are exhibited in a cross-section of its body. 

A tree should be cut as near to the ground as practicable, as the lower 
part furnishes best timber. 

Dressing Timber .— As soon as a tree is felled, it should be stripped of its 
bark, raised from the ground, reduced to its required dimensions, and its 
sap-wood removed. 

Inspection of Timber. —Quality of wood is in some degree indicated by its 
color, which should be nearly uniform, and a little deeper towards its cen¬ 
tre, and free from sudden transitions of color. White spots indicate decay. 
Sap-wood is known by its white color; it is next to the bark, and soon rots. 

IDefects of Timber. 

Wind-shakes are serious defects, being circular cracks separating the con¬ 
centric layers of wood from each other. 

Splits, Checks, and Cracks, extending toward centre, if deep and strongly 
marked, render timber unfit for use, unless purpose for which it is intended 
will admit of its being split through them. 

Brash is when wood is porous, of a reddish color, and breaks short, with¬ 
out splinters. It is generally consequent upon decline of tree from age. 

Belted is that which has been killed before being felled, or which has died 
from other causes. It is objectionable. 

Knotty is that containing many knots, though sound; usually of stinted 
growth. 

Twisted is when grain of it winds spirally; it is unfit for long pieces. 

Dry-rot is indicated by yellow stains. Elm and Beech are soon affected, 
if left with the bark on. 

Large or decayed knots injuriously affect strength of timber. 

Heart-shake. —Split or cleft in centre of tree, dividing it into segments. 

Star-shake. —Several splits radiating from centre of timber. 

Cap-shake. —Curved splits separating the rings wholly or in part. 

Rind-gall. —Curved swelling, usually caused by growth of layers over spot 
where a branch has been removed. 

Upset. —Fibres injured by crushing. 

Foxiness. —Yellow or red tinge, indicating incipient decay. 

Doatiness. —A speckled stain. 

Seasoning and. Preserving Timber. 

Seasoning is extraction or dissipation of the vegetable juices and moisture 
or solidification of the albumen. When wood is exposed to currents of air 
at a high temperature, the moisture evaporates too rapidly, and it cracks; 
and when temperature is high and sap remains, it ferments, and dry-rot 
ensues. 

Wood requires time in which to season, very much in proportion to density 
of its fibres. 

Water Seasoning is total immersion of timber in water, for purpose of 
dissolving the sap, and when thus seasoned it is less liable to warp and crack, 
but is rendered more brittle. 


WOOD AND TIMBER. 


867 

For purpose of seasoning, it should be piled under shelter and kept drj r ; 
should have a free circulation of air, without being exposed to strong cur¬ 
rents. Bottom pieces should be placed upon skids, which should be free 
from decay, raised not less than 2 feet from ground; a space of an inch 
should intervene between pieces of same horizontal layers, and slats or piling- 
strips placed between each layer, one near each end of pile, and others at 
short distances, in order to keep the timber from winding. These strips 
should be one over the other, and in large piles should not be less than 1 inch 
thick. Light timber may be piled in upper portion of shelter, heavy timber 
upon ground floor. Each pile should contain but one description of timber, 
and they should be at least 2.5 feet apart. 

It should be repiled at intervals, and all pieces indicating decay should be 
removed, to prevent their affecting those which are still sound. 

It requires from 2 to 8 years to be seasoned thoroughly, according to its 
dimensions, and it should be worked as soon as it is thoroughly dry, for it 
deteriorates after that time. 

Gradual seasoning is most favorable to strength and durability of timber. 
Various methods have been proposed for hastening the process, as Steaming , 
which has been applied with success; and results of experiments of various 
processes of saturating it with a solution of Corrosive sublimate and Anti¬ 
septic fluids are very satisfactory. Such process hardens and seasons wood, 
at the same time that it secures it from dry-rot and from attacks of worms. 

Woods are densest and strongest at the roots and at their centres. Their 
strength decreasing with the decrease of their density. 

Oak timber loses one fifth of its iceight in seasoning, and about one third 
in becoming perfectly dry. 

Pitch pine, from the presence of pitch, requires time in excess of that due 
to the density of its fibre. 

Mahogany should be seasoned slowly, Pine quickly. Whitewood should 
not be dried artificially, as the effect of heat is to twist it. 

Salt water renders wood harder, heavier, and more durable than fresh. 

Condition of timber, as to its soundness or decay, is readily recognized 
when struck with a quick blow. 

Timber that has been for a long time immersed in water, when brought 
into the air and dried, becomes brashy and useless. 

When trees are barked in the spring, they should not be felled until the 
foliage is dead. 

Timber cannot be seasoned by either smoking or charring; but when it 
is exposed to worms or to the production of fungi, it is proper to smoke or 
char it, and it may be partially seasoned by being boiled or steamed. 

Timber houses are best provided with blinds which keep out rain and 
snow, but which can be turned to admit air in fine weather, and the houses 
should be kept entirely free from any pieces of decayed wood. 

Kiln-drying is suited only for boards and pieces of small dimensions, as it 
is apt to cause cracks and to impair the strength, unless performed very 
slowly. 

Charring , Painting , or covering the surface is highly injurious to any but 
seasoned wood, as it effectually prevents drying of the inner part of the 
wood, in consequence of which fermentation and decay soon take place. 

Timber is subject to Common or Dry-rot , former occasioned by alternate 
exposure to moisture and dryness, and as progress of it is from the exterior, 
covering of the surface, if seasoned, with paint, tar, etc., is a preservative. 


868 


WOOD AND TIMBER. 


Common-rot is the consequence of its being piled in badly-ventilated slieds. 
Outward indications are yellow spots upon ends of pieces, and a yellowish 
dust in the checks and cracks, particularly where the pieces rest upon pil¬ 
ing-strips. 

Dry or Sap-rot is inherent in timber, and it is the. putrefaction of the veg¬ 
etable albumen. Sap wood contains a large proportion of fermentable ele¬ 
ments. 

Insects attack wood for the sugar or gum contained in it, and fungi subsist 
upon the albumen of wood; hence, to arrest dry-rot, the albumen must be 
either extracted or solidified. 

Most effective method of preserving timber is that of expelling or ex¬ 
hausting its fluids, solidifying its albumen, and introducing an antiseptic 
liquid. 

Strength of impregnated timber is not reduced, and its resilience is improved. 

In desiccating timber by expelling its fluids by heat and air, its strength 
is increased fully 15 per cent. 

The saturation of wood with creosote, tar, antiseptics, etc., preserves it 
from the attack of worms. Jarrow wood, from Australia, is not subjected 
to their attack. 

In a perfectly dry atmosphere durability of woods is almost unlimited. 
Rafters of roofs are known to have existed 1000 years, and piles submerged 
in fresh water have been found perfectly sound 800 years from period of 
their being driven. 

Resistance of Avoods to extension is greater than that of compression. 

Impregnation of 'Wood. 

Several of the successful processes are as folloAvs: 

Kyan, 1832.—Saturated AA'ith corrosive sublimate. Solution 1 lb. of chlo¬ 
ride of mercury to 4 gallons of water. 

Burnett ( Sir Wm.), 1838. — Impregnation with chloride of zinc by sub¬ 
mitting the wood eiKhvise to a pressure of 150 lbs. per sq. inch. Solution, 

1 lb. of the chloride to 4 gallons of water. 

Boucheri. —Impregnation by submitting the Avood endwise to a pressure 
of about 15 lbs. per sq. inch. Solution, 1 lb. of sulphate of copper to 12.5 
gallons of Avater. 

Bethel. —Impregnation by submitting the Avood endwise to a pressure of 
150 to 200 lbs. per sq. inch, with oil of creosote mixed with bituminous 
matter. 

Robbins , 1865.—Aqueous vapor dissipated by the Avood being heated in a 
chamber, the albumen solidified, then submitted to \-apor of coal tar, resin, 
or bituminous oils, which, being at a temperature not less than 325 0 , readily 
takes the place of the vapor expelled by a temperature of 212 0 . 

IIay ford , 187-.—Aqueous vapor dissipated by the wood being heated in a 
chamber to a temperature of from 250° to 270°, the albumen solidified, then 
air introduced to assist the splitting of the outer surfaces. When vapor is 
dissipated, dead oils are introduced under a pressure of 75 lbs. per sq. inch. 

Planks, Deals, and Battens. —When cut from Northern pine ( Pinus Sylve- 
stris ) are termed yellow or red deal, and when cut from spruce (Abies, alba, 
etc.) they are termed white deal. 

Desiccated wood, when exposed to air under ordinary circumstances, ab¬ 
sorbs 5 per cent, of water in the first three days; and avlII continue to absorb 
it until it reaches from 14 to 16 per cent., the amount varying according 
to condition of the atmosphere. 


WOOD AND TIMBER. 


869 


Durability- of Various Woods. 

Pieces 2 feet in Length , 1.5 ins. Square, driven 28.5 ins. into the Earth. 


After 5 Years. 


Wood. 

c 

After 2.5 Years. 

Acacia... 

Goo‘d. 

Ash, Amer. 

Much decayed. 

Cedar, Va. 

Verv good. 

“ Lebanon. 

Good. 

Elm, Eng. 

Much decayed. 

‘ 1 Can. 

u i r 

Fir. 

(l attacked.... . . 

Larch. 

Surface onlv attacked .... 

Oak, Can. 

Very much decayed. 

“ Memel. 

U t i a 

“ Dantzic. 

u u u 

u Chestnut. 

Verv irood. 

Pine, pitch. 

Surface only attacked. 

“ yellow.. 

Attacked. 

“ white. 

Veiy much decayed. 

Teak. 

Very good. 


(Externally decayed, rest per- 
l fectly sound. 

Decayed. 

Sound as when driven. 
Tolerable. 

Entirely decayed. 

Decayed. 

Much decayed. 

(Attacked in part only, rest fair 
{ condition. 

Very rotten. 


(Some moderately, most very 
( much, decayed. 

(Attacked in part only, rest fair 
| condition. 

Much decayed. 

Very rotten. 

Somewhat soft, but good. 


Kffect of Oreosoting. 

Results of Experiments with Various Woods ( E. R. Andrews). 


Wood. 

Water 

absorbed. 

Spruce . 

Oak.... 


( dried. 

Per cent. 
•2543 
.0261 

. 2 


1 creosoted... 

( dried. 


^ creosoted... 

( dried. 

.0 

.714 

•347 

oouon-\yooci 

{creosoted... 


Wood. 

Hard pine..... • 

dried. 

creosoted.. 

Gum, black .. • 

dried. 

creosoted.. 

Birch, white . ■ 

dried. 

creosoted.. 


Water 

absorbed. 


Per cent. 
. 16 
.0 

I 

.125 

•43 
. 124 


Sesquoia Gigantea of California, dried, .4722; creosoted, .0. 

Fluids will pass with the grain of wood with great facility, but will not 
enter it except to a very limited extent when applied externally. 


-Absorption. of Preserving Solvation by different Woods 
for a Period of V Days. Average Lbs. per Cube Foot. 

Black Oak. 3.6 I Hemlock... 2.6 I Rock Oak. 3.9 

Chestnut. 3 | Red Oak. 3.9 | White Oak. 3.1 


Proportion of Water in various Woods. 


Alder (Betula alnus) . 41.6 

Ash (Fraxinus excelsior) . 28.7 

Beech (Fagus sylvatica) . 33 

Birch (Betula alba) . 30.8 

Elm (Ulmus campestris) . 44.5 

Horse-chestnut ( AEsculushippocast .) 38.2 

Larch (Pinus larix) . 48.6 

Mountain Ash (Sorbus aucuparia).. 28.3 
Oak (Quercus robur) . 34.7 


Willow (Salis caprea) 


Pine (Pinus Sylvestris L.) . 

Red Beech (Fagus sylvatica) . 

Red Pine (Pinus picea dur) . 

Spruce (Abies, alba, nigra, rubra , i 

excel.sa) .) 

Sycamore (Acer pseudo-platanus).. 

White Oak (Quercus alba) . 

White Pine (Pinus abies dur) . 

White Poplar (Fopulus alba) . 

.. 26 


39-7 

39-7 

45.2 

35 

2 7 

36.2 
37-i 
50.6 


Decrease in Dimensions of Timber by Seasoning. 


Woods. 

Ins. 


Ins. 

Woods. 

Ins. 


Ins. 

Cedar, Canada. 

14 

to 

l 3- 2 5 

Pitch Pine, South. 

i 8 -375 

to 

18.25 

Elm. 

. II 

to 

10.75 

Spruce . 

8-5 

to 

8-375 

Oak, English. 

. 12 

to 

11.625 

White Pine, American. 

12 

to 

11.875 

Pitch Pine, North.. 

. ioXioto 9-75X9-75 

Yellow Pine, North. 

18 

to 

17-875 


Weight of a beam of English oak, when wet, was reduced by seasoning 
from 972.25 to 630.5 lbs. 






















































































870 


WOOD AND TIMBER. 


Weiglit of a. CuDe Foot of Oak and YTellovw Pine. 


Age. 


Green., 
x Year., 
2 Years 


White Oak, Va. 

Yellow Pine, Va. 

Round. 

Square. 

Round. 

Square. 

64.7 

67.7 

47.8 

39 - 2 

53-6 

53-5 

39.8. 

34-2 

46 

49.9 

34-3 

33-5 


Live Oak. 

78.7 

66.7 


In England, Timber sawed into boards is classed as follows: 

6.5 to 7 ins. in width, Battens; 8.5 to 10 ins., Deals; and 11 to 12 ins., 
Planks. ( See also page 62.) 


Distillation. —From a single cord of pitch pine distilled by chemical ap¬ 
paratus, following substances and in quantities stated have been obtained: 


Charcoal. 50 bushels. 

Illuminating Gas_about 1000 cu. feet. 

Illuminating Oil and Tar... 50 gallons. 
Pitch or Resin. 1.5 barrels. 


Pyroligneous Acid. 100 gallons. 

Spirits of Turpentine. 20 “ 

Tar. i barrel. 

Wood Spirit. 5 gallons. 


Strengtlx of Timber. 

Results of experiments have satisfactorily proved: That deflection was 
sensibly proportional to load; That extension and compression were nearly 
the same, though former being the greater ; That., to produce equal deflection, 
load, when placed in the centre, was to a load uniformly distributed, as .638 
to 1; That deflection under equal loads is inversely as breadths and cubes 
of the depths, and directly as cubes of the spans.* (M. Morin.) 

It has also been shown, that density of wood varies very little with its age. 
That coefficient of elasticity diminishes after a certain age, and that it de¬ 
pends also on the dryness and the exposure of the ground where the wood 
is grown. Woods from a northerly exposure, on dry ground, have a high 
coefficient , while those from swamps or low moist ground have a low one. 
That tensile strength is influenced by age and exposure. The coefficient 
of elasticity of a tree cut down in full vigor, or before it arrives at this 
condition, does not present any sensible difference. That there is no limit 
of elasticity in wood, there being a permanent set for every extension. 


Average Result of Erpcriments on Tensile Strength of Wood in Various A 
Positions per Sj. Inch. ( MM. Chemndier and Werlheim.) 

With the fibre, 6900 lbs. Radially, 683 lbs., and Tangentially, 723 lbs. 


To Compute Volume of aix Irregulbr Body. 
By “Simpsons Ride." 


Operation. —Take a right line in the figure for a base line, as A B, divide the fig¬ 
ure into any number of equal parts, and compute the areas of their plane sections 
as 1, 2, 3. etc., at the points of division, by rules applicable to area of a plane. Then, 
operate these areas as if they were the ordinates of a plane curve or figure of same 
length as the figure, and result will give volume required. 

Illustration. —Assume a figure having areas as follows, and A B = 24 feet. 


Sections, 1 Areas, 3 feet Multiplier, 1 



IB 


and 84 X 24 - 4 - 4 -r- 3 = 168 cube feet. 


Products, 3 
20 

14 

3 6 

11 


84 

























MISCELLANEOUS MIXTURES. 


871 


MISCELLANEOUS MIXTURES. 

Cements. 

Much depends upon manner in which a cement is applied as upon the 
cement itself, as best cement will prove worthless if improperly applied. 
Following rules must be rigorously adhered to to attain success: 

1. Bring cement into intimate contact with surfaces to be united. This is best 
done by heating pieces to be joined in cases where cement is melted by heat, as 
with resin, shellac, marine glue, etc. Where solutions are used, cement must be 
well rubbed into surfaces, either with a brush (as in case of porcelain or glass), 
or by rubbing the two surfaces together (as in making a glue joint between pieces 
of wood). 

2. As little cement as practicable should be allowed to remain between the united 
surfaces. To secure this, cement should be as liquid as practicable (thoroughly 
melted if used with heat), and surfaces should be pressed closely into contact until 
cement has hardened. 

3. Time should be allowed for cement to dry or harden, and this is particularly 
the case in oil cements, such as copal varnish, boiled oil, white lead, etc. When 
two surfaces, each .5 inch across, are joined by means of a layer of white lead 
placed between them, 6 months may elapse before cement in middle of joint be¬ 
comes hard. At the end of a month the joint will be wea-k and easily separated; at 
end of 2 or 3 years it may be so firm that the material will part anywhere else than 
at joint. Hence, when article is to be used immediately, the only safe cements 
are those which are liquefied by heat and which become hard when cold. A joint 
made with marine glue is firm an hour after it has been made. Next to cements 
that are liquefied by heat are those which consist of substances dissolved iu water 
or alcohol. A glue joint sets firmly in 24 hours; a joint made with shellac varnish 
becomes dry in 2 or 3 days. Oil cements, which do not dry by evaporation, but 
harden by oxidation (boiled ofl, white lead, red lead, etc.) are slowest of all. 

Stone. —Resin, Yellow Wax, and Venetian Red, each 1 oz.; melt and mix. 

Aquarium. 

Litharge, fine white dry Sand, and Plaster of Paris, each 1 gill; finely pulverized 
Resin, .33 gill. 

Mix thoroughly and make into a paste with boiled linseed oil to which drier has been added. Beat 
well, and let stand 4 or 5 hours before using it. After it has stood for 15 hours, however, it loses its 
strength. Glass cemented into a frame with this cement will resist percolation for either salt or fresh 
water. 

.A-dliesive for Fractures of all IYinds. 

White Lead ground with Linseed-oil Varnish, and kept from contact with the air. 

Requires a few weeks to harden. 

Stone or Iron. 

Compound equal parts of Sulphur and Pitch. 

Brass to Grlass. 

Electrical. —Resin, 5 ozs.; Beeswax, 1 oz.; Red Ochre or Venetian Red, in pow¬ 
der, 1 oz. Dry earth thoroughly on a stove at above 212 0 . Melt Wax and Resin 
together and stir in powder by degrees. Stir until cold, lest earthy matter settle 
to bottom. 

Used for fastening brass-work to glass tubes, flasks, etc. 

Chinese Waterproof. 

Schio-liao. —To 3 parts of Fresh Beaten Blood add 4 parts of Slaked Lime and a 
little Alum; a thin, pasty mass is produced, which can be used immediately. 

Materials which are to be made specially waterproof are painted twice, or at most three times. 
Wooden public buildings of China are painted with schio-liao , which gives them an unpleasant red¬ 
dish appearance, but adds to their durability. Pasteboard treated with it receives appearance and 
strength of wood. 

Oliina. 

Curd of milk, dried and powdered, ioozs.; Quicklime, i oz.; Camphor, 2 drachms. 

Mix, and keep air-tight. When used, a portion is to be mixed with a little water into a paste. 

Cisterns ancl Water-casks. 

Melted Glue, 8 parts; Linseed oil, boiled into a varnish with Litharge, 4 parts. 

This cement hardens in about 48 hours, and renders the joints of wooden cisterns and casks air and 
water tight. 


MISCELLANEOUS MIXTURES. 


872 ' 


Cloth, or Leather. 

Shellac, 1 part; Pitch, 2 parts; India Rubber, 4 parts; and Gutta Percha, 10 
parts; cut small; Linseed oil, 2 parts; melted together and mixed. 


Earthen and. Gflass Ware. 

Heat article to be mended a little above 212 0 , then apply a thin coating of gum 
Shellac upon both surfaces of broken vessel. 

Or, dissolve gum Shellac in alcohol, apply solution, and bind the parts firmly to¬ 
gether until cement is dry. 

Or, dilute white of egg with its bulk of water and beat up thoroughly. Mix to 
consistence of thin paste with powdered Quicklime. 

Use immediately. 

Entomologists’. 

Thick Mastic Varnish and Isinglass size, equal parts. 

Gutta Percha. 

Melt'together, in an iron pan, 2 parts Common Pitch and 1 part Gutta Percha. 

Stir well together until thoroughly incorporated, and then pour liquid into cold water. When cold 
it is black, solid, and elastic ; but it softens with heat, and at ioo° is a thin fluid. It may be used as a 
soft paste, or in liquid state, and answers an excellent purpose in cementing metal, glass, porcelain, 
ivory, etc. It may be used instead of putty for glazing. 


Grlass. 

SorePs .—Mix commercial Zinc White with half its bulk of fine Sand, add a solu¬ 
tion of Chloride of Zinc of 1.26 spec, grav., and mix thoroughly in a mortar. 

Apply immediately, as it hardens very quickly. 


Holes in Castings. 

Sulphur in powder, 1 part; Sal-ammoniac, 2 parts; powdered Iron turnings, 80 
parts. Make into a thick paste. 

Make only as required for immediate use. • 


Hydranlic Paint. 

Hydraulic cement mixed with oil forms an incombustible and waterproof paint 
for roofs of buildings, outhouses, walls, etc. 


Iron Ware. 

Sulphur, 2 parts; fine Black-lead. 1 part. Heat sulphur in an iron pan until 
it melts, then add the lead; stir well, and remove. When cool, break into pieces 
as required. Place upon opening of the ware to be mended, and solder with an 
iron. 

Kerosene Lamps, etc. 

Resin, 3 parts; Caustic Soda, 1; Water, 5, mixed with half its weight of Plaster 
of Paris. 

It, sets firmly in about three quarters of an honr. Is of great adhesive power, not permeable to kero¬ 
sene, a low conductor of heat, and but superficially attacked by hot water. 

Ijeatlier to Iron, Steel, or Gflass. 

1. —Glue, 1 quart, dissolved in Cider Vinegar; Venice Turpentine, 1 oz.; boil very 
gently or simmer for 12 hours. 

Or, Glue and Isinglass equal parts, soak in water 10 hours, boil and add tannin 
until mixture becomes “ropy;” apply warm. 

Remove surface of leather where it is to be applied. 

2. —Steep leather in an infusion of Nutgall, spread a layer of hot Glue on sur¬ 
face of metal, and apply flesh side of leather under pressure. 

Leather Eelting. 

Common Glue and Isinglass, equal parts, soaked for 10 hours in enough water to 
cover them. Bring gradually to a boiling heat and add pure Tannin until whole be¬ 
comes ropy or appears alike to white of eggs. 

Clean and rub surfaces to be joined, apply warm, and clamp firmly. 

HVIolding and Temporary A-dliesion. 

Soft .—Melt Yellow Beeswax with its weight of Turpentine, and color with finely 
powdered Venetian red. 

When cold it has the hardness of soap, but is easily softened and molded with the fingers. 


MISCELLANEOUS MIXTURES. 


873 


NEaltLa, or Greek NEastic. 

Lime and Sand mixed in manner of mortar, and made into a proper consistency 
with milk or size without water. 

jVLarble. 

Plaster of Paris, in a saturated solution of Alum, baked in an oven, and reduced 
to powder. Mixed with water, and color if required. 

NEetal to Glass. 

Copal Varnish, 15 parts; Drying Oil, 5; Turpentine, 3. Melt in a water bath and 
add 10 of Slaked Lime. 

IVE ending Sliells, etc. 

Gum Arabic, 5 parts; Rock Candy, 2; and White Lead, enough to color. 

Large Objects. 

Wollaston ' 1 s White. —Beeswax, 1 oz.; Resin, 4 ozs.; powdered Plaster of Paris, 5 
oz. Melt together. 

Warm the edges of the object and apply warm. 

By means of this cement a piece of wood may be fastened to a chuck, which will hold when cool ; and 
when work is finished it may be removed by a smart stroke with tool. Any traces of cement nmy be 
removed by Benzine. 

NEarble Workers and Coppersmiths. 

White of egg, mixed with finely-sifted Quicklime, will unite objects which are 
not submitted to moisture. 

Porcelain. 

Add Plaster of Paris to a strong solution of Alum till mixture is of consistency 
of cream. 

It sets readily, and is suited for cases in which large rather than small surfaces are to be united. 

Rust Joint. 

(Quick Setting.) — Sal-ammoniac in powder, 1 lb.; Flour of Sulphur, 2 lbs.; Iron 
borings, 80 lbs. Made to a paste with water. 

(Slow Setting.) —Sal-ammoniac, 2 lbs.; Sulphur, 1 lb.; Iron borings, 200 lbs. 

The latter cement is best if joint is not required for immediate use. 

Steam Boilers, Steam-pipes, etc. 

Finely powdered Litharge, 2 parts; very fine Sand, 1; and Quicklime slaked by 
exposure to air, 1. 

This mixture may be kept for any length of time without injuring. In using it, a portion is mixed 
into paste with linseed oil, boiled or crude. Apply quickly, as it soon becomes hard. 

Soft. —Red or White Lead in oil, 4 parts; Iron borings, 2 to 3 parts. 

Hard .—Iron borings and salt water, and a small quantity of Sal-ammoniac with 
fresh water. 

Transparent—Glass. 

India-rubber, 1 part in 64 of chloroform; gum Mastic in powder, 16 to 24 parts. 
Digest for two days, with frequent shaking. 

Or, pulverized Glass, 10 parts; powdered Fluor-spar, 20; soluble Silicate of Soda, 
60. Both glass and fluor-spar must be in finest practicable condition, which is best 
done by shaking each in fine powder, w T itli water, allowing coarser particles to de¬ 
posit, and then by pouring off remainder, which holds finest particles in suspension. 

The mixture must be made very rapidly, by quick stirring, and applied immediately. 

Uniting Beatlier and. NIetal. 

Wash metal with hot Gelatine; steep leather in an infusion of Nutgalls, hot, 
and bring the two together. 

Waterproof IVEastic. 

Red Lead, 1 part; ground Lime, 4 parts; sharp Sand and boiled Oil, 5 parts. 

Or, Red Lead, 1 part; Whiting, 5; and sharp Sand and boiled Oil, 10. 

Wood to Iron. 

Litharge and Glycerine.— Finely powdered Oxide of Lead (litharge) and Concen¬ 
trated Glycerine. 

The composition is insoluble in most acids, is unaffected by action of moderate heat, sets rapidly, 
and acquires an extraordinary hardness. 

Turner's .—Melt 1 lb. of Resin, and add .25 lb. of Pitch. 

While boiling add Brick dust to give required consistency. In winter it may be 
necessary to add a little Tallow. 


874 


MISCELLANEOUS MIXTURES. 


GLUES. 

Marine. 

Dissolve India Rubber, 4 parts, in 34 parts of Coal-tar Naphtha; add powdered 
Shellac, 64 parts. 

While mixture is hot pour it upon metal plates in sheets. When required for 
use, heat it, and apply with a brush. 

Or, India Rubber, 1 part; Coal Tar, 12 parts; heat gently, mix, and add powdered 
Shellac, 20 parts. Cool. When used, heat to about 250 0 

Or, Glue, 12 parts; Water, sufficient to dissolve; add Yellow Resin, 3 parts; and, 
when melted, add Turpentine, 4 parts. 

Strong Glue. —Add Powdered Chalk to common Glue. 

Mix thoroughly. 

Mucilage. 

Curd of Skim Milk (carefully freed from Cream or Oil), washed thoroughly, and 
dissolved to saturation in a cold concentrated solution of Borax. 

This mucilage keeps well, and, as regards adhesive power, far surpasses gtun Arabic. 

Or, Oxide of Lead, 4 lbs.; Lamp-black, 2 lbs.; Sulphur, 5 ozs.; and India Rubber 
dissolved in Turpentine, 10 lbs. 

Boil together until they are thoroughly combined. 

Preservation of Mucilage. —A small quantity of Oil of Cloves poured into a bottle 
containing Gum Mucilage prevents it from becoming sour. 

To Resist NToistnre. 

Glue, 15 parts; Resin, 4 parts; Red Ochre, 2 parts; mixed with least practicable 
quantity of water. 

Or, Glue, 4 parts; Boiled Oil, 1 part, by weight, Oxide of Iron, 1 part. 

Or, Glue, 1 lb., melted in 2 quarts of skimmed Milk. 

Parchm ent. 

Parchment Shavings, 1 lb.; Water, 6 quarts. 

Boil until dissolved, then strain and evaporate slowly to proper consistence. 

Rice, or Japanese. 

Rice Flour; Water, sufficient quantity. 

Mix together cold, then boil, stirring it during the time. 

Eiqnid. 

Glue, Water, and Vinegar, each 2 parts. Dissolve in a water-bath, then add Al¬ 
cohol, 1 part. 

Or, Cologne or strong Glue, 2.2 lbs.; Water, 1 quart; dissolve over a gentle heat; 
add Nitric Acid 36°, 7 ozs., in small quantities. 

Remove from over tire, and cool. 

Or, White Glue, 16 ozs.; White Lead, dry, 4 ozs.; Rain Whiter, 2 pints. Add Al¬ 
cohol, 4 ozs., and continue heat for a few minutes. 

Elastic and. Sweet.—Stamps or Rolls. 

Elastic. —Dissolve good Glue in water by a water-bath. Evaporate to a thick con¬ 
sistence, and add equal weight of Glycerine to Glue; submit to heat until all water 
is evaporated, and pour into molds or on plates. 

Sweet. —Substitute Sugar for the Glycerine. 

To Adhere Engravings or Eitliograplis -upon Wood. 

Sandarach, 250 parts; Mastic in tears, 64 parts; Resin, 125 parts; Venice Tur¬ 
pentine, 250 parts; and Alcohol, 1000 parts by measure. 

BROWNING, OR BRONZING, LIQUID. 

Sulphate of Copper, 1 oz.; Sweet Spirit of Nitre, 1 oz.; Water, 1 pint. 

Mix. Let stand a few days before use. 


MISCELLANEOUS MIXTURES. 


875 


Grviii Barrels. 

Tincture of Muriate of Iron, i oz.; Nitric Ether, i oz.; Sulphate of Copper, 4 
scruples; rain water, 1 pint. If the process is to be hurried, add 2 or 3 grains of 
Oxymuriate of Mercury. 

When barrel is finished, let it remain a short time in lime-water, to neutralize any acid which may 
have penetrated , then rub it well with an iron wire scratch-brush. 

After Browning. —Shellac, 1 oz.; Dragon’s-blood, .25 oz.; rectified Spirit, 1 qt. 
Dissolve and filter. 

Or, Nitric Acid, spec. gray. 1.2; Nitric Ether, Alcohol, and Muriate of Iron, each 1 
part. Mix, then add Sulphate of Copper 2 parts, dissolved in Water 10 parts. 

LACQUERS. 

Small Arms, or Waterproof Paper. 

Beeswax, 13 lbs.; Spirits Turpentine, 13 gallons; Boiled Linseed Oil, 1 gallon. 

All ingredients should be pure and of best quality. Heat them together in a copper or earthen ves¬ 
sel over a gentle fire, in a water-bath, until they are well mixed. 

Briglit Iron. Work. 

Linseed Oil, boiled, 80.5 parts; Litharge, 5.5 parts; White Lead, in oil, 11.25 parts; 
Resin, pulverized, 2.75 parts. 

Add litharge to oil; simmer over a slow fire 3 hours ; strain, and add resin and white lead , keep it 
gently warmed, and stir until resin is dissolved. 

Or, Amber, 6 parts; Turpentine, 6 parts; Resin, 1 part; Asphaltum, 1 part; and 
Drying Oil, 3 parts; heat and mix well. 

Or, Shellac, 1 lb.; Asphaltum, 6 lbs.; and Turpentine, 1 gallon. 

Iron and Steel. 

Clear Mastic, 10 parts; Camphor, 5 parts; Sandarac, 15 parts; and Elitni Gum, 
5 parts. Dissolve in Alcohol, filter, and apply cold. 

Brass. 

Shellac, 8 ozs.; Sandarac, 2 ozs.; Annatto, 2 ozs.; and Dragon’s-blood Resin, .25 
oz.; and Alcohol, 1 gallon. 

Or, Shellac, 8 ozs.; and Alcohol, 1 gallon. Heat article slightly, and apply lacquer 
with a soft brush. 

"Wood, Iron, or Walls, and rendering Clotli, Baper, etc., 

W ate rp roof. 

Heat 120 lbs. Oil Varnish in one vessel, 33 lbs. Quicklime in 22 lbs. water in an¬ 
other Soon as lime effervesces, add 55 lbs. melted India Rubber. Stir mixture, 
and pour into vessel of hot Varnish. Stir, strain, and cool. 

When used, thin with Varnish and apply, preferably hot. 

To Clean Soiled Engravings. 

Ozone Bleach, 1 part; Water, 10; well mixed. 

INKS. 

Indelible, for Marking Linen, etc. 

1. —Juice of Sloes, 1 pint; Gum, .5 oz. 

This requires no “ preparation ” or mordant, and is very durable. 

2. —Nitrate of Silver, 1 part; Water, 6 parts, Gum, 1 part; Dissolve. 

3. —Lunar Caustic, 2 parts; Sap Green and Gum Arabic, each 1 part; dissolve with 
distilled water. 

“ Preparation. ’’—Soda, 1 oz.; Water, 1 pint; Sap Green, .5 drachm. Dissolve, 
and wet article to be marked, then dry and apply the ink. 

Perpetual, for Tombstones , Marble, etc. — Pitch, n parts; Lamp-black, 1 part; 
Turpentine sufficient. Warm and mix. 

Copying Inlc.— Add 1 oz. Sugar to a pint of ordinary Ink. 

• SOLDERING. 

Base for Soldering. 

Strips of Zinc in diluted Muriatic, Nitric, or Sulphuric Acid, until as much is de¬ 
composed as acid will effect. Add Mercury, let it stand for a day; pour off the 
Water, and bottle the Mercury. 

When required, rub surface to be soldered with a cloth dipped in the Mercury. 


876 


MISCELLANEOUS MIXTURES. 


VARNISHES. 

W aterproof. 

Flour of Sulphur, 1 lb.; Linseed Oil, 1 gall.; boil them until they are thoroughly 
combined. 

Good for waterproof textile fabrics. 

Harness. 

India Rubber, .5 lb.; Spirits of Turpentine, 1 gall.; dissolve into a jelly; then mix 
hot Linseed Oil, equal parts with the mass, and incorporate them well over a slow' fire. 

Fastening Beatker 01a Top Rollers. 

Gum Arabic, 2.75 ozs., and a like volume of Isinglass, dissolved in Water. 

To Preserve GJTass from tlae Sun. 

Reduce a quantity of Gum Tragacanth to fine powder, and dissolve it for 24 hours 
in white of egg well beat up. 

Water-color Drawings. 

Canada Balsam, 1 part; Oil of Turpentine, 2 parts. 

Mix and size drawing before applying. 

Objects of Natural History, Sliells, Fisk, etc. 
Mucilage of Gum Tragacanth and of Gum Arabic, each 1 oz. 

Mix, and add spirit with Corrosive Sublimate, to precipitate the more stringy por¬ 
tion of the Gum. 

Iron and Steel. 

Mercury, 120 parts; Tin, 10 parts; Green Vitriol, 20 parts; Hydrochloric Acid of 
1.2 sp. gr., 15 parts, and pure Water, 120 parts. 

Blackboards. 

Shellac Varnish, 5 gallons; Lamp-black, 5 ozs.; fine Emery, 3 ozs.; thin with 
Alcohol, and lay in 3 coats. 

Black. 

Heat, to boiling, Linseed Oil Varnish, 10 parts, with Burnt,Umber, 2 parts, and 
powdered Asphaltum, 1 part. 

When cooled, dilute with Spirits of Turpentine as may be required. 

Balloon. 

Melt India Rubber in small pieces w T ith its weight of boiled Linseed Oil. 

Thin with Oil of Turpentine. 

Transfer. 

Alcohol, 5 ozs.; pure Venice Turpentine, 4 ozs.; Mastic, 1 oz. 

To render Canvas Waterproof and Pliable. 

Yellow Soap, 1 lb , boiled in 6 pints of Water, add, w r hile hot, to 112 lbs. of oil Paint. 

"Waterproof Bags. 

Pitch, 8 parts, Wax and Tallow, each 1 part. 

To Clean Varnish. 

Mix a lye of Potash or Soda, with a little powdered Chalk. 

STAINING. 

Wood and Ivory. 

Yellow .—Dilute Nitric Acid will produce it on wood. 

Red .—An infusion of Brazil Wood in Stale Urine, in the proportion of 1 lb. to a 
gallon, for wood, to be laid on when boiling hot, also Alum water before it dries. 
Or, a solution of Dragon’s-blood in Spirits of Wine. 

Black .—Strong solution of Nitric Acid. 

Blue .—For Ivory: soak it in a solution of Verdigris in Nitric Acid, which will turn 
it green; then dip it into a solution of Pearlash boiling hot. 

P«?2>Ze.—Soak Ivory in a solution of Sal-ammoniac into four times its weight of 
Nitrous Acid. 

jlahogany— Brazil. Madder, and Logwood, dissolved in water and put on hot. 


MISCELLANEOUS MIXTURES. 


87; 


MISCELLANEOUS. 

Blacking for Harness. 

Beeswax, .5 lb. ; Ivory Black, 2 ozs.; Spirits of Turpentine, 1 oz.; Prussian Blue 
ground in oil, 1 oz.; Copal Varnish, .25 oz. 

Melt wax and stir it into other ingredients before mixture is quite cold; make it 
into balls. Rub a little upon a brush, and apply it upon harness, then polish lightly 
with silk. 

To Clean Brass Ornaments. 

Brass ornaments that have not been gilt or lackered may be cleaned, and a very 
brilliant color given to them, by washing them in Alum boiled in strong Lye, in the 
proportion of an ounce to a pint, and afterwards rubbing them with strong Tripoli. 

To Harden Brills, Ckisels, etc. 

Temper them in Mercury. 

To Clean Coral. 

Brush with equal parts Spirits of Salts and cold water. 

Or, dip in a hot solution of Potash or Chloride of Lime. If much discolored, let 
it remain in solution for a few hours. 

Blacking, without Bolislring. 

Molasses, 4 ozs.; Lamp-black, .5 oz.; Yeast, a table-spoonful; Eggs, 2; Olive Oil, 
a teaspoonful; Turpentine, a teaspoonful. Mix well. 

To be applied with a sponge, without brushing. 

Dubbing. 

Resin, 2 lbs.; Tallow, 1 lb.; Train-oil, 1 gallon. 

Anti-friction Grease. 

Tallow, 100 lbs.; Palm-oil, 70 lbs. Boiled together, and when cooled to 8o°, strain 
through a sieve, and mix with 28 lbs. of Soda, and 1.5 gallons of Water. 

For Winter, take 25 lbs. more oil in place of the Tallow. 

Or, Black Lead, 1 part; Lard, 4 parts. 

To Attack Bair Belt to Boilers. 

Red Lead, 1 lb.; White Lead, 3 lbs.; and Whiting, 8 lbs. Mixed with boiled Lin¬ 
seed Oil to consistency of paint. 

Bas tils for Fumigating. 

Gum Arabic, 2 ozs.; Charcoal Powder, 5 ozs.; Cascarilla Bark, powdered, .75 oz.; 
Saltpetre, .25 drachin. Mix together with water, and make into shape. 

Bor "Writing upon Zinc Labels. — Borticultural. 

Dissolve 100 grains of Chloride of Platinum in a pint of water; add a little Mu¬ 
cilage and Lamp-black. 

Or, Sal-ammoniac, 1 dr.; Verdigris, 1 dr.; Lamp-black, .5 dr.; Water, 10 drs. Mix. 

To Ltemove old. Ironmold. 

Remoisten part stained with ink, remove this by use of Muriatic Acid diluted by 
5 or 6 times its weight of water, when old and new stain will be removed. 

To Cut India Rubber. 

Keep blade of knife wet with water or a strong solution of Potash. 

Adhesive for Rubber Belts. 

Coat driving surface with Boiled Oil or Cold Tallow r , and then apply powdered 
Chalk. 

Biard. 

50 parts of finest Rape-oil, and 1 part of Caoutchouc, cut small. Apply heat until 
it is nearly all dissolved. 

To Preserve Leatker Belting or Bose. 

Apply warm Castor Oil. For hose, force it through it. 

To Oil Leatker Belting. 

Apply a solution of India Rubber and Linseed Oil. 

4 E 


8/8 


MISCELLANEOUS MIXTURES. 


Dressing for Leather Belts. 

1. —Beef Tallow, i part, and Castor Oil, 2 parts. Apply warm. 

2. —Beef Tallow, 3 lbs.; Beeswax, 1 lb. Heated and applied warm to both sides. 

Files. 

Lay dull files in diluted Sulphuric Acid until they are bitten deep enough. 

To Remove Oil from Leather. 

Apply Aqua-ammonia. 

To Clean Baint. 

Wash with a solution of Pearlash in water. If greasy, use Quicklime. 

Or, Extract of Litherium diluted with from 200 to 300 parts of water. 

To Remove Faint. 

Mix Soft Soap, 2 ozs., and Potash, 4 ozs., in boiling Water, with Quicklime, .5 lb. 
Apply hot, and let remain for 1 day. 

Or, Extract of Litherium, thinly brushed over the surface 2 or 3 times. 

To Clean UVEarhle. 

Chalk, powdered, and Pumice-stone, each 1 part; Soda, 2 parts. Mix with water. 

Wash the spots, then clean and wash off with Soap and Water. 

Paste for Cleaning NEetals. 

Oxalic Acid, 1 part; Rottenstone, 6 parts. Mix with equal parts of Train Oil and 
Spirits of Turpentine. 

Watchmaker’s Oil, which never Corrodes or Thickens. 

Place coils of thin Sheet Lead in a bottle with Olive Oil. Expose it to the sun for 
a few weeks, and pour otf the clear oil. 

Durable Paste. 

Make common Flour paste rather thick (by mixing some Flour with a little cold 
water until it is of uniform consistency, and then stir it well while boiling water is 
being added to it); add a little Brown Sugar and Corrosive Sublimate, which will 
prevent fermentation, and a few drops of Oil of Lavender, which will prevent it be¬ 
coming moldy. When dried, dissolve in water. 

It will keep for two or three years in a covered vessel. 

To Extract Grease from Stone or Alarble. 

Soft Soap, 1 part; Fuller’s Earth, 2 parts; Potash, 1 part. Mix with boiling water. 

Lay it upon the spots, and let it stand for a few hours. 

Stains. 

To Remove.— Stains of Iodine are removed by rectified Spirit; Ink stains by Ox¬ 
alic or Superoxalate of Potash; Ironmolds by same; but if obstinate, moisten them 
with Ink, then remove them in the usual way. 

Red spots upon black cloth, from acids, are removed by Spirits of Hartshorn, or 
other solutions of Ammonia. 

Stains of Marking-ink, or Nitrate of Silver. —Wet stain with fresh solution of 
Chloride of Lime, and, after 10 or 15 minutes, if marks have become white, dip the 
pai’t in solution of Ammonia or of Hyposulphite of Soda. In a few minutes wash 
with clean water. 

Or, stretch the stained linen over a basin of hot water, and wet mark with Tinc¬ 
ture of Iodine. 

Preservative Paste for O Ejects of NTat-ural History. 

White Arsenic, 1 lb.; Powdered Hellebore, 2 lbs. 

To Preserve Bottoms of Iron. Steam-boilers. 

Red Lead, 75 parts; Venetian Red, 17 parts; Whiting, 6.5 parts; and Litharge. 
1.5 parts by weight. 

To Preserve Sails. 

Slacked Lime, 2 bushels. Draw off the lime-water, and mix it with 120 gallons 
water, and with Blue Vitriol, .25 lb. 


MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 879 


Wliitewasli. 

For outside exposure, slack Lime, .5 bushel, iu a barrel; add common Salt, 1 lb.; 
Sulphate of Zinc, .5 lb.; and Sweet Milk, 1 gallon. 

To Preserve Woodwork. 

Boiled Oil and finely powdered Charcoal, each 1 part; mix to the consistence of 
paint. Apply 2 or 3 coats. 

This composition is well adapted for casks, water-spouts, etc. 

To dPolish. "Wood. 

Rub surface with Pumice Stone and water until the rising of the grain is removed. 
Then, with powdered Tripoli and boiled Linseed Oil, polish to a bright surface. 

3 ?aint for Window Gflass. 

Chrome Green, .23 oz.; Sugar of Lead, 1 lb.; ground fine, in sufficient Linseed Oil 
to moisten it. Mix to the consistency of cream, and apply with a soft brush. 

The glass should be well cleansed before the paint is applied. The above quantity is sufficient for 
about 200 feet of glass. 

To Nlalxe Drain Tiles Porous. 

Mix sawdust with the clay before burning. 


MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 

1.—it is required to lay out a tract of land in form of a square, to be en¬ 
closed with a post and rail fence, 5 rails high, and each rod of fence to con¬ 
tain 10 rails. What must be side of this square to contain just as many 
acres as there are rails in fence ? 

Operation, i mile = 320 rods. Then 320 X 320 -f- 160, sq. rods in an acre = 640 
acres ; and 320 X 4 sides and X 10 rails = 12 800 rails per mile. 

Then as 640 acres : 12800 rails :: 12800 acres : 256000 rails , which will enclose 
256 ooo acres, and 1/256000X 69.5701 = number of yards in side of a sq. acre, and 
-4- 1760, yards in a mile — 20 miles. 


2 .—How many fifteens can be counted with four fives? 

4X3X2X1 24 


Operation. 


1X2X3 


— 4 - 


3._What are the chances in favor of throwing one point with three dice ? 

Operation.— Assume a bet to be upon the ace. Then there will be 6 X 6 X 6 = 216 
different icays which the dice may present themselves , that is, with and without an ace. 

Then, if the ace side of the die is excluded, there will be 5 sides left, and 5X5X5 
= 125 ways without the ace. 

Therefore, there will remain only 216 —125 = 91 ways in which there could be an 
ace. The chance, then, in favor of the ace is as 91 to 125; that is, out of 216 throws, 
the probability is that it will come up 91 times, and lose 125 times. 


4,—The hour and minute hand of a clock are exactly together at 12; 
when are they next together ? 

Operation. —As the minute hand runs u times faster than the hour hand, then, 


as 11 : 60 :: 1 : 5 min. 27-^y sec. = time past 1 o'clock. 


r —Assume a cube inch of glass to weigh 1.49 ounces troy, the same of 
sea-water .59, and of brandy .53. A gallon of this liquor in a glass bottle, 
which weighs 3.84 lbs., is thrown into sea-water. It is proposed to deter¬ 
mine if it will sink, anil, if so, how much force will just buoy it up . 


30.92 cube ins. of glass in bottle. 
- 122.43 ounces of brandy. 


Operation. 3.84 x 12 = i -49 = 

231 cube ins. in a gallon X -53 : 

Then, bottle and brandy weigh 3.84 X 12 + 122.43 = l68 ' 5 i ounces, and contain 
261.92 cube ins., which X -59 = 154-53 ounces, weight of an equal bulk ofsea-watei. 
And, 168.51_154-53 : 13 98 ounces, weight necessary to support it in the water. 





880 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 


6.—A fountain has 4 supply cocks, A, B, C, and D, and under it is a cis¬ 
tern, which can be tilled by the cock A in 6 hours, by B in 8 hours, by C in 
10 hours, and by D in 12 hours; now, the cistern lias 4 holes, designated E, 
F, G, and II, and it can be emptied through E in 6 hours, F in 5 hours, G in 
4 hours, and H in 3 hours. Suppose the cistern to be full of water, and that 
all the cocks and holes were opened together, in what time would the cistern 
be emptied? 

Operation.— Assume the cistern to hold 120 gallons. 


hrs. gall. hrs. gall. 

If 6 : 120 11 1 : 20 at A. 

8 : 120 1 : 15 at B. 

10 : 120 ;ii : 12 at C. 

12 : 120 II 1 : 10 tt( D. 

Run in in 1 hour, 57 gallons. 


hrs. 

gall. 

brs. 

gall. 

If 6 : 

120 ; 

!i 1 : 

: 20 at E. 

5 : 

120 : 

\\ 1 : 

: 24 at F. 

4 •' 

120 : 

Hi : 

: 30 at G. 

3 : 

120 

Hi : 

: 40 at H. 


Run out in 1 hour, 114 gallons. 

57 


Run out in 1 hour more than run in, 57 gallons. 
Then, as 57 gallons : 1 hour :: 120 gallons : 2.158-)- hours. 


7. —A cistern, containing 60 gallons of water, has 3 cocks for discharging 
it; one will empty it in 1 hour, a second in 2 hours, and a third in 3 hours; 
in what time will it be emptied if they are all opened together? 

Operation.— 1st, .5 would run out iii 1 hour by the 2d cock, and .333 by the 3d; 
consequently, by the 3 would the reservoir be emptied in 1 hour. .5 -)- .333 + 1 — 
-f- | -j- jr , being reduced to a common denominator, the sum of these 3 = LA; whence 
the proportion, n : 60 II 6 : 32^ minutes. 

8. —A reservoir has 2 cocks, through which it is supplied; by one of them 
it will till in 40 minutes, and by the other in 50 minutes; it has also a dis¬ 
charging cock, by which, when full, it may be emptied in 25 minutes. If 
the 3 cocks are left open, in ivhat time would the cistern be tilled, assuming 
the velocity of the water to be uniform ? 


Operation. —The least common multiple of 40, 50, and 25, is 200. 

Then, the 1st cock will fill it 5 times in 200 minutes , and the 2d, 4 times in 200 
minutes, or both, 9 times in 200 minutes ; and, as the discharge cock will empty it 
8 times in 200 minutes, ' hence 9 — 8 = 1, or once in 200 minutes = 3.2 hours. 

9.—The time of the day is between 4 and 5, and the hour and minute 
hands are exactly together; what is the time ? 

Operation. —Difference of speed of the hands is as 1 to 12 = n. 

4 hours X 60 = 240, which -1-11 = 21 min. 49.09 sec., which is to be added to 4 hours. 


10.—Out of a pipe of wine containing 84 gallons, 10 were drawn off, and 
the vessel refilled with water, after which 10 gallons of the mixture were 
drawn off, and then 10 more of water were poured in, and so on for a third 
and fourth time. It is required to compute how much pure wine remained 
in the vessel, supposing the two fluids to have been thoroughly mixed. 

Operation. 84 —10 = 74, quantity after the 1 st draught. 

Then, 84 : 10 II 74 : 8.8095, and 74 — 8.8095 = 65.1905, quantity after 2d draught. 

84:101165.1905: 7.7608, and 65.1905 — 7.7608 = 57.4297, quantity after 3 d draught. 

84:10:157.4297:6.8367, and 57.4297—6.8367 = 50-593) quantity after 4th draught , 
= result required. 


11.—A reservoir having a capacity of 10000 cube feet, has an influx of 
750 and a discharge of 1000 cube feet per day. In what time will it be 

emptied? IOO oo 

Operation. - - = 40 days. 

1000 — 750 

Contrariwise: The discharge being 1000 and the influx 1250 cube feet per hour. 
In what time will it be filled? 


10000 


Operation. 


1250 —1000 


40 hours = 1 day 16 hours. 






MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 88 I 


12. —A son asked his father how old he was. His father answered liim 
thus: If you take away.5 from my years, and divide the remainder by 8, 
the quotient will be one third of your age ; but if you add 2 to your age, and 
multiply the whole by 3, and then subtract 7 from the product, you will have 
the number of years of my age. What were the ages of father and son? 

Operation. —Assume father’s age 37. 

Then 37 — 5 = 32, and 32 - 4 - 8 = 4, and 4X3 = 12, son's age. Again: 12 + 2 = 14, 
and 14 X 3 = 42, and 42 — 7 = 35. Therefore 37 — 35 = 2, error too little. 

Again: Assume father’s age 45; then 45 — 5 = 40, and 40 4- 8 = 5. Therefore 
5 X 3 — 15, son's age. Again: 15 + 2 = 17, and 17 x 3 = 5 L and 51 — 7 = 44. There¬ 
fore 45 — 44 = 1, error too little. 

Hence (45 sup. X 2 error) — (37 sup. X 1 error) = 90 — 37 = 53, and 2 — 1 = 1. 

Consequently, 53 is father's age. Then 53 — 5=148, and 484-8 = 6 = .333 of son's 
age, and 6X3 = 18 years , son's age. 

13. —Two companions have a parcel of guineas. Said A to B, if you will 
give me one of your guineas I shall have as many as you have left. B re¬ 
plied, if you will give me one of your guineas I shall have twice as many as 
you will have left. How many guineas had each of them ? 

Operation. —Assume B had 6. 

Then A would have had 4, for 6 — 1 = 4 -f-1 = 5. Again: 4 (A’s parcel) —1 = 3, 
and 6 -j- x = 7, and 3X2 = 6. Therefore 7 — 6 = 1, error too little. 

Again: Assume B had 8. 

Then A would have 6, for 8 — 1 = 6 -|- 1 = 7. Again: 6 (A’s parcel) — 1 = 5, and 
8 -f- x = 9, and 5 X 2 = 10. Therefore 10 — 9 = 1, error too great. 

Hence 8X1 = 8, and 6X1 = 6. Then 8 + 6 = 14, and 1 + 1 = 2. Whence, di¬ 
viding products by sum of errors, 144-2 = 7 = B’s ’parcel, and 7 — 1=5+1=6 

for A when he had received 1 o/B; also 5 — 1 X 2 = 7 + x = 8 = B’s parcel when he 
had received 1 of A. 

14. —If a traveller leaves New York at 8 o’clock in the morning, and walks 
towards New London at the rate of 3 miles per hour, without intermission; 
and another traveller starts from New London at 4 o’clock in the evening, 
and walks towards New York at the rate of 4 miles per hour continuously ; 
assuming distance between the two cities to be 130 miles, whereabouts upon 
the road will they meet? 

Operation. — From 8 to 4 o’clock is 8 hours; therefore, 8 X 3 = 24 miles, per¬ 
formed by A before B set out from New London ; and, consequently, 130 — 24 = 106 
are the miles to be travelled between them after that. 

Hence, as (3 + 4) 7 : 3 :: 106 : AT 8 _ ^3. more m n es travelled by A at the meeting; 
consequently, 24 + 457- = 6 g^ miles from New York is place of their meeting. 

15. —If from a cask of wine a tenth part is drawn out and then it is filled 
with-water; after which a tenth part of the mixture is drawn out; again 
is filled, and again a tenth part of the mixture is drawn out: now, assume 
the fluids to mix uniformly at each time the cask is replenished, what frac¬ 
tional part of wine will remain after the process of drawing out and replen¬ 
ishing has been repeated four times? 

Operation. —Since .1 of the wine is drawn out at first drawing, there must remain 
.9. After cask is tilled with water, .1 of whole being drawn out, there will remain 
.9 of mixture ; but .9 of this mixture is wine ; therefore, after second drawing, there 

will remain .9 0/. 9 of wine, or ; and after third drawing, there will remain .9 
9 s 

of .9 Of .9 of wine, or —3. 

Hence, the part of wine remaining is expressed by the ratio .9, raised to a power 
exponent of which is number of times cask has been drawn from. 

q4- 

Therefore, fractional part of wine is — - = .6561. 

4°E* 



882 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 


16. —There is a fish, the head of which is 9 ins. long, the tail as long as 
the head and half the body, and the body as long as both the head and tail. 
Required the length of the fish. 

Operation.— Assume body to be 24 ins. in length. Then 24=24-9 = 21, length 
of tail. 

Hence 21 4-9 = 30, length of body, which is 6 ins. too great. 

Again: assume the body to be 26 ins. in length. Then 26 = 24-9 = 22, length of 
tail. Hence 22 -f- 9 = 31, length of body, which is 5 ins. too great. 

Therefore, by Double Position , divide difference of products (see rule, page 99) 

by difference of errors (the errors being alike), 26 X 6 — 24 X 5 = 36 = difference of 
products, and 6 — 5 = 1= difference of errors. 

Consequently, 36 = 1 = 36, length of body , and 36-7-24-9 = 27, length of tail , and 
36 -)- 27 -j- 9 = 72 ins ., length required. 

17. —A hare, 50 leaps before a greyhound, takes 4 leaps to the greyhound’s 
3, but 2 leaps of the hound are equal to 3 of the hare’s. How many leaps 
must the greyhound take before he can catch the hare? 

Operation.—A s 2 leaps of the greyhound equal 3 of the hare, it follows that 6 of 
the greyhound equal 9 of the hare. 

While the greyhound takes 6 leaps, the hare takes 8; therefore, while the hare 
takes 8, the greyhound gains upon her 1. 

Hence, to gain 50 leaps, she must take 50 X 8 = 400 leaps ; but , while hare takes 
400 leaps , greyhound takes 300, since number of leaps taken by them are as 4 to 3. 


18.—If a basket and 1000 eggs were laid in a right line 6 feet apart, and 
10 men (designated from A to J) were to start from basket and to run alter¬ 
nately, collect the eggs singly, and place them in basket as collected, and 
each man to collect but 10 eggs in his turn, how many yards would each 
man run over, and what would be entire distance run over? 

Operation. —A’s course would be 6 x 2 feet {first term) 4- 10 X 6 X 2 feet ( last 
term) = 132 = sum of first and last terms of progression. 

Then 132-4-2 X 10 = 660 feet = number of times X half sum of extremes = sum of 
all the terms, or the distance run by A in his first turn. 

B’s course would be n X6X2 = 132 feet (first term) -j- 20 X 6 X 2 = 240 feet (last 
term) = 372 = sum of first and last terms. 

Then 372-7-2 X 10 = i860 = sum of all the times, or B 's first turn. 

A’s last course would be 901 x 6 X 2 = 10812 feet for the first term , and 910X6X2 
= 10920 feet for the last term of his last turn. 


Then 10 812 4- 10 920 - 1 - 2 X 10 = 108 660 = sum of the terms, or distance run. 


B’s last course would be 911 x 6 X 2 = 10932 feet for the first term, and 920X6X2 
= 11 040 feet for the last term of his last turn. 

Then 10 932 -j- 11 040 = 2X10 = 109 860 = sum of the terms or distance run. 

Therefore, if A’s first and last runs = 660 and 108660 feet, and the number of 
terms 10, then, by Progression, the sum of all the terms = 546 600 feet. 

And if B’s first and last runs= i860 and 109 860 feet, and the number of terms 10, 
then the sum of all the terms = 358600 feet. 

Consequently, 558 600 — 546 600 = 12 000 = common difference of runs, which, be¬ 
ing added to each man’s run = sum of all runs, or entire distance'run over. 


A’s run, 546 600 = 182 200 yds. 
B’s “ 558600 = 186200 “ 

C’s “ 570600 = 190200 “ 

D’s “ 582600 = 194200 “ 
E’s “ 594600 = 198200 “ 


F’s run, 606 600 = 202 200 yds. 


G’s 

H's 

I’s 

J’s 


618 600 = 206 200 
630 600 = 210 200 
642 600 = 214 200 
654 600 = 2l8 200 


6006 000 feet, which=5280: 


: 1137.5 miles. 

19.—If, in a pair of scales, a body weighs 90 lbs. in one scale, and but 40 
lbs. in the other, what is the true weight? 

-J (40 x 90) = 60 lbs. 

















MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 883 


20.—If a steamboat, running uniformly at tlie rate of 15 miles per hour 
through the water, were to run for 1 hour with a current of 5 miles per hour, 
then to return against that current, what length of time would she require 
to reach the place from whence she started ? 


Operation. 15 -|- 5 •= 20 mites, the distance run during the hour. 

Then 15 — 5 = 10 miles is her effective velocity per hour when returning, and 
20-7-10^:2 hours, the time of returning, and 2 -f-1 = 3 hours, or the whole time oc¬ 
cupied. 


Or, Let d represent distance in one direction, t and t' greater and less times of run¬ 
ning in hours, and c current or tide, 
t-ft' 


d 

Then, 


txt' 


; velocity of boat through the water, and - —- = c. 


21.—Flood-tide wave in a given river runs 20 miles per hour, current of 
it is 3 miles per hour. Assume the air to he quiescent, and a floating body 
set free at commencement of flow of the tide; how long will it drift in one 
direction, the tide flowing for 6 hours from each point of river ? 


Operation. —Let x be the time required; 20 x = distance the tide has run up, to¬ 
gether with the distance which the floating body has moved; 3X = whole distance 
which the body has floated. 


Then 20 x — 3 x = 6 X 20, or the length in miles of a tide. 



X 6 = 7 hours, 3 minutes, 31.765 seconds. 


22.—A steamboat, running at the rate of 10 miles per hour through the 
water, descends a river, the velocity of which is 4 miles per hour, and re¬ 
turns in 10 hours ; how far did she proceed? 


Operation. —Let x — distance required, —— = time of going, —-— == time of 

10-(-4 10 — 4 

returning. Then, — -f- ~ = 10; 62: f- 14X = 840; 2001 = 840; 840 = 20 = 42 miles. 

14 6 


23.—From Caldwell’s to Newburgh (Hudson River) is 18 miles; the cur¬ 
rent of the river is such as to accelerate a boat descending, or retard one 
ascending, 1.5 miles per hour. Suppose two boats, running uniformly at the 
rate of 15 miles per hour through the water, were to start one from each 
place at the same time, where will they meet? 

Operation.— Let x = the distance from N. to the place of meeting; its distance 
from C., then , will be 18 — x. 


Speed of descending boat, 15-f-1.5 = 16.5 miles per hour ; of ascending boat, 15 — 

rQ 


— x 


i -5 


13.5 miles per hour. ■ = time of boat descending to point of meet ing. 

10.5 ^ 3 * 5 

time of boat ascending to point of meeting. 


x 10 — 

These times are of course equal; therefore, —— =- 

16.5 13.5 


Then, 13.521 = 297 — 


16.5a;, and 13.5x4-16.50; = 297, or 30X7=297. 


Hence x = — = 9.9 miles , the distance from Neivburgh. 

30 

24.—There is an island 73 miles in circumference; 3 men start together 
to walk around it and in the same direction: A walks 5 miles per day, B 8, 
and C 10; when will they all come aside of each other again? 

Operation. —It is evident that A and C will be together every round gone by A; 
hence it remains to ascertain when A and B will be in conjunction at an even round, 
as 3 miles are gained every day by B. Therefore, as 3 : 1 :: 73 : 24.33-P; but, as 
the conjunction is a fractional number, it is necessary to ascertain what number of 
a multiplier will make the division a whole number. 

73 24.33-)- — 3, the number of days required in which A will go round 5 times, 

B 8, and C 10 times. 









884 miscellaneous operations and illustrations. 


25. —Assume a cow, at age of 2 years, to bring forth a cow-calf, and then 
to continue yearly to do the same, and every one of her produce to bring 
forth a cow-calf at age of 2 years, and yearly afterward in like manner; 
how many would spring from the cow and her produce in 40 years ? 

Operation.— The increase in 1st year would be o, in 2d year r, in 3d 1, in 4th 2, 
in 5th 3, in 6th 5, and so on to 40 years or terms, each term being = sum 0/ the two 
preceding ones. The last term, then, will be 165580141, from which is to be sub¬ 
tracted 1 for the parent cow, and the remainder, 165 580140, will represent increase 
required. 

26. —The interior dimensions of a box are required to be in the propor¬ 
tions of 2, 3, and 5, and to contain a volume of 1000 cube ins.; what should 
be the dimensions ? 


„ /1000X2 3 /1000X3 3 , , , /1000X5 3 , . 

Operation. — 3 /-=6.43; 3 /- ^- = <5.65; and 3 / - — = 1 6 ms . 

V2X3X5 V 2X3X5 y V 2X3X5 

And what for a box of one half the volume, or 500 cube ins., and retaining 
same proportionate dimensions ? 

°o 

Operation.—2 X 3 X 5 = 30, and — = 15. 

2 

/15 X 6.433 /15 X 9.65 s , /15 X 16 3 

~ 5. 1; 3 / - = 7.66; and 3 /-- = 12 ms . 

V 3 ° V 30 


Then, 3 


30 


27.—The chances of events or games being equal, what are the odds for 
or against the following results? 


Five Events. 

Four Events. 

Odds. 

Against. 

In favor. 

Odds. 

Against. 

In favor. 

31 to I 

4-33 t0 1 

All the 5 

4 out of 5 

1 out of 5 

2 OUt Of 5 

15 to 1 
2.2 tO I 

All the 4 

3 out of 4 

1 out of 4 

2 out of 4 

5 to 3 in favor of the 5 events result¬ 
ing 3 and 2. 

5 to 3 against 2 events only, or that 
the 4 events do not result 2 and 2. 


TT 

Odds. 

tree Even 

Against. 

its. 

In favor. 

T 

Odds. 

wo Even 
Against. 

ts. 

In favor. 

7 to 1 

Even 

3 to 1 in fa 
ing 2 and 1. 

All the 3 
(2 or all out 

1 of 3 

ivor of the 3 e 

I out of 3 
(2 or all out 
l of 3 

vents result- 

3 to 1 

Even 

Even that 

Both events 
(1 only out 
[ Of 2 

the events res 

1 out of 2 
(1 only out 
\ of 2 

>ult i and 1. 


28.—Required the chances or probabilities in events or games, when the 
chances or probabilities of the results, or the players, are equal. 


Events 

or 

Games. 

That a 
named event 
occurs a 
majority or 
more of 
times. 

Against a 
named event 
occurring 
an exact 
majority of 
times. 

Against each 
event occur¬ 
ring an equal 
number 
of times. 

Events 

or 

Games. 

That a 
named event 
occurs a 
majority or 
more of 
times. 

Against a 
named event 
occurring 
an exact 
majority of 
times. 

Against each 
event occur¬ 
ring an equal 
number of 
times. 

21 

Even 

5 to 1 

— 

II 

Even 

3.4 to 1 

_ 

20 

1.33 to I 

— 

4.66 to 1 

IO 

1.7 to I 

— 

3.06 to 1 

19 

Even 

4.5 to 1 

— 

9 

Even 

3 to 1 

— 

18 

1.55 to I 

— 

4.4 to 1 

8 

1.75 to X 

— 

2.66 to 1 

17 

Even 

4.4 to I 

— 

7 

Even 

2.7 to I 

— 

l6 

1.5 to 1 

— 

4. X to I 

6 

2 to I 

— 

2.2 to I 

15 

Even 

4 to 1 

— 

5 

Even 

2.2 tO I 

— 

14 

1.5 to I 

— 

3.8 to I 

4 

2.2 to 1 

— 

1.66 to 1 

13 

Even 

3.7 to I 

— . 

3 

Even 

1.66 to 1 

— 

12 

1.6 to I 

— 

3.44 to 1 

2 

3 to I 

— 

Even. 


29.—The chances of consecutive events or results are as follows: 

11.—2047 to 1. | 10.—1023 to 1. | 9.—sutoi. | 8.—255 to 1. | 7.—127 to 1. | 6.—63 to 1. 

Hence it will be observed that the chances increase with the number of events 
very nearly in a duplicate ratio. 

Illustration. —The chances of n consecutive events compared with 10, are as 
2047 to 1023, or 2 to 1. 














































MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 885 


3°-—Required the chances or probabilities of events or results in a given 
number of times. 

The numerator of a fraction expresses the chance or probability either for the re¬ 
sult or event to occur or fail, and the denominator all the chances or probabilities 
both for it to occur or fail. 


Thus, in a given number of events or games, if the chances are even, the proba¬ 
bility of any particular result is as—; —— ; ——— , etc., being 1 out of 

i+r 2 2+2 3+3 

2, 2 out of 4, etc., or even. 

If the number of events or games are 3, then the probability of any par¬ 
ticular result, as 2 and 1, or 1 and 2, is determined as follows: * 

Number of permutations of 3 events are 1 x 2 x 3 = 6, which represents number 
of times that number of events can occur, 2 and 1, or 1 and 2, to which is to be 
added the 2 times Or chances they can occur all in one way or the reverse thereto. 

6333 

Hence, —— = — =- = — , or 3 to 1 in favor of result: and probability ot 

2+6 4 4—3 1 

one party naming or winning two precise events or results, as winning 2 out of 3, 
is determined as follows: Number of permutations and chances, as before shown, 


are 8. Hence, number of his chances being 3, 


3 

3 + 5 


= “ 5 or 3 to 5 in 

favor of result; and probability of one party naming or winning all, or 3 events 
or results, is determined as follows: Number of permutations and chances being 
also, as before shown, 8. Hence, as there is but one chance of such a result, 


1 -j- 7 8 8 - 


— = — , or 1 to 7 in favor of result. 
1 7 ' 


If number of events, etc., are 4, then probability of any particular result, 
as 2 and 2, or of winning 2 or more of them, is determined as follows: 


Number of permutations and chances of 4 events are 16. Hence, as number of 

II ^ I 

chances of such a result are n, —-— -.= - 

5 + ii 1 

of the result, and that the results do not occur precisely 2 and 2. The number of 
chances of such a result being 10, 


-= —, or as 11 to 5 in favor 

16 —11 5 3 


6 -j- 10 8 8 — 5 


, or s to 3 against it. 
3 


If number of events, etc., are 5, then probability of any particular result, 
as 3 and 2, is determined as follows: 

Number of permutations and chances being 32, and number of chances of such 

a result being 20, —--= = —— ■= — = — , or as 5 to 3 in favor of the 

124-20 16 16 —10 6 3 

result; and that it may occur precisely 3 out of 5, the number of chances are 

10 10 3 5 5 , . .. 

- = — = + = —-— = — , or 11 to 5 against it. 

JO —p 22 32 10 ID 5 II 


31. —What is the dilatation of the iron in a railway track per mile, be¬ 
tween the temperatures of —20° and +130 0 ? 

Operation;- 20 0 + 130 0 = 150 0 . The dilatation of wrought iron (as per table, 

page 519) is, from 32 0 to 212 0 = 180 0 — .001 257 5 times its length. 

Hence, as 180 : 150 .001 257 5 : .0010479 = - 47 - of 5280 (feet in a mile) = 

5.53 feet per mile. 

32. —A steamer having an immersed amidship section of 125 sq. feet, has 
a speed of 15 miles per hour with 300 IP. What power would be required 
for one of like model, having a section of 150 sq. feet for a speed of 20 miles? 

As power required for like models is as cube of speeds 

ieo . .. , 2o‘ j = 8ooo , ,. 

Then — = x.2 relative sections, and —— —— =2.37 relative powers. 

125 i 5 3 = 3375 

Hence, 1 : 1.2 :: 2.37 : 2.844 limes IP. 














886 


MARINE STEAMERS AND ENGINES. 


MARINE STEAMERS AND ENGINES. 

Iron Cruiser (Propeller). 

“ Zabiaca,” I. R. N.—Vertical Direct Engine (Compound).—Length between 
perpendiculars , 228 feet; at water-line of 12 feel, 220 feet; beam , 30 feet; hold, 17.5 
feet. 

Displacement at load draught of 12.58 and 14.58 feet, 1202 tons. Per inch at load- 
line, 11.58 tons. Areas. — Of Load-line, 4867 sq. feet; of Sails, 12 312 sq. feet. 

Coefficients. — Of Total Displacement, .5; of Surface, .74; of Cylindroid from cyl¬ 
inder', .61; of Cylindroid from parallelopipedon, .475. 

Cylinders .—34 and 59 ins. in diam. by 36 ins. stroke of piston. 

Pressure of Steam. —78 lbs. per sq. inch, cut off at 23 ins. full throttle. Revolu¬ 
tions, 89.4 per minute. IIP, 1400. Pitch of Propeller, 19 feet. Speed, 14 knots 
per hour. Fuel. —Anthracite coal, 1.6 lbs. per IIP per hour. 

Centres oj Gravity. —Forward of after perpendicular, 100 feet; below meta-centre 
at draught of 10.46 and 12.21 feet, 2.81 feet, and at load-line 3.12 feet. Of Buoyan¬ 
cy, below load-line, 4.97 feet. Of Engines, Boilers , Water, etc., aft of centre of length, 
25.25 feet; do. above top of keel, 9.17 feet. 

Meta-centre .—Above centre of buoyancy for mean draught of 11.3 feet = 4 feet. 

Iron PT*eiglit and. [Passenger (Propellers). 

“ Orient.’’—Vertical Direct Engine (Compound,).—Length upon deck, 460 feet; 
beam, 46.35 feet; depth to main deck, 27.1 feet; to spar deck, 35.1 feet. 

Immersed section at load-line, 1094 sq.feet. Displacement at load draught 0/26.5 
feet, 9500 tons ; per inch, 40 tons. Tons, 3440-5380. 

Cylinders .—1 of 60 ins. in diam., and 2 of 85 ins., by 5 feet stroke of piston. Con¬ 
denser. — Surface, 12000 sq. feet. Propeller. — 4 blades, 22 feet in diam. Pitch, 30 
feet. Shaft, 20 ins. in diam. 

Boilers.— 4. (cylindrical tubular), 15.5 feet in diam. by 17.5 feet in length; 6 fur¬ 
naces, 4 feet in diam. by 6 feet in length. Pressure of Steam, 75 lbs. per sq. inch. 
Revolutions, 60 per minute. IIP, 5400. Bulkheads, 12. Decks, 3 of iron. 

Capacity. —3000 tons coal, 3600 tons (measurement) cargo, 120 1st class passen¬ 
gers, 130 2d, and 300 3d class, or 3000 troops and 406 horses. 

Water Ballast. —Aft, 82 feet in length. Rig, 4-masted bark. Passage, 35 days, 
Plymouth to Australia. Weights.— Hull, Engines, and Boilers, 4940 tons.’ 

‘‘Arizona.”—Vertical Direct Engine (Compound). — Lenqth between perpen¬ 
diculars and for tonnage, 450 feet; breadth, 45.5 feet; depth, 35.7 feet; Tons, 5146.55. 

Cylinders. — 1 of 62 ins. and 2 of 90 ins. in diam., by 5.5 feet stroke of piston. 
Condenser. —Surface, 12 540 sq. feet. 

Propeller (Cast Steel).—Diam., 23 feet; weight, 27 tons. 

Boilers.—6 of 13.5 feet in diam., 3 of 10 feet in length, and 3 of 18 feet. Heating 
Surface , 19 500 sq. feet. Grate, 780 sq. feet. Pressure of Steam, 86 lbs. per sq inch 
Revolutions, 55 per minute. IIP, 6306. Speed, 17 knots per hour. 

“Normandie.”—Vertical Direct Engines (Compound). — Length, 459 feet 11 
ins. ; beam, 49 feet 11 ins. Hold, 37 feet 5 ins. Mean draught at trial, 20 feet. 

Displacement , 7656 tons. Immersed Section at load-draught of 24.25 feet 1060 
sq. feet. 

Cylinders. 3 of 35.4375 ins., and 3 of 74.875 ins. in diam. ; stroke of pistons, 
67 ins.; ratio of low to high pressure, 1 to 4.46. 

Condensers. —3, surface, n 682 sq. feet. Air-pumps. —3, single acting, 34 ins. diam.; 
stiolce of piston, 32 ins. Centrifugal Pumps. —3, 12.5 ins. in diam., driven by three 
11 by 11 inch engines. 

Boilers. 8 (cylindrical tubular), 4 double end, 13.3 feet in diam., 18.5 feet in 
length, 4 single end, 13.75 feet in diam., 9.5 feet in length. Grates, 808.5 sq. feet. 
Heating Surface , 21 405 sq. feet. Steam Room , 3950 cube feet. 

Pressure of Steam, 85 lbs., cut off at .75 stroke. Revolutions, 59 per minute IIP 
8006. Shaft, 23,625 ins. in diam. Propeller, 22 feet in diam. Pitch, 31 feet. ’ 

Speed, 17.25 knots per hour. Weight of Engines, Boilers, and Water in boilers 
complete, 1376 tons. ’ 


MARINE STEAMERS AND ENGINES. 


887 


“City of San Francisco.” — Vertical Direct Engine {Compound). — Length 
over all , 352 feet; for tonnage , 339 feet; beam, 40.2 feet; hold , 28 feet 10 ins.; Load 
draught , 22 feet. 

Cylinders , 2.— 51 and 88 ins. in diam. by 5 feet stroke of piston. Condenser. — 
Surface, 6425 sq. feet. 

Pressure of Steam, 80 lbs. per sq. inch. Revolutions, 55. Speed , 14 knots per hour. 
Propeller, 4 blades, 20 feet in diam. by 25 feet pitch. 

Boilers. —6 (cylindrical tubular), 13 feet in diam. Heating Surface, 10650 sq. feet. 
Grates, 378 sq. feet. Ratio of Grate to heating surface, 1 to 28; to tube area, 9 to 1 ; 
to smoke-pipe area, 6.66 to 1. 

Iron -Auxiliary- ITreigTvt. 

Vertical Direct Engine {Compound).—Length on deck, 135 feet; beam, 22.5 feet; 
hold, 11 feet. 

Load-draught , 4 feet 10 ins. and 10 feet 6 ins. Free board, 1.5 feet. 

Cylinders.—- 21 and 40 ins. in diam. by 27 ins. stroke of piston. Condenser. — Sur¬ 
face, 617 sq. feet. 

Boiler (cylindrical tubular).—12 feet in diam. by 9.5 feet in length. Heating 
surface, 1205 sq. feet. Grates, 38.5 sq. feet. 

Pressure of Steam, 80 lbs. per sq. inch. Speed, 10. 8 knots per hour. IIP, 370. 
Consumption of coal, 8.5 tons in 24 hours. Rig .— Schooner. 

“Isle of Dursey.”—Vertical Direct Engines {Compound Triple Expansion). 
—Length on deck, 2x0 feet ; beam, 31.25 feet ; hold, 1 4.1 feet. Tons, 620.963. 

Cylinders. — 2, each 15.75 and 22 ins., and 44.33 ins. in diam.; stroke of piston, 
2.75 feet. • Condenser. —466 .75 inch tubes, No. 18 B W G. Surface, 792 sq. feet. 
Propeller .— 4 blades, 12.5 feet in diam. Pitch, 14.5 feet. Surface, 38.5 sq. feet. 
Pressure of Steam, 150 lbs. per sq. inch. Revolutions, 73 per minute. 

Boiler .— 1 (horizontal tubular). Heating surface, 1650 .sq. feet.. Grate, 42 sq. feet. 
IIP per sq foot of grate, 12.3; of heating surface, .374. Total, 500. 

Fuel.— Bituminous, 1.5 lbs. per IIP per hour. Rig.— Fore-topsail schooner. 

Iron ITire-boat. 

“Zophar Mills.”—Vertical Direct Engine.— Length on load-line, 115 feet; 
beam, molded , 24 feet; hold at side, 8 feet 8 ins. 

Immersed section at load-line of 7 feet, 150 sq.feet. 

Cylinder. — 30 ins. in diam. by 30 ins. stroke of piston; volume of piston space, 

12.25 cube feet. Condenser. — Surface, 900 sq. feet. 

Boilers (return tubular).—Two, 8 feet in width by 14 feet in length. Heating sur¬ 
face, 2120 sq. feet. Grates, 80 sq. feet. 

Pressure.— 70 lbs. per sq. inch, cut ofTat. 5 stroke. Revolutions, 84 at 45 lbs. press¬ 
ure, cut off at .5. Speed. — 12.5 miles per hour. 

Propeller.— 4-bladed, 8 feet 9 ins. in diam. 

Pumps, Vertical Duplex. Steam cylinders, 4. — 16 ins. in diam. by 9 ins. stroke. 

Pumps , 4._7.5 ins. in diam. by 9 ins. stroke. Receiving pipes, 8.5 ins. in diam. 

Revolutions, no per minute. 

Discharge .— 2200 gallons per minute; or, 8 streams, 2.5 ins. to 3.25 ins. hose, 
average 75 feet in length of hose each, and 1.5 ins. nozzles, 160 feet. Or, 4 streams, 

3.25 ins. hose, 100 feet in length of hose each, connected to one length of 16 feet of 
4-inch hefee, and 3.25 ins. nozzle, 280 feet. 

Steel Launch. 

Inverted Direct Engine {Non-condensing).— Length, 25 feet; beam, 5 feet; hold, 

2.5 feet. 

Cylinder.— 5 ins. in diam. by 5 ins. stroke of piston. 

Hull. — Frame, .75 X -75 inch, No. 12 W G. Keel, stem and stern-post, each, 

1.5 X 1-25 ins. 


888 MARINE STEAM VESSELS AND ENGINES. 


Steel Yachts. (Propellers). 

“Lady Torfrida.”—Vertical Direct Engines (Compound). — Length, 200 feet 
8 ins.; beam, 25 feet 7 ins.; hotd, 15 feet 7 ins. Tons, 611. 

Cylinders. —3, one of 24 ins. in diam., and two of 34 ins. by 30 ins. stroke of piston. 

Condenser. — Surface, 1978 sq. feet. Circulating Pump , double, 12 by 17 ins. Air- 
pump, single, 20 by 17 ins. 

Boilers (return tubular).—14.5 feet in diam. by 9 feet in length. Heating Sur¬ 
face, 1887 sq. feet. Grate, 77 sq. feet. 

Pressure of Steam, no lbs. Vacuum, 28.5 ins. IIP, 1020. Propeller, Manganese 
bronze, 11 feet in diam. Pitch, 14.5 feet. Speed. —15 knots per hour. 

Iron. 

“Isa.” — Vertical Direct Engines (Compound).—Length of heel, 118.66 feet; 
beam, 18.75 feet; hold, 10 feet. Tons, 248. 

Cylinders, 3.—10,15, and 28 ins. in diam. by 2 feet stroke of piston. Condenser.— 
Surface, 350 sq. feet. Circulating Pump , 6 ins. in diam. by 12 ins. stroke. 

Pressure of Steam. —120 lbs. per sq. inch full stroke. Revolutions, 112 per minute. 
Speed, 12 knots per hour. 

Propeller. —2 blades, 8.5 feet in diam. Pitch, 12.25 feet. 

Composite. 

“Radha.” — Vertical Direct Engines (Compound). — Length for tonnage , 142 
feet; beam, 20 feet; depth of hold, 8 feet 8.5 ins. Tons, 77.04 and 149.15. 

Immersed section at load-draught of 8.25 feet, 115 sq.feet. 

Cylinders, 3.—1 of 20 ins. in diam., and 2 of 26 by 2 feet stroke of piston. 

Condenser. — Surface, 80o sq. feet. 

Boiler (flue and return tubular).—9 feet 8 ins. wide, and 15 feet in length. Heat¬ 
ing surface, 1947 sq. feet. Grate, 48 sq. feet. 

Propeller, 7.5 feet in diam. Pitch, 12 feet. Revolutions, 135 per minute. 

Pressure of Steam. —100 lbs., cut off at .5. Blast draught. 

“Siesta.”—Vertical Direct Engine (Compound), Herreshoff.— Length on deck 
over all, 98 feet; at water-line, 90.3 feet; beam at deck, 17 feet; at water-line, 15.16 
feet; depth of hull from rabbet of keel to top of shear plank, 8.33 feet; draught of 
water at load-line, 5.66 feet. * 

Immersed section at load-line, 43 sq. feet. Displacement at load-draught, 63.83 tons. 

Area of water section, 878.7 sq. feet, and of immersed surface of hull, 1438 sq.feet. 

Ratio of water surface to its circumscribing parallelogram, .64 ; of immersed trans¬ 
verse section to its do. do., .584; and of displacement of immersed hull above lower 
edge of rabbet of keel to its circumscribing parallelopiped, .5677. 

Cylinders, 2.—10.5 and 18 ins. in diam. by 18 ins. stroke of piston. Volume of 
piston space, 3.45 cube feet. Relative volumes of displacement of cylinders, 1 to 
2.96. Air-pump, single acting. 6 ins. in diam. by 6.25 ins. stroke. 

Circulating and Feed Pumps, single acting, 1.125 ins. in diam. by 18 ins. stroke. 

Condenser, External. — Surface, 731, 5 ins. by 29.5 ins. tubes; condensing surface, 
235 sq. feet. 

Propeller, 4 blades, 4 feet 7 ins. in diam. Pitch, 8 feet. Helicoidal area of blades, 
9.46 sq. feet. Transverse area, 6.59 sq. feet. 

Shaft. —Journal, 3.875 X 8 ins.; stress, 3.75 ins. Engine space, 3 feet by 5.5 feet 
in length. 

Boiler (vertical double coil).—Diam. outside of casing, 6.66 feet; height, 8 feet 
10 ins. Heating surface, 558 sq. feet. Grates, 26 sq. feet. 

Smoke-pipe, 23.5 ins. in diam. by 25 feet above grates. Steam room, 5.7 cube feet. 

Heating surface to Grate, 21.5 to 1. 

Pressure of Steam, 60.7 lbs. per sq. inch, cut off in small cylinder at .88 of stroke, 
and in large at .3 stroke. In small cylinder at end of stroke, 55.2 lbs.; in large 
cylinder at commencement of stroke, 50.6; and at end of stroke, 15.6 lbs. Mean 
back pressure in small cjdinder, 47.36 lbs.; and in large, 5.77. 

Revolutions, 193 per minute. Speed, 12.75 miles (u.06 knots) per hour. Slip of 
Propeller, 27.3 per cent. 


MARINE STEAM VESSELS AND ENGINES. 


889 


Herresiioff.—Vertical Direct Engine {Compound).—Length on deck, 100 feet; 
beam, 12.5 feet. 

Cylinder .—1 of 12.5 and 21.5 ins. in diam. by 16 ins. stroke of piston. 

Pressure of steam , 120 lbs. per sq. inch. Revolutions , 480 per minute. Speed, 
22.5 knots per hour. 

Thrust of Propeller at 15.73 knots, 4080 lbs. 

Torpedo Boats. (Propellers.) 

Iron. 

Vertical Direct Engine [Compound). — Length , no feet; beam , 12.5 feet. 

Displacement, 52 tons. 

Cylinders, 1.—12.5 ins. and 21.5 ins.; stroke of piston, 16 ins. 

Boiler .—(Horizontal tubular.) Diam., 4.75 feet. Tubes, 125 of 2 ins. in diam. 
Heating surface , 1016 sq. feet. Speed, 20.3 knots per hour. 

Steel. Composite. 

“Torpedo Boat,” R. N.—Vertical Direct Engine [Compound), Herreshoff 
Mfg. Co.— Length, 59.5 feet; beam, 7.5 feet. 

Cylinders .—6 and 10.5 ins. in diam. by 10 ins. stroke of piston. 

Condenser, External. — Surface. Boiler (vertical coil).—Tubes 2 ins. in diam. and 
300 feet in length. 

Propeller .—4 blades, 38 ins. in diam. by 5 feet pitch. 

Weight at load-draught of hull of 1.5 feet; armament and stores, 8 tons. 

Iron. Side "W'h.eels. 

“Princess Marie and Elizabeth.”—Oscillating Engine [Compound).—Length 
on load-line, 274.8 ins.; beam, 34.75 feet; hold, 24.25 feet. Tons, 1606. 

Cylinders, 2.—60 and 104 ins. in diam. by 3.5 feet stroke of piston. 

Pressure of Steam. — 70 lbs. per sq. inch, cut off at .6 stroke. Revolutions, 32.75 
per minute. Speed, 17.12 knots per hour. IIP, 3543. 

Consumption of fuel, 1.92 lbs. per IIP per hour. Cost, £54900 sterling. 

Cutter (Corrugated.). 

“La Bonita.”—Inclined Engine [Non-condensing).—Length upon deck, 42 feet; 
beam, 9 feel; hold, 3 feet. 

Immersed section at load-line, 8.75 sq. feet. Displacement at load-draught of 1.3 
feet, 8386 lbs. Tons, 9.65, 0 . M. 

Cylinder .—8 ins. in diam.; stroke of piston, 1 foot; volume ofpiston space, .35 cube 
foot. Water-wheels. —Diam. 5.66 feet. Blades, 7; breadth, 2.3 feet; depth, 7 ins. 

Boiler .—(Horizontal tubular). Heating surface , 95 sq. feet. Grates, 6 do. Fuel, 
coal or wood. Exhaust draught. 

Pressure of steam .—65 lbs. per sq. inch. Revolutions, 54 per minute. IIP, 9. 

Hull .—Corrugated and galvanized plates, .0625 inch thick. Weights. —Hull, 2876 
lbs.; Engine and wheels, 2400 lbs.; Boiler, 2260 lbs.; pipes, grates, etc., 750 lbs. 

Steel. 

Ferry Boat.—Inclined Engine [Surface Condensing). — Length, 78 feet; beam, 
15 feet; hold, 8 feet amidships and 5 feet at ends. 

Load-draught, 2.25 feet. 

Cylinder , 15.5 ins. by 24 ins. stroke ofpiston. Boiler (cylindrical tubular).—Steel; 
heating surface, 220 sq. feet. 

Pressure of Steam., 80 lbs. per sq. inch. Plates, .125, .1875, and .25 ins. 

Liglit Draught. 

“Ho-nam.”—Vertical Beam Engine [Compound).—Length upon deck, 280 feet; 
beam, 73 feet; depth from hold to upper deck, 30 feet. Tons , 2364. 

Cylinders , 1.—40 and 72 ins. in diam.; stroke of piston, 10 feet. IIP, 3000. 

Speed, 16.14 knots per hour. Passengers, 2000. 

Decks, 3: Main, Saloon, and Promenade. Rig. —Schooner. 

4 F 


89O MARINE STEAM VESSELS AND ENGINES. 


Wood Side Wheels. Passenger and H>ec!k Cargo. 

“City of Fall River,” New York to Fall River, Mass., Vertical Beam En¬ 
gine (Compound).—From notes of James E. Sague and John B. Adger, Jr. Length 
on water-line, 260 feet; overall, 273 feet; beam, 42 feet; over guards, 73 feet; hold, 
18 feet; 1723 ions N. M. 

Immersed section at load-line of 9 feet 3 ins. (1750 tons), 365 sq.feet; and at load- 
draught of 12 feet, 480 sq.feet. Displacement at load-draught, 2350 tons. 

Cylinders , 2.—1 of 44 ins. in diam. by 8 feet stroke of piston, and 1 of 68 ins. in 
diam. by 12 feet stroke. Clearance at each end of high pressure, 4.6 per cent., and 
at low pressure, 3 per cent. Volumes, 85 and 303 cube feet. 

Receiver, 89.13 cube feet. Air-pump, 37 ins. in diam. by 4.75 feet stroke of pis¬ 
ton. Condenser.—Surface, 4067 sq. feet. 

Water-wheels, 25.5 feet in diam. Blades , feathering, 12 of 40 ins. in depth by 10 
feet in length. Centre of Pressure on Blades, 11.22 feet from axis of shaft. 

Boilers. — 2 (flue and return tubular), 17.5 feet in width by 15 feet in length, 220 
tubes 3.5 ins. in diam. and 12 feet in length. Grates, 230 sq. feet. 

Fuel. —Anthracite. Natural Draught. Consumption. 1463 lbs. per hour; refuse, 
281 lbs. = 19-23 per cent, per sq. foot of grate, and 12.73 lbs. 

Pressure of Steam. — High-pressure cylinder, throttle open and cut off' at .445, 
mean in boiler per guage, 70 lbs.; in receiver, u lbs.; mean effective pressure, 41.8 
lbs. per sq. inch. 

Low-pressure cylinder, at point of cutting off of .45, 17.42 lbs. above zero; mean 
effective pressure, 12.4 lbs. per sq. inch. Expansion of steam, 6.99 times. Vacuum, 
28.4 ins. 

LP.—High-pressure cylinder, 783; low-pressure, 840. 

Revolutions, 25.8 per minute. 

Feed Water, 27 854 lbs. per hour; per IIP, 17.17 lbs. Temperatures. —Feed water, 
97 0 ; sea water, 49.4 0 ; water of condensation, 90 0 ; heat units per hour per IIP, 
19090. 

Stress of wheels, 20.4 per cent. 

Condensing water, per IIP per hour, 407 lbs. 

Consumption.— Compound engine, 2.03 lbs. per IIP; and as a simple condensing 
engine, without high-pressure cylinder, 2.84 lbs. 

Evaporation per hour, 1208 lbs. water; per lb. of combustible from 212 0 , 11.75 
lbs.; from temperature of feed (97 0 ), 10.22 lbs.; from feed per lb. of coal, 826 lbs. 
Temperature of gases in chimney 485°. 

Heating Surface, 29 sq. feet to 1 of grate. 

Weights, Engine, and Frame, 250 tons; boilers complete, 120 tons; water, 50 tons. 

Speed, 14.14 knots per hour; and IIP, 1623. 

Draft of water, 10.65 feet; Displacement, 1938 tons. 

“City of Boston,” New York and Norwich. — Vertical Beam Engine (Con¬ 
densing). — Length upon load-line , 320 feet; beam , 3 9 feet; hold, 12.6 feet. 

Immersed section at load-line, 288 sq.feet. Displacement 1450 tons, at load-draught 
of 8.25 feet. 

Cylinder.— 80 ins. in diam. by 12 feet stroke of piston. Volume, 419 cube feet. 

Water-wheels. —Diam. 37 feet 8 ins. Arms, 36. Blades, 37; breadth of do., 10 
feet; depth of do., 30.5 ins. Dip at load-line, 4.25 feet. 

Boilers. — 2 (flue and return tubular), Shells, 12.5 feet in diam., and in length 
26. 5 feet. Heating Surface, 10120 sq. feet. Grates, 184 sq. feet. 

Pressure of Steam.— 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions (maxi¬ 
mum), 19.75'per minute. IIP, 2500. 

Fuel. —Anthracite; Blast. Consumption , at ordinary speed, 5200 lbs. per hour. 

Weights of Engine, Boilers, etc., 263 tons. 

Hull. — Weight, 800 tons. Light draught of hull without fuel, water, or furni¬ 
ture, 7 feet. 


MARINE STEAM VESSELS AND ENGINES. 89 1 
~W ood Propellers. 

Herreshoff, R. N., Vertical Direct Engine {Compound).—Length on deck, 46 
feet; over all, 48 feet ; beam, 9 feet; hold, 5 feet. 

Displacement at load-line, 7.44 tons. Area of section at load-line, 217.8 sq. feet. 
Area of wetted surface, 365.5 sq. feet. Coefficient of fineness, .396. 

Cylinder. —8 and 14 ins. in diam. by 9 ins. stroke of piston. 

Condenser, External. — Surface. 

Propeller. —4 blades, 3 feet in diam. by 4 feet 1 inch pitch. 

Blower, 42 ins. in diam. 

Boiler (vertical coil). Heating surface, 174 sq. feet. Grates, 12.5 sq. feet. 
Pressure of Steam, 53 lbs. per sq. inch. Revolutions, 333 per minute. IIP. 68.4. 
Speed, 10. iS knots per hour. With 129 lbs. and 466 revolutions, 14.26 knots. IIP, 
169.5. Weight of Engines, Boiler, and Water, 5300 lbs. 

Herreshoff, Vertical Direct Engine {Compound).—Length over all, 86 feet; 
beam, n feet. Displacement, 27 tons. 

Cylinder. —13 and 22 ins. in diam. by 12 ins. stroke of piston. 

Surface Condensing. 

Pressure, 130 lbs. per sq. inch. 

Revolutions, 460 per minute. Speed, 20 knots per hour. IIP, 425. 

Propeller, 3 blades. Pitch, 5 feet. 

Herreshoff, R. I. N.— Vertical Direct Engine (Compound).—Length over all, 
60 feet; beam, 7 feet; hold, 5.5 feet. Displacement at load-draught of 32 ins., 7 tons 
(2240 lbs.). 

Cylinders. —8 and 14 ins. in diam. by 9 ins. stroke of piston. Surface condenser. 
Pressure of Steam. —140 lbs. per sq. inch, cut off at .5. 

Revolutions, 600 per minute. Speed, 19.875 knots per hour. 

Cable or Rope Towing. 

“Nyitra.” — Horizontal Direct Engines ( Condensing).—Length of boat, iffifeet; 
beam, 24.5 feet; hold, 7.5 feet. 

Immersed section, 74.4 sq. feet. Displacement, 200 tons at load-line of 3.75 feet. 
Immersed section, 263.7 sq. feet. Displacement, 949 tons. Tow.— 3 barges. 

Cylinders. —2 of 14.18 ins. in diam. by 23.625 ins. stroke of pistbn. 

IIP, net effective, 100. Speed, 7.73 miles per hour. 

Propellers.— Twin, 4 feet 2 ins. in diam. 

Stress. — Cable, 7485 lbs. Per ton of displacement, 6.5 lbs.; per sq. foot of im¬ 
mersed section, 22 lbs. 

Fuel. —Per mile and ton of displacement (1149), .078 lbs. 

Towing. Wood Side Wheels. 

“Wm. H. Webb.”—Harbor and Coast.—Vertical Beam Engines ( Condensing ). 
— Length upon deck, 185 .5 feet; beam, 30.25 feet; hold, 10.8 feet. 

Immersed Section at load-line, 194 sq. feet. Displacement 498.25 tons, at load- 
draught of 7.25 feet. 

Cylinders. —2, of 44 ins. in diam. by 10 feet stroke of piston; volume, 211 cube feet. 
Condensers. — Jet, 2, volume 105 cube feet. Air-pumps. —2, volume 45 cube feet. 

Water-wheels.— Diam., 30 feet. Blades (divided), 21; breadth of do., 4.6 feet; 
depth of do., 2.33 feet. Dip at load-line, 3.75 feet. 

Boilers. — 2 (return flue). Healing surface, 3280 sq. feet. Grates, 147.5 sq. feet. 
Smoke-pipe.— Area, 11.6 sq. feet, and 35 feet in height above the grate level. 
Pressure of Steam.— 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions, 22 per 
minute. IIP, 1500. 

Fuel. —Anthracite or Bituminous. Consumption, 1680 lbs. per hour. 

Speed. —20 miles per hour. 

Weights. —Engines, Wheels, Frame, and Boilers, 310579 lbs. 


892 


RIVER STEAMBOATS AND ENGINES. 


Wood. Side Wheels. Passenger. 

“ Daniel Drew.” New York to Albany.—Vertical Beam Engine ( Condensing ). 
—Length upon deck, 251.66 feet; at load-line , 244 feet; beam, 31 feet; hold, 9.25 feet. 

Immersed section at load-line, 136 sq.feet. Displacement 380 tons, at load-draught 

of 4-83 feet. 

Cylinder.— 60 ins. in diam. by 10 feet stroke of piston; volume , 196 cube feet. 
Condenser. — Jet, volume 68 cube feet. Air-pump, volume 26 cube feet. 

Water-wheels. — Diam. 29 feet. Arms, 24. Blades, 24; breadth of do., 9 feet; 
depth of do., 26 ins. Dip at load-line, 2.33 feet. 

Boilers.— 2 (return flue), 29 feet in length by 9 feet in width at furnace. Shell, 
diam. 8 feet. Heating surface, 3350 sq. feet. Grates , 105 sq. feet. Cross area of 
lower flues, 15.5 sq. feet; of upper, 13 sq. feet. Weight, 80650 lbs. 

Smoke-pipes. —2, area 25.13 sq. feet, and 32 feet in height above the grate level. 

Pressure of Steam .— 35 lbs. per sq. inch, cut off at .5 stroke. Revolutions ( maxi¬ 
mum), 26 per minute. IBP, 1720. 

Fuel. —Anthracite; Blast. Consumption, 3800 lbs. per hour. 

Speed , 22.3 miles per hour. Slip of Wheels from Centre of Pressure, 12.5 per cent. 

Frames. —Molded, 15.75 ins.; sided, 4 ins.; and 20 ins. apart at centres. 

“Mary Powell,” Hudson River.—Vertical Beam Engine ( Condensing).—Length 
on water-line, feet; over all , 294 feet; beam, 34 feet 3 ins. ; over all , 64 feet; hold, 

9 feet. Deck to promenade deck, 10 feet. 

Immersed section at load - line of 6 feet, 200 sq. feet. Displacement, 800 tons at 
mean load-draught of 6 feet. 

Area of transverse head surface of hull above water, 2000 sq. feet. 

Cylinder. — 72 ins. in diam. by 12 feet stroke of piston; volume, 338 cube feet. 
Clearance at each end, 12.5 cube feet. 

Steam and Exhaust Valves, 14.75 ins. in diam. Air-pump, 40 ins. in diam. by 5 
feet 2 ins. stroke of piston. Condenser.—Jet, 128 cube feet. Crank-pin, 8.75 ins! in 
diam. x 10.75 ins. 

Beam, 22.5 feet in length; centre, 9.75 in diam. 

Water-wheels —Diam. 31 feet; blades (divided), 26; breadth of do., 10 feet 6 ins. • 
width, 1 foot 6 ins.; immersion, 3 feet 6 ins. Shafts.— Journal, 15.625 ins. by 17 ins.’ 

Boilers. — 2 (flue and return tubular), of steel, n feet front by 26 feet in length; 
shell, 10 feet in diam. and 16 feet 1 inch in length. Furnaces, 2 in each, of 4 feet 

10 ins. by 8 feet in length. Heating Surface, 2660 sq. feet; and Superheating, 340 
sq. feet in each. Grates, 152 sq. feet. Flues, 10 in each, transverse area, u feet 
7 ins. Tubes, 80 in each, 4.5 ins. in diam., 6 feet 6 ins. in length, and 8 feet 7 ins 
in transverse area. 

Steam Chimneys. 8 feet in diam. X 12 feet in height. Smoke-pipe, 4 feet 6 ins. in 
diam. and 68 feet in height from grates. 

Combustion, Blast. Blowers, 4 feet in diam. and 3 feet in width. Revolutions 78 
per minute. Fuel (anthracite), 6280 lbs. per hour, or 40 lbs. per sq foot of grate 
per hour. Per sq. foot of heating surface, 2.25 lbs. 

Speed , 23.65 miles per hour. 

Pressure of Steam., 28 lbs. per sq. inch, cut off at .47 stroke; terminal pressure 
16.4 lbs.; throttle, .625 open. Vacuum, 25 ins. Revolutions, 22.75 per minute. 

Temperatures.— Reservoir, 120 0 . Feed water, 120 0 . Chimney, jao°. IR.—Total 
1900. IEP, 1560. Net, 1450. 

Evaporation.—Water per lb. of coal, from 120 0 , 7 lbs.; per lb. of combustible 
from 120 0 , 8.2 lbs. Steam per total HP per hour, 21.1 lbs. Coal per do. do., 3.14 lbs.’ 

Weights. Engine. — Frame, keelson, out-board wheel-frames donkey engine 
and boiler, blower engines and blowers, all complete, 360000 lbs. Boilers —Iron 
return flue. 120000 lbs. Steel return tubular, 116000 lbs. Water, 128000 lbs. 

Capacity.— 2000 passengers and their baggage. 

Memoranda. —Tbis vessel was originally but 266 feet in length, and when length¬ 
ened the cylinder of 62 ins. in diam. was removed and replaced with one of 72 ins 
Engine designed throughout for original cylinder and a pressure of from ko to qs 
lbs., cutting off at .625 of stroke, with throttle wide open. 33 

Engines and Boilers built by Fletcher, Harrison, & Co., New York, 1861 and 1875. 



EIVER STEAMBOATS AND ENGINES. 


893 

“Solano,” Ferry Boat.—Vertical Beam Engines [Condensing).—Length over 
all, 424 feet; on keel, 406 feet; beam (molded), 64 feet; hold at £gj, 18.5 feet; at ends, 
15 feet 10 ins. ; width over guards, 116 feet. 

Light draught, 5 feet; loaded, 6.5 feet. Tons, 3541. 

Cylinders.— 2 of 60 ins. diam. by u feet stroke of piston. 

Wheels, 34 feet in diam. by 17 feet face. Blades , 24. 

Boilers, 8.—Steel; 7 feet in diam. by 28 feet in length. Heating surface, 19640 
sq. feet. Grates, 288 sq. feet. IIP, 4000. 

Passenger and. XAglit ITreiglit. 

“Setii Grosvenor.”—Steeple Engine (Condensing).—Length upon deck qk feet ■ 
beam, 17.2 feet; hold, 5 feet. 

Immersed section at load-line, 43 sq. feet. Displacement 73 tons, at load-draught 
of 3.25 feet. 

Cylinder.— 28 ins. in diam. by 3 feet stroke of piston; volume, 12.8 cube feet. 

Waterwheels. —Diam. 13.5 feet. Blades, 14; breadth of do., 3 feet; depth of do. 

1.25 feet. 

Boiler (flue and return tubular).— Heating surface , 540 sq. feet. Grates, 22.5 sq. 
feet. Area of tubes, 367 sq. ins. IIP, 90. 

Weights. —Engine, Wheels, Frame, and Boiler, 61 556 lbs. = 27.4 tons. 

The operation of this vessel was in every way successful, being very fast, economical in fuel, etc., 
and she would have been improved if the hull Uad had 15 feet additional length, all other dimension's 
and capacities remaining the same. 

Wood Stem W'lieels. 

Passenger a,n.cl Decik IFreiglit. 

“ Montana.’’—Horizontal Engines (Non-condensing).—Length upon deck (over 
all), 248 feet; at water-line, 245 feet; beam, 48 feet 8 ins. (over all, 50 feet 4 ins.) ; 
hold, 6 feet; draught of water at load-line , 5.5 feet. 

Immersed section at load-line, 244 sq. feet. Displacement at mean light draught 
of 22 ins., 594 tons (2000 lbs. ) 

Cylinders.— Two, 18 ins. in diam. by 7 feet stroke of piston. 

Valves, 4.5 and 5 ins. in diam. Piston-rod, 4 ins. Steam-pipe, 4.5 ins. Connect¬ 
ing-rod, 30 feet in length. 

Water-wheel, 19 feet in diam. by 35 feet face; blades, 3 feet in depth. Shaft, 

10.25 ’ ns - i u diam. 

Boilers. —Four (horizontal tubular), 42 ins. in diam. by 26 feet in length. Two 
flues in each, 15 ins. in diam. Heating surface, effective, 1023, total 1431 sq. feet. 
Furnace, 6.5 X 17 feet. Grates, 4.16 X 17 feet; surface, 70.8 sq. feet. Smoke-pipes. 
—Two, 3 feet in diam. by 55 feet 3 ins. in height. Exhaust or Blower draught. 

Calorimeter. —Of Bridge, 15.27; of Flues, 9.82; and of Chimneys, 14.14 sq. feet. 
Areas of grate, compared to calorimeter of flues, 7.2; to ditto, of chimneys, 5; and 
of bridge, 4.6 sq. feet. 

Steam-room, 562; and water space, 294 cube feet. 

Hull. — Frames, 4X6 ins. and 15 ins. apart at centres. Intermediate do., 4X6 
ins., and running for 7.5 feet each side of keelson. Planking. —Bottom, oak, 4 ins.; 
side do., 2.5 to 4 ins. Deck beams, pine, 3X6 ins. Deck plank, 2.5 ins. Keelson, 
oak; side do., eight each side, one each 7, 8.75, and 9 ins., and five 6.75 ins. Wales. 
one each side, 9 and 7 ins. by 3, and one 10 X 2.5 ins. Deck posts, 3.5 X 3 ins. and 4 
feet apart. Deck beams, 5.5 X 3 ins. Knuckles, oak, 6 X 12 ins. Bulkheads, one 
longitudinal and one athwartship at shear of stern. Sheathing of wrought iron, 
.0625 to .125 inch from just below light to load-line. 

Hog Posts. —White pine, 8.5 and n ins. square. Chains , 1.5 ins. in diam. 

Weights. —Boilers, 29 264; water, 18 351; and boilers, chimneys, grates, and water, 
55672 lbs. Hull, oak, 520560; Pine, 91437; Bolts, spikes, etc., 8000, and Deck and 
guards, 76000 lbs.; Hull alone, 310 tons. 

Weight of hull compared to one of iron as 8 to 5, effecting a difference of about 
100 tons. 

4 F * 


894 


RIVER STEAMBOATS.—SAILING VESSELS. 


IPassenger and. Deck ITreiglit. 

“ Pittsburgh. ” —Horizontal Engines (Non-condensing).—Length on deck, 252 
feet; beam, 39 feet; hold, 6 feet; draught of water at load-line, 2 feet. 

Immersed section at load-line, 75 sq.feet. Displacement at load-draught of 2 feet, 
380 tons (2000 lbs.). 

Cylinders.— Two, 21 ins. in diam. by 7 feet stroke of piston. 

Water-wheel. —21 feet in diam. by 28 feet face. 

Boilers. — 2 (horizontal tubular), 47 ins. in diam. by 28 feet in length. Two fires 
in each. 

Iron. Stern Wheels. 

Horizontal Engines (Non-condensing). — Length upon deck, no feet; beam, 14 
feet (deck projecting over, 4 feet ); hold, 3.5 feet. 

Immersed section at load-line, 10.25 sq.feet. Displacement at load-draught of 1.1 
feet, 33 tons. 

Cylinders.— Two, of 10 ins. in diam. by 3 feet stroke of piston; volume of piston 
space, 1.6 cube feet. 

Wheel .—Diam. 13 feet. Blades, 13; breadth of do., 8.5 feet; depth of do., 8 ins. 

Revolutions, 33 per minute. Boiler. —One (horizontal tubular). Tubes, 100 of 2 
ins. in diam. 

Fuel. —Bituminous coal. Consumption, 4480 lbs. in 24 hours. 

Hull. — Plates, keel, No. 3; bilges, No. 4; bottom, No. 5; sides, Nos. 6 and 7. 
Frames, 2.5 X -5 ins., and 20 ins. apart from centres. 

Steel. 

“Chattahoochee.” — Inclined Engines (Non-condensing). — Length on deck, 157 
feet; beam, 31.5 feet; hold, 5 feet. 

Immersed section at load-line, 153 sq.feet. Freight capacity, 400 tons (2000 lbs.). 

Cylinders .—Two, 15 ins. in diam. by 5 feet stroke; volume of piston space, 12.26 
cube feet. 

Wheel. — One, 18 feet in diam.; blades, 2 feet in depth. 

Boilers .—Three (cylindrical Sued). Diam. 42 ins.; length, 22 feet; 2 flues of 10 
ins. in each. Heating surface, 690 sq. feet. Grates, 48 sq. feet. 

Pressure of Steam, 160 lbs. per sq. inch, cut off at .375. Revolutions, 22 per min. 

Consumption of Fuel, 12 tons (2000 lbs.) in 24 hours. Plating of Hull, 1875 to 
.25 inch. Light draught, 21 ms. 

Iron. Propellers. 

Vertical Direct Engines (Non-condensing). — Length on deck, 70 feet; beam, 10.5 
feet; draught, 12 ins. 

Propellers, 2.—2 blades, 16. ins. in diam., set n ins. below water-line. 

Boiler (tubular coil). Revolutions, 480 per minute. 

Speed, 10.49 m 'l ef s P er hour. 

Water led to propellers through tunnels in bottom at sides. 

“Louise.”—Vertical Tandem Engines (Compound).—Length, 60 feet; beam , 12 
feet; hold, 4.25 feet. 

Displacement at load-draught of 2.5 feet, 8 tons. 

Cylinders , 5 and 10 ins. in diam. by 8 ins. stroke of piston. 

Surface Condenser.—Boiler (vertical tubular), 4 feet in diam. by 8.5 in length. 

Iron Sailing Vessels. 

Passenger and. Freiglrt. 

English. — Ship. — Length upon deck, 178 feet; do. at mean load-line of 19. 16 feet, 177 
feet; keel, ijifeet; beam, 32.88 feet; depth of hold, 21.75 feet; keel (mean), 2.75 feet. 

Immersed section at load-line, 387 sq. feet. Displacement at load-draught of 19.16 
feet, 1385 tons ; at deep load-draught of 20 feet, 1495 tons; and, in proportion to its 
circumscribing parallelopipedon, .524. 

Load-line.—Area at load-draught, 4557 sq. feet. Angle of entrance, 57 0 ; of clear¬ 
ance, 64°. Area in proportion to its circumscribing parallelogram, .784. 



YACHTS.-CUTTERS.—PILOT BOAT. 


895 . 

Centre of Gravity , 6.416 feet below iTica.ii load-line. Centre of Displacement (grav¬ 
ity of), 6.25 feet below load-line; and 4.33 feet before middle of length of load-line. 

Immersed Surface.—Bottom, 7370 sq. feet. Keel, 1130 sq. feet. Sails, 13 282 sq. feet. 

Meta-centre, 6.66 feet above centre of gravity of displacement. Centre of Effort 
before centre of displacement, 3.5 feet; height of do. above mean load-line, 55.5 feet. 

Launch. Wood. 

Steam Launch “ Herreshoff.”—Vertical Engine (Compound).—Lenatli, aa feet 
1 inch; beam, 8.75 feet. 1 

Displacement at mean load-draught of (to rabbet of keel) 19 ins., 8929 lbs. 

Weights.—Hull and Machinery, 6555 lbs. Coal, 1120 lbs. 

Yachts. Wood. 

“America,” Schooner.— Length over all , 98 feet; upon deck, 94 feet; at load-line, 
90.5 feet ; beam, 22.5 feet; at load-line, 22 feet; depth of hold, 9.25 feet. Height at 
side from under side ofgarboard slralce, 11 feet. Sheer, forward, 3 feet; aft , 1.5 feet, 

Immersed section at load-line , 121.8 sq. feet Displacement at load-draught 0/8.5 
feet, from under side ofgarboard strake and of u feet aft, 191 tons; anil, in pro¬ 
portion to Volume of circumscribing parallelopipedon, .375. 

Displacement at 4 feet (from garboard strake ), 43 tons ; at 5 feet, 66 tons ; at 6 
feet, 93 tons ; at 7 feet, 127 tons; and at 8 feet, 167 tons. 

Centre of Gravity.— Longitudinally, 1.75 feet aft of centre of length upon load- 
line. Sectional, 2.58 feet below load-line. Of Fore body, 14.25 feet forward; and 
of After body, 19 feet aft. Meta-centre , 6.72 feet above centre of gravity. 

Centre of Effort, 31 17 feet from load-line. Centre of Lateral Resistance, 6.33 feet 
abaft of centre of gravity. Area of Load-line, 1280 sq. feet. Mean girths of im¬ 
mersed section to load-line , 25 feet. 

Load-draught.— Forward, 4.91 feet; aft, it. 5 feet. Rake of Stem, 17 feet. 

Spars.—Mainmast, 81 feet in length by 22 ins. in diam. Foremast , 79.5 feet in 
length by 24 ins. in diam. Main boom, 58 feet in length. Main gaff, 28 feet. Fore 
gaff, 24 feet. Rake, 2.7 ins. per foot. Drag of Keel, 3 feet. Tons, 170.56. 

“Julia,” Sloop .—Length for tonnage, 72.25 feet; on water-line, 70 feet 7 ins.; 
beam, 19 feet 8 ins.; hold , 6 feet 8 ins. Tons , 0 . M. 83.4; N. M. 43.98. 

Load-draught, 6.25 feet. 

Sails. — Mainsail , hoist, 49.75 feet, foot 54.25, and gaff 27.66; Jib, hoist, 49.75 feet, 
foot 39.5, and stay 63.5. Gaff topsail, hoist, 24.5 feet. 

Areas. —Mainsail, 2322 sq. feet. Jib, 986, and Topsail, 454. 

Cmtters. 

“Tara” (English) Sloop .—Length on load-line , 66 feet; beam, 11.5 feet. 

Immersed section at load-line, 11.5 sq.feet. Displacement, 75 tons. 

Spars. — Mast, deck to hounds, 42 feet. Boom, 58 feet. Gaff, 39 feet. Bowsprit 
outside of stem, 30 feet. Mast to stem, 26 feet. Topmast, foot to hounds, 25 feet. 
Balloon topsail yard, 46 feet. Canvas , area, 3450 sq. feet. Tons, C. H., 90. 

Ballast.—At Keel, 38.5 tons. Hull, 1.5 tons. 

“Mischief” (English), Sloop .—Length on load-line, 61 feet; beam, ig.gfeet. 

Immersed section at load-line, 60 sq.feet. Displacement, 55 tons. 

(Pilot Boat. 

“Wm. H. Aspinwall,” Schooner .—Length of keel, 74 feet; upon deck, 80 feet; 
beam, ig feet; hold, 7.6 feet. Draught of water, 6 feet forward ; aft, 9.5 feet. 

Keel, 22 ins. in depth. False keel, 12 ins. in depth at centre. 

Spars.—Mainmast. 77 feet in length. Foremast, 76 feet. Main boom, 46 feet. 
Main gaff, 21 feet. Fore gaff, 20 feet. 

Tons. —N. M., 46.32. 


PASSAGES OF STEAMBOATS.-ICE-BOATS. 


PASSAGES OF STEAMBOATS. 

Distances in Statute Miles. 

1807, Clermont , of N. Y., New York to Albany, 145 miles, in 32 hours == 4.53 miles 
per hour, neglecting effect of the tide. 

1811, New Orleans , of Pittsburgh, Penn, (non-condensing and stern-wheel), Pitts¬ 
burgh to Louisville, Ky., 650 miles, in 2 days 22 hours. 

1849, Alula , of N. Y., Caldwell’s, N. Y., to Pier 1, North River, 43.25 miles, in 1 
hour 42 min., ebb tide = 2.75 miles per hour. Speed = 22.19 wiles per hour, i860, 
30th Street, N. Y., to Cozzens’s Pier, West Point, 50.5 miles, in 2 hours 4 min., and 
to Poughkeepsie, 74.25 miles, in 3 hours 27 min., 5 landings, flood tide. And 1853, 
Robinson Street to Kingston Light, 90.375 miles, in 4 hours, making 6 landings, 
flood tide. 

1850, Buckeye State, of Pittsburgh, Penn, (non - condensing), Cincinnati to Pitts¬ 
burgh, 500 miles (200 passengers), 53 landings, in 1 day 19 hours ; 4 miles per hour 
adverse"current. Speed = 15.63 miles and 1.23 landings per hour. Average depth 
of water in channel 7 feet. 


1852, Reindeer , of N. Y.. New York to Hudson, 116.5 miles, in 4 hours 57 min., 
making 5 landings. Flood tide. 

1853, Shotwell, of Louisville, Ky. (non-condensing), New Orleans to Louisville, 
1450 miles, 8 landings, in 4 days q hours ; 4.5 to 5.5 miles per hour adverse cur¬ 
rent. Speeds 18.81 miles per hour. 


Note. —In 1817-18 the average duration of a passage from New Orleans to Louisville was 27 days, 
12 hours ; the shortest, 25 days. 

1855, New Princess, of New Orleans (non-condensing). New Orleans, La., to Natchez, 
Miss., 310 miles, in 17 hours 30 min.; 3.5 to 4 miles per hour adverse current. 
Speed = 20.98 miles per hour. 

1864, Daniel Drew, of N. Y., Jay Street. N. Y., to Albany, 148 miles, in 6 hours 51 
min., 9 landings. Flood tide. Speed of boat = 22.6 miles per hour. 

1867, Mary Powell , of N. Y., Desbrosses Street, N. Y., to Newburgh, 60.5 miles, in 
2 hours 50 min. , 3 landings; from Poughkeepsie to Rondout Light, 15.375 miles, in 
39 min., flood tide. 1873, Milton to Poughkeepsie, light draught and flood tide, 4 
miles, in 9 min.; and 1874, Desbrosses Street to Piermont, 24 miles, in 1 hour ; to 
Caldwell’s, 43.25 miles, in 1 hour 50 min. Speed = 22.77 to 23 miles per hour. 


Runs f rom New York to Albany , 146 miles , by different Boats. 


1826 , Sun . 12 hours 16 min. 

1826 , North America* . 10 “ 20 “ 

1841, Troy f. 8 “ 10 “ 

1841’ South America t. 7 “ 28 “ 


1852, Fr. Skiddy §. 7 hours 24 min. 

i860, ArmeniaW . 7 “ 42 “ 

1864 , Daniel Drewi. ... 6 “ 51 “ 

1864, Cldncey VibbardX. 6 “ 42 “ 


* 7 landings. + 4 landings. i 9 landings. § 6 landings. |] 11 landings. 

Timing Distance .—From 14th St., Hudson River, N. Y., to College at Mount St. Vincent, 13 miles. 


Note. —Where landings have been made, and the river crossed, the distance between the points 
given is correspondingly increased. 

1870, R. E. Lee, of St. Louis (non-condensing), New Orleans to St. Louis, Mo., 1180 
miles (without passengers or freight), 4 to 5 miles per hour adverse current; Vicks¬ 
burg, 1 day 38 min. ; Memphis, 2 days 6 hours 9 min.; Cairo, 3 days 1 hour.; and to 
St. Louis, 3 days 18 hours 14 min., inclusive of all stoppages. 

1870, Natchez, of Cincinnati, Ohio, from New Orleans to Baton Rouge, 120 miles, 
in 7 hours 40 min. 42 sec. 


Runs from Nero Orleans to Natchez , 295 miles, by different Boats. 

1814, Orleans , 6 days 6 hours 40 min. I 1856, New Princess, 17 hours 30 min. 

1840, Edward Shippen, 1 day 8 hours. | 1870, R. E. Lee, 16 hours 36 min. 47 sec. 

Ice-boats. 

Distances in Statute Miles. 

1872, Haze, of Poughkeepsie, N. Y., to buoy off' Milton, 4 miles, in 4 min. 

1872, Whiz, of Poughkeepsie, N. Y., to New Hamburg, 8.375 miles, in 8 min. 







PASSAGES OF STEAMERS AND SAILING VESSELS. 897 

PASSAGES OF STEAMERS AND SAILING VESSELS. 

Distances in Geographical Miles or Knots. 

Steamers. Sid.e-wh.eels. 

1807. Phoenix, of Hoboken, N. J. (John Stevens), New York, N.Y., to Philadelphia, 
Penn. First passage of a steam vessel at sea. 

1814, Morning Star , of Eng., River Clyde to London, Eng. First passage of an 
English steamer at sea. 

1817, Caledonia , of Eng., Margate, Eng., to Cassel, Germ., 180 miles, in 24 hours. 

1819, Savannah, of N.Y., about 340 tons 0 . M., Tybee Light, Savannah River, Ga., 
to Rock Light, Liverpool, Eng., 3640 miles, in 25 days 14 hours; 6 days 21 hours of 
which were under steam. 

1825, Enterprise , of Eng., 500 tons, Falmouth, Eng., to Table Bay, Africa, in 57 
days; and to Calcutta, India, in 113 days. First passage of a steamer to India. 

1830, Hugh Lindsay , 4x1 tons, 80 BP, Bombay, India, to Suez, Egypt, 3103 miles, 
in 31 days running time. 

1837, Atlanta, of Eng., 650 tons, Falmouth, Eng., to Calcutta, in 91 days. 

1839, Great Western , of Eng., Liverpool to New York, N. Y., 3017 miles, in 12 
days 18 hours. 

1870, Scotia, of Eng., Queenstown, Ireland, to Sandy Hook, N. J., 2780 miles, in 
8 days 7 hours 31 min. 1866, New York to Queenstown, 2798 miles, in 8 days 2 
hours 48 min.; thence to Liverpool, Eng., 270 miles, in 14 hours 59 min.; total, 8 
days 17 hours 47 min. 

Screw. 

1874, India Government Boat , Steel, length 87 feet, beam 12 feet, draught of water 
3.75 feet, mean speed for one mile 20.77 miles per hour, and maintained a speed of 
18.92 miles in 1 hour. 

1877, Lusitania , of Eng., London to Melbourne, Australia, via Cape, n 445 miles, 
in 38 days 23 hours 40 min. 

Sailing "Vessels. 

1851, Chrysolite (clipper ship), of Eng., Liverpool, Eng., to Anjer, Java, 13000 
miles, in 88 days. The Oriental, of N. Y., ran the same course in 89 days. 

1833, Trade Wind (clipper ship), of N. Y., San Francisco, Cal., to New York, N. Y., 
13610 miles, in 75 days. 

1854, Lightning (clipper ship), of Boston, Mass., Melbourne, Australia, to Liver¬ 
pool, Eng., 12 190 miles, in 64 days. 

1854, Comet (clipper ship), of N. Y., Liverpool, Eng , to Hong Kong, China, 13040 
miles, in 84 days. 

1854, Sierra Nevada (schooner), of N. H., Hong Kong, China, to San Francisco, 
Cal., 6000 miles, in 34 days. 

1854, Red Jacket (clipper ship), of N. Y., Sandy Hook, N. J., to Melbourne, Aus¬ 
tralia, 12 720 miles, in 69 days u hours 1 min. 

1855, Euterpe (half-clipper ship) of Rockland, Me., New York to Calcutta, India, 
i2 500 miles, in 78 days. 

i860, Andrew Jackson (clipper ship), of Boston, New York, N. Y., to San Fran¬ 
cisco, Cal., 13 610 miles, in 80 days 4 hours. 

1865, Dreadnought (clipper ship), of Boston, Honolulu, Sandwich Islands, to New 
Bedford, Mass., 13470 miles, in 82 days; and 1859, Sandy Hook, N. J., to Rock 
Light, Liverpool, Eng., 3000 miles, in 13 days 8 hours. 

1865, Sovereign of the Seas (medium ship), of Boston, Mass., in 22 days sailed 
5391 miles = 245 miles per day. For 4 days sailed 341.78 miles per day, and for 1 
day 375 miles. 

1866, Henrietta (schooner yacht), of N. Y., Sandy Hook, N. J., to the Needles, 
Eng., 3053 miles, in 13 days 21 hours 55 min. 16 sec. 

1866, Ariel and Serica (clipper ships), of England, Foo-chou-foo Bar, China, to 
the Downs, Eng., 13 500 miles, in 98 days. 

1869, Sappho (schooner yacht), of N. Y., Light-ship ofT Sandy Hook, N. J., to 
Queenstown, Ireland, 2857 miles, in 12 days 9 hours 34 min. 


898 


ELEMENTS OF MACHINES AND ENGINES. 


ELEMENTS OF MACHINES AND ENGINES. 

BLOWING ENGINES. 

Furnaces. — Two. Fineries. — Two. ( England !.) 

240 Tons Forge Pig Iron per Week. 

Engine (non-condensing).— Cylinder , 20 ins. in diam. by 8 feet stroke of piston. 

Boilers .—Six (plain cylindrical), 36 ins. in diam. and 28 feet* in length. Grates, 
100 sq. feet. 

Blowing Cylinders. —Two, 62 ins. in diam. by 8 feet stroke of piston. Pressure , 
2.17 lbs. per sq. inch. Revolutions , 22 per minute. 

Pipes, 3 feet in diam.= 168 area of cylinder. 

Tuyeres .—Each Furnace, 2 of 3 ins. in diam.; 1 of 3.25 ins.; and 1, 3 of 3 ins. 
Each Finery, 6 of 1.33 ins.; and 1, 4 of 1.125 ins. 

Temperature of Blast, 6oo°. Ore, 40 to 45 per cent, of iron. 

Furnaces. — Eight, diam. 16 to 18 feet. Dowlais Iron Works (England). 

1300 Tons Forge Iron per Week; discharging 44000 Cube Feet of Air per 

Minute. 

Engine (non-condensing).— Cylinder, 55 ins. in diam. by 13 feet stroke of piston. 

Pressure of Steam .—60 lbs. per sq. inch, cut off at .33 the stroke of piston. Valves, 
120 ins. in area. 

Boilers .—Eight (cylindrical flued, internal furnace), 7 feet in diam. and 42 feet in 
length; one flue 4 feet in diam. Grates, 288 sq. feet. 

Fly Wheel. —Diam., 22 feet; w’eight, 25 tons. 

Blowing Cylinder, 144 ins. in diam. by 12 feet stroke of piston. 

Revolutions , 20 per minute. Blast , 3.25 lbs. per sq. inch. Discharge pipe, diam. 
5 feet, and 420 feet in length. Valves. —Exhaust, 56 sq. feet; Delivery, 16 sq. feet. 

Furnaces .— Lackenby (England). 

800 Tons Iron per Week. 

Engine (horizontal, compound condensing).— 32 and 60 ins. in diam. by 4.5 
feet stroke of piston. „ 

Blowing Cylinders. —Two, 80 ins. in diam. by 4.5 feet stroke of piston. Pressure, 
4.5 lbs. per sq. inch. Revolutions, 24 per minute. 

Pipe, 30 ins. in diam.; volume, 12.25 times that of blowing cylinders. 

IP.—Engine, 290 lbs.; Blowing cylinders, 258; efficiency, 89 per cent. 

Valves .—Area of admission, .16 of area of piston; of exit, .125. 

Volume .—190000 cube feet of air are supplied per ton of air. 


Blower and. Exhausting Fan. (Sturtevant’s.) 


Blower. 

Grate 

Surface. 

Inlet. 

Outlet. 

Diam. of 
Pulley. 

Face of 
Pulley. 

Revolu¬ 

tions. 

Air per 
Minute. 

IP. 

No. 

Sq. Feet. 

Diam. Ins. 

Diam. Ins. 

Ins. 

Ins. 

Per Min. 

Cube Feet. 


OO 

5 

5 

4 

2-75 

2 

3000 

500 

— 

O 

6 

5-75 

4-75 

3 

2.25 

2600 

600 

— 

I 

8 

6-5 

5-75 

3-5 

3 

2200 

764 

.6 

2 

IO 

7-5 

7-5 

3-75 

3-5 

1928 

I OI9 

•79 

3 

14 

9 

9 

4-25 

4 

1638 

1 427 

I .11 

4 

20 

10.5 

10.5 

5 

5 

1410 

1936 


5 

27 

12 

12 

6 

5-23 

1194 

2 7OI 

2. I 

6 

36 

M 

14 

7 

6-5 

1018 

3669 

2.86 

7 

48 

16 

l6 

9 

7-5 

878 

4847 

3-77 

8 

62 

18 

l8 

10 

8-5 

766 

6115 

4.76 

9 

80 

21 

21 

12 

10.5 

671 

8 154 

6-35 

IO 

IOO 

24 

24 

*4 

12 

59 8 

10 702 

8.34 


* 40 feet would have afforded economy in fuel. 




















ELEMENTS OF MACHINES AND ENGINES. 


COTTON FACTORIES. (English.) 

For driving 22060 Hand-mule Spindles, with Preparation, and 260 Looms , 

wtf/t common Sizing. 

Engine (coudensing). — Cylinder, 37 ins. in diam. by 7 feet stroke of piston; 
volume of piston space, 53.6 cube feet. 

Pressure of Steam .—(Indicated average) 16.73 lbs. per sq. inch. Revolutions, 17 
per minute. 

Friction of Engine and Shafting. —(Indicated) 4.75 lbs. per sq. inch of piston. 

IIP, 125. Total power = 1. Available, deducting friction = .717. 

f 305 hand-mule spindles, with preparation, 

Notes.—E ach IIP will drive J or 230 self-acting “ ‘‘ 

| or 104 throstle “ “ 

(_ or 10.5 looms, with common sizing. 

Including preparation: 

1 throstle spindle = 3 hand-mule, or 2.25 self-acting spindles. 

1 self-acting spindle = 1.2 hand-mule spindles. 

DREDGING MACHINES. 

Dredging 20 Feet from Water-line, or 180 Tons of Mud or Silt per Hour 

11 Feet from Water-line. 

Length upon deck, 123 feet; beam , 26 feet. Breadth over all, 41 feet. 

Immersed section at load-line, 60 sq. feet. Displacement, 141 tons, at load-draught 
of 2.83. feet. 

Engine (non-condensing).— Cylinders, two, 12.125 ' us - > n diam. by 4 feet stroke 
of piston. 

Boilers .—Two (cylindrical flue), diam. 40.5 ins., and length, 20 feet 3 ins.; two 
flues, 14.625 ins. in diam. Heating surface, 617 sq. feet. Grates, 37 sq. feet. 

Pressure of Steam, 25 lbs. per sq. inch; throttle .25 open, cut off at .5 the stroke 
of piston. Revolutions, 42 per minute. 

Buckets .—Two sets of 12, 2.5 feet in length by 15 ins. at top and 2 feet deep; vol¬ 
ume, 6.25 cube feet. Chain Links , 8 ins. in length by .5 inch diam. 

Scows or Camels. —Four, of 40 tons capacity each. 

STEAM HOPPER DREDGER. (Wm. Simons Co.) 

Iron. 

“Neptune” (English). — Length, 150 feet; breadth, 32 feet. 

Dredge from 6 Ins. to 25 Feet. Capacity of Hopper, 500 to 600 Tons. 
Engines.—Two (compound), 375 BP, for dredging and propulsion, and one for 
raising bucket-frame and anchor-posts. 

A like designed dredger of iooq tons capacity has dredged 10000 tons silt per 
week and transported it 7.5 miles. 

Dredging 400 Tons of Mud or Silt per Hour, 5 to 35 Feet in Depth 
Capacity of Hopper, 1300 Tons. 

Engines .—Two (compound), IP 700. Speed. —7.5 knots per hour. 

Steam Dredging Crane. (English.) 


Lift, 30 Feet per Hour. 


'be § 

So 

^■3 

Lifting 

Power. 

Volume 

of 

Bucket. 

Mud or 
Silt. 

T 3 

S H 3 
*3 3 

0 C/2 

0 

Excava¬ 

tion 

Ground. 

Weight 
of Crane. 

Lifting 
j Power. 

Volume 

of 

Bucket. 

Mud or 
Silt. 

Coal and 
Sand. 

| Excava¬ 
tion 

! Ground. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Tons. 

C. Yds. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Tons. 

C. Yds. 

21 280 

2.5 

1120 

25 

20 

20 

18 000 

5 

2240 

50 

40 

30 

24640 

3 

1680 

37-5 

3 2 

25 

33 4 So 

7 

33 6 ° 

60 

54 

40 




















900 


ELEMENTS OF MACHINES AND ENGINES, 


Iron. 

Dredger and Hopper Barge (Compound ; English). — Length, extreme, 120 feet; 
beam , 32 feet; hold, 10.5 feet. ^Breadth of bucket well, 6.75 feet. Load-draught, 6 feet. 

Cylinders. —21.5 and 40 ins. in diam. by 2.5 feet stroke of piston. Condenser. — 
Surface, 600 sq. feet. Circulating rump, single acting.—15 ins. in diam. by 15 ins. 
stroke of piston. 

Boiler. — Heating surface, 1150 sq. feet. Grates, 40 sq. feet. Steam-room, 300 
cube feet. 

Shaft.— 7.75 ins. in diam. 

Bucket ladder. — Of wrought iron, 74 feet in length, 5 feet in depth at centre, and 
2 feet 2 ins. at ends. Buckets. —34; volume, 15 cube feet each. 

Excavation and Delivery. —For a transit of 7.5 miles, 3000 tons per day. 

Hopper Barge.— Length between perpendiculars, 115 feet; beam, 32 feet; 
hold, 9 feet n ins. Load-line with 400 Tons dredge, 8 feet. 

Hoppers Length, 50 feet; breadth at top , 22 feet; at bottom, gfeet. 

Cost. — Dredge, $ 90000; Hoppers, $ 18 000 each. 

Maintenance. — Dredger, 1.75 cents; Hopper, 1.7 cents; Towing, 1.2 cents, per ton 
of dredge excavated and delivered. 

“Hercules,” Panama Canal. — Length on deck, 100 feet; beams, 40, 60, and 45 
feet; depth of hold, 12 feet. Slot, 36 feet in length by 6 feet 7 ins. in width. 

Ways. —Two. one 40 feet and one 60 feet, by 5 feet in width. 

Buckets. —38; volume, 1.33 cube yards. Spuds, 2 feet in diam. and 60 in length. 

Engines. —Two of 100 IP each, and two of 40 IP each. 

Boilers. —Three (horizontal tubular), 16 feet in length. 

Elevator and Discharge. —Maximum, 24 cube yards per minute. 

Crane. ('W ood.) 

Hull.—Length on deck, 100 feet; beam, 44 feet; load-draught, 4.5 feet. 

Radius of crane, 46 feet; height, 70 feet; counter-balance , 70 tons. 

Boiler. — Heating surface, 500 sq. feet. Pressure of Steam, 80 lbs. per sq. inch. 
HP, 150. 

Propellers.— Two, 4.25 feet in diam. Speed, 5 miles per hour. 

Engine to operate crane. Cylinder. —10 ms. in diam. by 12 ins. stroke of inslon. 

FLOUR MILLS. 

30 Barrels of Flour per Hour. 

Water-wheels, Overshot.— 5, diam. 18 feet by 14.5 feet lace. Buckets, 15 
! ns. in depth. Water.—Head, 2.5 feet. Opening, 2.5 ins. by 14 feet in length over 
each wheel. 

5 Barrels of Flour per Hour , and Elevating 400 Bushels of Grain 36 Feet. 

Water-wheel, Overshot. — Diam. 22 feet bj r 8 feet face. Buckets, 52 of 1 
foot in depth. Water. — Head, from centre of opening, 25 ins. Opening, 1.75 ins. 
by 80 ins. in length. 

Revolutions, 3.5 per minute. Stones, three of 4.5 feet; revolutions, 130. 

Three Run of Stones, Diameter 4 Feet. 

"Water- wlieel. Overshot .— Diam. 19 feet by 8 feet face. Buckets, 14 ins. in 
depth. 

Or, 

Steam-engine (non-condensing).— Cylinder, 13 ins. in diam.by 4 feet stroke. 

Boiler (cylindrical flued).—Diam. 5 feet by 30 in length; two flues 20 ins. in d : am. 


ELEMENTS OF MACHINES AND ENGINES. QOI 


HOISTING ENGINES. 

For 3 ?ile Driving, Hoisting, NLining, etc. 


Lidgerwood IMariLif’g Co., IN"ew York. 



Single Cylinders. 



Double Cylinders. 










FP 

Cylinder. 

Capacity. 

Boiler.* 

IP 

Cylinder. 

Capacity. 

Boiler.* 

No. 

Ins. 

Lbs. 

$ 

No. 

IllS. 

Lbs. 

$ 

4 

5X5 

IOOO 

600 

8 

5 X 8 

2000 

950 

6 

6X8 

1250 

675 

12 

6 X 8 

2500 

1050 ■ 

IO 

7 X 10 

1800 

825 

20 

7 X 10 

3500 

1350 

i 5 

8 X 10 

2800 

1050 

30 

8 X 10 

6000 

1550 

20 

9 X 12 

4000 

1275 

40 

9 X 12 

8000 

2000 

25 

10 X 12 

5000 

I 375 

50 

10 X 12 

9000 

2350 


* Complete. 


Details and. Operation. 




Boiler. 


1 

Leaders. 

Lift. 

Blows 

Piles 

Fuel 

Engine. 

Drum. 

Dimen¬ 

sions. 

Tubes. 

Ram. 

Hoist. 

Ram. 

per 

Minute. 

per 10 
Hours. 

per 

Hour. 

IP 

Ins. 

Ins. 

No. 

Lbs. 

Feet. 

Feet. 

No. 

No. 

Lbs. 

10* 

12 X 24 

32 X 75 

48 of 2 in. 

1953 

40 

8 to 12 

25 

50 

70 

20 

14 x 26 

40 X 84 

80 of 2 in. 

2700 

75 

8 to 11 

29 

IOO 

80 


* Weight complete, 8500 lbs. 

NEining Engines and Boilers. [Various Capacities.) 
Engine , Boiler, etc., as given for Pile Driving, page 902. 

Operation. — 250 to 300 tons of coal in 10 hours. Fuel, 40 lbs. coal per hour. 
Water , 20 gallons per hour. 

Weight of Engine and Boiler , 4500 lbs. 


Hancock Inspirator. For a Lift of Water of 25 Feet. 


No. 

Diameter. 

Discharge 
at Pressure 

No. 

Diameter. 

Discharge 
at Pressure 

Steam-pipe. 

Suction. 

of 60 Lbs. 

Steam-pipe. 

Suction. 

of 60 Lbs. 


Ins. 

Ins. 

G’lls.perh’r. 


Ins. 

Ins. 

G’lls.perh’r. 

IO 

•375 

•5 

120 

30 

1.25 

1-5 

1260 

12.5 

•5 

•75 

220 

35 

1.25 

1-5 

1740 

15 

•5 

•75 

300 

40 

1-5 

2 

2230 

20 

•75 

I 

540 

45 

1-5 

2 

2820 

25 

I 

1.25 

900 

5o 

2 

2-5 

3480 


Temperature of water not over 145 0 for a low lift, and ioo° for a high lift. 


HYDROSTATIC PRESS. {Cotton.) 

30 Bales of Cotton per Hour. 

Engine (non-condensing).— Cylinder , 10 ins. in diam. by 3 feet stroke of piston. 

Pressure of Steam, 50 lbs. per sq. inch, full stroke. Revolutions, 45 to 60 per 
minute. 

Presses. —Two, with 12-inch rams; stroke, 4.5 feet. 

Pumps. —Two, diam. 2 ins.; stroke, 6 ins. 

For 83 Bales per Hour. 

Engine (non-condensing).— Cylinder, 14 ins. in diam. by 4 feet stroke of piston. 

Boilers. —Three (plain cylindrical), 30 ins. in diam. and 26 feet in length. Grates , 
32 sq. feet. Pressure of Steam , 46 lbs. per sq. inch. Revolutions, 60 per minute. 

Presses. —Four, geared 6 to 1, with two screws, each of 7.5 ins. in diam. by 1.625 
in pitch. 

Shaft (wrought iron). —Journal, 8.5 ins. Fly Wheel, 16 feet in diam.; weight, 
8960 lbs. 



















































ELEMENTS OF MACHINES AND ENGINES. 


LOCOMOTIVE. 

“ Experiment ” {Compound). — Cylinders, one each, 12 and 26 ins. in diam., and 
one 26 ins. by 2 feet stroke of piston. 

Boiler.—Heating surface, 1083.5 sq. feet. Grate, 17.1 sq. feet. Pressure of Steam, 
150 lbs. per sq. inch, cut off at .35. Speed , 50 miles per hour. Weight. —Empty, 
34.75 tons. 

Street Railroad or Tramway Engine. 

Cylinder , 7 ins. in diam. by n ins. stroke of piston. 

Boiler , 78 tubes 1.75 ins. in diam. by 4 feet in length. Heating surface, 160 sq. 
feet. Grate, 4.25 sq. feet. W/ieels, 2.33 feet in diam, Base, 4.5 feet. Gauge , 4 
feet 8.5 ins. 

Cost.— Average per mile in England, 2.52 pence sterling = 4.48 cents. 

PILE-DRIVING. 

Driving One Pile. 

Engine (non-condensing).— Cylinder , 6 ins. in diam. by 1 foot stroke of piston. 

Boiler (vertical tubular).—32 ins. in diam., and 6.166 feet in height. Grates, 3.7 
sq. feet. Furnace, 20 ins. in height. Tubes , 35, 2 ins. in diam., 4.5 feet in length. 

Revolutions, 150 per minute. Drum, 12 ins. in diam., geared 4 to 1. Leader, 40 
feet in height. Ram .—2000 lbs., 2 blows per minute. Fuel, 30 lbs. coal per hour. 

Driving Two Piles. 

Engine (non-condensing).— Cylinders, two, 6 ins. in diam. by 18 ins. stroke of 
piston. 

Boiler (horizontal tubular).— Shell, diam. 3 feet, and 6 feet in length. Furnace 
end 3.75 feet in width, 3.5 feet in length, and 6 feet in height. 

Pressure of Steam, 60 lbs. per sq. inch. Revolutions, 60 to 80 per minute. 

Frame, 8.5 feet in width by 26 feet in length. Leaders, 3 feet in width by 24 feet 
in height. Rams. —Two, 1000 lbs. each, 5 blows per minute. 

PUMPING ENGINES. 

Corliss Steam-engine Co., Providence, R. I. —Vertical - Beam Engine (Com¬ 
pound).— Cylinders .—18 and 36 ins. in diam. by 6 feet stroke of piston. 

Pumps .—Four plunger, 19 ins. in diam. by 3 feet stroke of piston. Displacement 
per revolution of engine, 84.96 cube feet. 

Boilers. —Three, vertical lire tubular. Grate .—93 sq. feet. Heating surface, 1680 
sq. feet. Pressure of Steam. 125 lbs. per sq. inch, cut off at .22 feet. Revolutions, 
36 per minute. IH? 313. Fly-wheel .—25 feet in diam., weight 62000 lbs. 

Fuel .—Cumberland coal, 486 lbs. per hour, inclusive of kindling and raising steam. 
Ash and Clinkers, 9.4 per cent. Duty for one week, 113 271 000 foot-lbs. 

Water delivered, 17 621 gallons per minute, against head of 180 feet. 

Duty, average for 1883, per 100 lbs. anthracite coal, 106 048 000 foot-lbs. 

For Elevating 200000 Gallons of Water per Hour. 

Lynn, Mass. —Engine (Compound). — Cylinders, 17.5 and 36 ins. in diam. by 7 feet 
stroke of piston; volume of piston space, 61.2 cube feet. Air Pump (double act¬ 
ing), 11.25 ' ns - hi diam. by 49.5 ins. stroke of piston. 

Pump Plunger, 18.5 ins. in diam. by 7 feet stroke. 

Boilers .—Two (return flued), horizontal tubular; diam. of shell, 5 feet; drum, 3 
feet; tubes, 3 ins. Length of shell, 16 feet. Grates, 27.5 sq. feet. 

Pressure of Steam, 90.5 lbs.; average in high-pressure cylinder, 86 lbs., cut off at 
1 foot, or to an average of 44.5 lbs.; average in low-pressure cylinder, 27 lbs., cut 
off at 6 ins., or to an average of 10.8 lbs. 

Revolutions, 18.3 per minute. Fly Wheel. —Weight, 24000 lbs. 

Evaporation of Water, 4644 lbs. per hour. Loss of action by Pump , 4 per cent. 

Consumption of Coal. —Lackawanna, 291 lbs. per hour. 

Duty, 205 772 gallons of water per hour, under a load and frictional resistance of 
73.41 ibs. per square inch, equal to 103 923 217 foot-lbs. for each 100 lbs. of coal. 


ELEMENTS OF MACHINES, MILLS, ETC. 


903 


“ Gaslall,” at Saratoga , N. Y. 

Engine ( Horizontal Compound). Cylinders.— High pressure, 2 of 21 ins. diam. 
Low pressure, 2 of 42 ins. diam., all 3 feet stroke of piston. Pumps. —Two of 20 ins. 
diam. by 3 feet stroke of piston. 

Fly Wheel, 12.33 feet in diam.; weight, 12000 lbs. 

Boilers (horizontal tubular).—Two of 5.5 feet in diam. by 18 feet in length. Heat¬ 
ing surface , 2957 sq. feet. Grates , 51 sq. feet of grate; to heating surface, 1 to 58, 
and to transverse section of tubes, 1 to 7. Chimneys , 75 feet. 

Pressure of Steam. —Mean of 20 hours, 74.25 lbs. per sq. inch. Revolutions , 17.87 
per minute. HP.—High-pressure cylinders, 109.2; low-pressure, 76.65. Total, 185.8. 

Fuel. —Anthracite, 6.9 lbs. per sq. foot of grate per hour. Evaporation, per sq. 
foot of heating surface per hour, 1.175 lbs.; per lb. of coal, 9.25 lbs.; per cent, of 
non-combustible, 3.2. 

Duty, 1x2 899993 foot-lbs. per 100 lbs. coal. Heating surface per HP, 14.9. 

Steam per sq. foot of surface per hour, 1.19 lbs.; per sq. foot of surface per lb. of 
coal per hour from 212 0 , 11.28 lbs. 


Ericsson’s Caloric. For an Elevation of 50 Feet. 


Dimen- 

Space 

Volume 

Pipes, 

Suction 

Fuel 

per Hour. 



COST. 

Deep Well Pump. 

occupied. 

per 

and 

Furnace. 


Extra. 


sions. 



Hour. 

Dis- 

Nut 




Pump. 

Pipes per Foot. 


Floor. 

Height. 


charge. 

Anthr. 

Gas. 

Gas. 

Coal. 

Plain. 

Galvan. 

Ins. 

Ins. 

Ins. 

Gall. 

Ins. 

Lbs. 

Cub. ft. 

$ 

$ 

$ 

$ 

$ 

5 

34 Xi 8 

48 

150 

•75 

— 

15 

150 

— 

— 



6 

39X20 

51 

200 

•75 

2-5 

l8 

200 

210 

— 

— 

— 

8 

48X21 

63 

350 

I 

3-3 

25 

235 

250 

IO 

.64 

,86 

12 

54 X 27 

63 

800 

i -5 

6 

— 

— 

320 

15 

.80 

i-i 5 

12* 

42X52 

65 

1600 

2 

12 

— 

— 

450 

25 t 

.92 

1.25 


* Over 90 feet, 92 cents. + Duplex. 


Including engine and pump, oil-can and wrench, complete in all but suction and 
discharge-pipe. 

SUGAR MILLS. 

Expressing 40000 lbs. Cane-juice per day, or for a Crop of 5000 Boxes of 
450 lbs. each in four Months' Grinding. 

Engine (non-condensing).— Cylinder, 18 ins. in diam. by 4 feet stroke of piston. 
Boiler (cylindrical flued).—64 ins. in diam. and 36 feet in length; two return flues, 
20 ins. in diam. Heating surface, 660 sq. feet. Grates, 30 sq. feet. 

Pressure of Steam, 60 lbs. per sq. inch, cut off at .5 the stroke of piston. Revolu¬ 
tions, 40 per minute. 

Rolls. — One set of 3, 28 ins. in diam. by 6 feet in length; geared 1 to 14. Shafts, 
ii and 12 ins. in diam. Spur Wheel, 20 feet in diam. by 1 foot in width. Fly 
Wheel, 18 feet in diam.; weight, 17 40x1 lbs. 

Weights. —Engine, 61 460 lbs.; Sugar Mill, 65 730 lbs.; Spur Wheel and Connect¬ 
ing Machinery to Mill, 28 680 lbs.; Boiler, 18 520 lbs.; Appendages, 6730 lbs. Total, 
181120 lbs. 


STONE AND ORE BREAKERS. (Blake’s.) 


No. 

Re¬ 

ceiver. 

Pul 

D’m. 

ley. 

Face. 

V’locity 

per 

Minute. 

Power 

re¬ 

quired. 

Weight. 

No. 

Re¬ 

ceiver. 

Pul 

D’m. 

ley. 

Face. 

V’locity 

per 

Minute. 

Power 

re¬ 

quired. 

Weight 


Ins. 

Feet. 

Ins. 

Feet. 

IP. 

Lbs. 


Ins. 

Feet. 

Ins. 

Feet. 

HL 

Lbs. 

A 

4X10 

1.66 

6 

250 

4 

4 000 

5 

9 X 15 

2-5 

9 

250 

9 

13360 

I 

5 Xio 

2-75 

6 

180 

5 

6 700 

6 

11X15 

2-33 

6 

180 

9 

11 600 

2 

7X10 

2 

7-5 

250 

6 

8 000 

7 

I 3 XI 5 

2-33 

8 

180 

9 

11 760 

3 

5 Xi 5 

2-33 

8 

180 

9 

9 100 

8 

15X20 

3-5 

IO 

150 

12 

32 600 

4 

7 Xi 5 

2-33 

9 

180 

9 

IO 49O 

9 

18X24 

6 

12 

125 

12 

37 5oo 


Note.— Amount of product depends on distance jaws are set apart, and speed. 
Product given in Table is due when jaws are set 1.5 ins. open at bottom, and ma¬ 
chine is run at its proper speed and diligently fed. It will also vary somewhat with 
character of stone. Hard stone or ore will crush faster than sandstone. 

A cube yard of stone is about one and one third tons. 












































904 


ELEMENTS OF MACHINES.-CHIMNEYS, 


STEAM FIRE-ENGINE. 

Amoskeag, IN'. II. 1 st Class. 

Steam Cylinder. —Two of 7.625 ins. in tliam. by 8 ins. stroke of piston. 

Water Cylinder. —Two of 4.5 ins. in diam. 

Boiler (vertical tubular).— Heating surface, 175 sq. feet. Grates , 4.75 sq. feet. 
Pressure of Steam. —100 lbs. per sq. inch. Revolutions, 200 per minute. 
Discharges.— Two gates of 2.5 ins., through hose, one of 1.25 ins. and two of 1 inch. 
Projection. — Horizontal, 1.25 ins. stream, 311 feet; two 1 inch streams, 256 feet. 
Vertical, 1.25 ins. stream, 200 feet. Water Pressure.— With 1.125 ins. nozzle, 200 lbs. 
Time of Raising Steam. —From cold water, 25 lbs., 4 min. 45 sec. 

Weights. —Engine complete, 6000 lbs.; water, 300 lbs. 

SAW-MILL. 

Two Vertical Saws , 34 Ins. Stroke, Lathes, etc. 

Engine (non-condensing). Cylinder. —ioins. in diam. by 4 feet stroke of piston. 
Betters. —Three (plain cylindrical), 30 ins. in diam. by 20 feet in length. 

Pressure of Steam. —90 lbs. per sq. inch. Revolutions , 35 per minute. 

Note.—T his engine has cut, of yellow-pine timber, 30 feet by 18 ins. in 1 minute. 

STONE SAWING. 

Emerson Stone Saw Co. (Diamond Stone Saw, Pittsburgh, Penn.).— 
20 IP, 150 sq. feet of Berea sandstone, inclusive of both sides of cut, in 1 hour. 

CHIMNEYS. 

Lawrence, Mass. Octagonal, 222 Feet above Ground, and 19 Feet below. 
Foundation , 35 Feet square and of Concrete 7 Feet deep. (Hiram F. Mills.) 

Shaft.— 234 feet in height, 20 feet at base, and 11.5 at top; 28 ins. thick at base 
and 8 at top. Core. —2 feet thick for 27 feet, and 1 foot for 154. 

Horizontal Flues. — 7.5 feet square, and Vertical flue or cylinder of 8.5 feet, 234 
high, with walls 20 ins. thick for 20 feet, 16 for 17 feet, 12 for 52 feet, and 8 for 145 feet.. 
Purpose .—For 700 sq. feet grate surface. Weight. —2250 tons. Briclcs, 550000. 

New York Steam Heating Co. Quadrilateral, 220 Feet above Ground 
and 1 Foot below. ( Chas. E. Emery, Ph D.) 

Shaft..— 220 feet in height, and 27 feet 10 ins. by 8 feet 4 ins. in the clear inside. 
Foundation.— 1 foot below high water. Capacity. —Boilers of 16000 IP. 


Cost of Steam-Engines and Boilers complete, and of 
Operation per Bay- of IO Hours, inclusive of Labor, 
Enel, and Repairs. (Chas. E. Emery, Ph.D.) 


IIP. 

Engine. 

Water 

orate( 

IIP per 
Hour. 

Evap- 

per 

Lb. of 
Coal. 

Coa 

IIP. 

1 per 

Day. 

Labor. 

Sup¬ 
plies 
and Re¬ 
pairs. 

Cost 

of 

Coal.* 

Total 
Cost of 
Operat’n, 
including 
Coal. 



Lbs. 

Lbs. 

Lbs. 

Lbs. 

$ 

$ 

$ 

$ 

6.25 

Portable Vertical! ,, ti 

42 

7-5 

56 

394 

i -75 

•33 

•73 

2.86 

12.5 

U U ) o'CS 

3 8 

7-5 

51 

717 

i -75 

.41 

1-33 

3-56 

29 

Horizontal.(** 8 

32 

8 

40 

1 308 

2.25 

.60 

2 -43 

5-45 

112 

Single Condensing... 

23 

8.8 

26.1 

3 3 oo 

3-75 

1.17 

6.14 

11.66 

276 

it U 

22. 2 

8.8 

25.2 

7 8 3 i 

4-25 

2. 12 

14.58 

22.27 

552 


22.2 

8.8 

25.2 

15663 

6 

4.02 

29.16 

4 i -52 


* $ 4.42 per ton (2240 lbs.), including cartage. 



















GRAPHIC OPERATION. 


905 


GRAPHIC OPERATION. 

Solutions of QtxestioiiS by a Grraplaic Operation. 

1. If a man walks 5 miles in 1 hour, how far will he walk in 4 hours? 

Operation. —Draw horizontal line, divide it into equal parts, 
as 1, 2, 3, and 4, representing hours. From each of these 
points let fall vertical lines AC, u, etc., and divide A C into 
miles, as 5, 10, 15, and 20, and from these points draw equi¬ 
distant lines parallel to the horizontal. 

Hence, the horizontal lines represent time or hours, and 
the vertical, distance or miles. 

Therefore, as any inclined line in diagram represents both 
time and distance, course of man walking 5 miles in an hour 
is represented by diagonal Ae; and if he walks for 4 hours, 
continue the time to 4, and read off from vertical line A C the distance = 20 miles. 

2. How far will a man walk in 2 hours at rate of 10 miles in 1 hour ? 

His course is shown by the line A 0, representing 20 miles. 

3. If two men start from a point at the same time, one walking at the 
rate of 5 miles in an hour and the other at 10 miles, how far apart will they 
be at the end of 2 hours ? 

Their courses being shown by the lines A r and A 0, the distance r 0 represents 
the difference of their distances, io A. 20 = 10 miles. 

4. How long have they been walking? 

Their courses are now shown by the lines A 0 and A 4, the distance 2 4 represents 
the difference of their times, or 2 4 = 2 hours. 

5. When they are 10 miles apart, how long have they been walking? 

Their courses are again shown by the lines A r and A 0, the distance r 0 repre¬ 
sents the difference of their distances of 10 miles, and A 2, 2 hours. 

6 . If a man walks a given distance at rate of 3.5 miles per hour, and then 
runs part of distance back at rate of 7 miles, and walks remainder of dis¬ 
tance in 5 minutes, occupying 25 minutes of time in all, how far did he run ? 

C Operation.— Draw horizontal line, as A C, 
representing wdiole time of 25 minutes; set 
off point e representing a convenient fraction 
of an hour (as 10 minutes), and a i equal to 
corresponding fraction of 3.5 miles (or .5833); 
draw diagonal A n, produced indefinitely to 0 , 
and it will represent the rate of 3.5 miles per 
hour. 

Set off C r equal to 5 minutes, upon same 
scale as that of A C; let fall vertical r s, and 
draw diagonal C u at same angle of inclination 
as that of An; then from point u draw diagonal u 0 , inclined at such a rate as to 
represent 7 miles per hour; thus, if in represents rate of 3.5 miles, s 0, being one 
half of the distance, will represent 7 miles. 

The whole distance between the two points Is thus determined by C x, and dis¬ 
tance ran by u s, measured by scale of miles employed. 

Verification. —The distances A e and A i are respectively 10 minutes =. 166 of an 
hour, and .5833 mile = .166 of 3.5 miles. Hence, Ca; = .875 mile, and us = .5833 
mile. Consequently, the man walked A 0^.875 mile — 15 minutes, ran 0 u — 
.5833 mile = 5 minutes, and walked u C = .2916 mile. 

7. If a second man were to set out from C at same time the man referred 
to in preceding question started from A, and to walk to A and return to C, 
at a uniform rate of speed and occupying same time of 25 minutes, at which 
points and times will he meet the first man ? 

Operation. —As A C represents whole time, and C x distance between the two 
points, v z and 2 x will represent course of second man walking at a uniform rate, 
and he will meet the first man, on his outward course, at a distance from his start¬ 
ing-point of A, represented by A 0, and at the time A a; and on his return course 
at distance A v, x m. and at the time A c. 

4 G* 
















go6 


MISCELLANEOUS 


MISCELLANEOUS. 


No . 9 Diameter, and Number of Sliot. (American Standard.) 


Compressed Bvich: Sliot. 


No. 

Diam. 

Shot 
per Lb. 

No. 

Diara, 

Shot 
per Lb. 

No. 

Diam. 

Shot 
per Lb. 


Inch. 

No. 


Inch. 

No. 


Inch. 

No. 

3 

•25 

284 

1 

•3 

173 

00 

•34 

H 5 

2 

.27 

232 

0 

•32 

140 

000 

•36 

98 


Balls, .38 Inch, 85 No. per lb.; .44 Inch, 50 No. per lb. 


Chilled Shot. 


No. 

Diam. 

Shot 
per Oz. 

No. 

Diam. 

Shot 
per Oz. 

No. 

Diam. 

Shot 
per Oz. 

No. 

Diam. 

Shot 
per Oz. 


Inch. 

No. 


Inch. 

No. 


Inch. 

No. 


Inch. 

No. 

12 

•PS 

2385 

9 

.08 

585 

6 

.11 

223 

1 

.16 

73 

11 

.06 

1380 

8 

Trap 

495 

5 

.12 

172 

B 

•*7 

61 

10 

Trap 

1130 

8 

.09 

4°9 

4 

•13 

136 

BB 

.18 

52 

10 

•°7 

868 

7 

Trap 

345 

3 

.14 

109 

BBB 

.T9 

43 

9 

Trap 

7l6 

7 

. 1 

299 

2 

•15 

85 





Drop Sliot. 


No. 

Diam. 

Pellets 
per Oz. 

No. 

Diam. 

Pellets 
per Oz. 

No. 

Diam. 

Pellets 
per Oz. 

No. 

Diam. 

Pellets 

perOz. 


Inch. 

No. 


Inch. 

No. 


Inch. 

No. 


Inch. 

No. 

Extra Fine Dust 

.015 

84 021 

9 

Trap 

688 

5 

.12 

H 

o\ 

00 

BBB 

.19 

42 

Fine Dust 

•03 

10 784 

9 

.08 

568 

4 

•13 

132 

T 



Dust 

.04 

4565 

8 

Trap 

472 

3 

• J 4 

106 


. 2 

3° 

12 

•05 

2 326 

8 

.O9 

399 

2 

•15 

86 

TT 

. 21 

3 1 

11 

.06 

1 34 6 

7 

Trap 

338 

1 

.16 

7 1 

y 



10 

Trap 

I O56 

7 

. I 

29I 

B 

• *7 

59 


. 22 

27 

10 

•°7 

848 

6 

. II 

2l8 

BB 

.18 

50 

FF 

•23 

24 


The scale of the Le Roy standard (adopted by the Sportsman’s Convention) com¬ 
mences with .21 inch for TT shot, and reduces .01 inch for each size to .05 inch for 
No. 12. The number of pellets per oz. being the actual number in perfect shot. 

The number of pellets by this standard is nearly identical with that of the Amer¬ 
ican Standard. 

Tatham’s scale is same as Le Roy’s, but number of pellets is deduced mathemat¬ 
ically, by computing them from the specific gravity of the lead. 


Drains, Diameter and Grade of, to Discharge Rainfall. 


Diam. 

Grade 

1 Inch. 

Acres. 

Diam. 

Grade 

1 Inch. 

Acres. 

Diam. 

Grade 

1 Inch. 

Acres. 

Diam. 

Grade 

1 Inch. 

Acres. 

Ins. 



Ins. 



Ins. 



Ins. 



4 

30 

•5 


40 

1.2 


60 

2.1 


80 

5-8 


20 

.6 


20 

x -5 

9 

120 

2.1 

J 5 

240 

7.8 

5 

80 

•5 

7 

20 

1.2 


80 

2.5 


120 

7.8 


60 

.6 


60 

x -5 


60 

2-75 


80 

9 


20 

I 

8 

120 

i -5 

12 

120 

4-5 


60 

10 

6 

60 

I 


80 

1.8 


80 

5-3 

18 

240 

10 


British and IVEetric Measures, Commercial Equivalents 

of. ( G . Johnstone Stones , F. R. S.) 


Length. Millimeters. 

Inch. 914.4 

Foot. 304.8 

Yard. 25.4 


Weight. Grammes. 

Pound. 453.6 

Ounce. 28.35 

Grain.0648 


Volume. Cube Centimeter. 

Gallon. 4554 

Quart. 1136 

Ounce. 28.4 















































































































MEMORANDA, 


907 


MEMORANDA. 

[Physical and. [Mechanical Elements, Constructions, 

and Results. 

Belting. Double. — 600 IP (to be transmitted) - 4 - velocity of belt in feet per 
minute, or 191 H? 4 - number of revolutions per minute X diameter of pulley in feet 
= width in ins. Machine Belts. —1500 to 2000 IP 4-velocity of belt in feet per 
minute width in ins. (Edward Sawyer.) 

Blast Pipe of a Locomotive. Best height is from 6 to 8 diameters 
of pipe, and best effect when expanded to full diam. of pipe at 2 diameters from base. 

Boiler Riveting. A riveting gang (2 riveters and 1 boy) will drive in shell, 
furnace, etc., a mean of 12.5 rivets per hour. 

Brick or Compressed Fuel is composed of coal dust agglomerated 
by pitchy matter, compressed in molds, and subjected to a high temperature in an 
oven, in order to expel the moisture or volatile portion of the pitch and any fire¬ 
damp that may exist in the cells of the coal. 

Bridge, Highest. At Garabil, France, 413 feet from floor to surface of water, 
and 1800 feet in length. 

Bronze, JVlallea'ble. P. Dronier, in Paris, makes alloys of copper and 
tin malleable by adding from .5 per cent, to 2 per cent, quicksilver. 

Bu-ilcling Department, Requirements of. (New York.) 

Furnace Flues of Dwelling Houses hereafter constructed at least 8-inch walls on 
each side. The inner 4 ins. of which, from bottom of flue to a point two feet above 
2d story floor, built of fire-brick laid with fire-clay mortar; and least dimensions of 
furnace flue 8 ins. square, or 4 ins. wide and 16 ins. long, inside measure; and when 
furnace flues are located in the usual stacks, side of flue inside of house to which it 
belongs may be 4 ins. thick. If preferred, furnace flues may be made of fire-clay 
pipe of proper size, built in the walls, with an air space of 1 inch between them, 
and 4 ins. of brick wall on outside. 

Boiler Flues to be lined with fire-brick at least 25 feet in height from bottom, 
and in no case walls of said flues to be less than 8 ins. thick. 

All flues not built for furnaces or boilers must be altered to conform to the above 
requirements before they are used as such. 

Buildings, Protection of, from Lightning. A wire rope of 
4 lbs. per yard is held to be the most efficient. 

Single Conductors, weighing 8 lbs. per yard and 4 lbs. for duplicated and all otheis, 
may be located 50 feet apart, thus bringing every portion of the building to which 
they are applied within 25 feet of their protection. 

Iron is the best material for a conductor; it should be continuous, and all joints 
soldered Several points are preferable to one, and greater surface should be given 
to connections with the earth than usually practised. (Sir W. Thompson.) 

For other information, see Van Noslrand's Magazine , N. Y., Aug. 1882, page 154. 

Cement Iron to Stone.— Fine iron filings, 20 parts, Plaster of Paris, 60, and 
Sal Ammoniac, 1; mixed fluid with vinegar, and applied forthwith. 

Chimnev Draught. W — w h = T>. W and w representing weights of a 
cube foot of air at external and internal temperatures, h height of chimney or pipe in 
feet, and D value of draught. See Weight of Air, page 521. 

Chinese or India Ink improves with age, should be kept in dry air, 
and in rubbing it down, the movement should be in a right line and with very little 
pressure. 



MEMORANDA. 


908 

Coal, Effective "Value of. Theoretical quantity' of heat per IP is 
2564 units per hour, and average quantity of heat in a lb. of coal that is utilized 
in the generation of steam in a boiler is 8500 units; hence, theoretical quautity 

of coal required per IP per hourr= = .3 lbs., after the water has been heated 

8500 

into atmospheric steam, being theoretically nearly 7.5 per cent, of total heat re¬ 
quired to change 30 lbs. water at 6o° into steam of 60 lbs. effective pressure. 

The total heat developed by the combustion of coal, when utilized evaporatively, 
ranges from .55 to .8, but in practice it does not exceed 65 per cent. 

Coast and. I 3 ay- Service. A velocity of current of 2.5 feet per second 
will scour and transport silt, and 5 to 6.5 feet sand. For river scour the velocities 
are very much less. 

Cold, Greatest. —220 0 , produced by a bath of Carbon, Bisulphide, and 
liquid Nitrous Acid. 

Corrosion of Iron and Steel. The corrosion of steel over iron is, 
as a mean, fully one third greater. 

Cost of Family of Mechanics in France ranges from $220 
to $600 per annum, of which clothing costs 16 parts, food 61, rent 15, and mis¬ 
cellaneous 8. 

Crushing Resistance of Briclu. A pressed brick of Philadelphia 
clay withstood a pressure of 560000 lbs. for a period of 5 minutes. 

Earthwork. Shovelling. —Horizontal, 12 feet. Vertical, 6 feet. When 
thrown horizontal, 12 to 20 feet, 1 stage is required, and from 20 to 30, 2 stages. 
When vertical, 6 to 10 feet, 1 stage is required. 

Wheelbarrow. —Proper distance up to 200 feet. 


Nnmloer of Loads and Volume of Earth per Day*. 
One Laborer. (C. Herschell , C. E.) 


Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Feet. 

No. 

Cub. Yds. 

Feet. 

No. 

Cub. Yds. 

Feet. 

No. 

Cub. Yds. 

20 

120 

23-5 

150 

96 

13-3 

350 

88 

11.6 

50 

no 

16.9 

200 

94 

12.8 

400 

86 

11.2 

70 

IOO 

14.4 

250 

9 2 

12-4 

450 

84 

10.9 

IOO 

98 

13.8 

300 

90 

12 

500 

82 

10.5 


Volume of a barrow load, 2.5 cube feet. 

Portable Railroad and Hand Cars .—For a distance of 550 feet, 60 cube yards can 
be transported per day. 


Horse Cart .—Volume of Eartli transported per Day. 

One Laborer. 


Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Feet. 

No. 

Cub. Yds. 

Feet. 

No. 

Cub. Yds. 

Feet. 

No. 

Cub. Yds. 

3 °° 

86 

17.1 

IOOO 

43 

8.6 

2000 

25 

5 

500 

67 

13.6 

1500 

3 i 

6.4 

2500 

21 

4-3 


Volume of each load, 8 cube feet. 

Ox Cart is less in cost at expense of time. 

Electric Eight, Candle Power of. Maxim Incandescent Lamp .— 
Current with 30 Faure cells, 74 volts, 1.81 Amperes, 16 standard candles. With 50 
like cells, 124 volts, and 3.2 Amperes, 333 candles. (Paget Hills , LL.D.) 

The elavated electric lights at Los Angeles, Cal., are distinctly visible at sea for a 
distance of 80 miles. 

Engine and Sugar IVIill, Weights of. Engine (non-condensing). 

— Cylinder. —30 ins. in diam. by 5 feet stroke of piston. Boilers (cylindrical flue)._ 

70 ins. in diam. by 40 feet in length. Weights.—Engine. 105000 lbs.; Boilers , com¬ 
plete, 75 000 lbs. ; Sugar-mill. 40 ins. by 8 feet, 220050 lbs.; Connecting Machinery, 
137 179 lbs. Cane carriers, etc., 46 787 lbs. 






































MEMORANDA. 


909 


Filtering Stone. Artificial. —Clay, 15 parts; Levigated Chalk, 1.5; and 
Glass Sand, coarse, 83.5. Mixed in water, molded, and hard burned. 

Fire-engine, Steam. Relative effect for equal cost compared with a 
hand engine, as 1 to 1x3. Each IIP requires about 112 weight of engine. 

Floating Booties, Velocities of. At low speeds resistance increases 
somewhat less than square of velocity. In a Canal, at a speed of 5 miles per hour, 
a large wave is raised, which at a speed of 9 miles disappears, and when speed is 
superior to that of the wave, resistance of boat is less in proportion to velocity, and 
immersion is reduced. 

Length of Vessel .—The proper length for a vessel in feet (upon the wave-line 
theory) is fifteen sixteenths of square of her speed in knots per hour. 

Flow of -A_ir. 67 y/h= Velocity per second X C. h representing column 
of water in ins., and C a coefficient ranging from 56 to 100. 

Circular orifices, thin plate..56 to .79 

Cylindrical mouth-pieces, short.81 “ .84 

do. do. rounded at inner end.. . 92“ .93 

Conical converging mouth-pieces.9 “ 1 

Conoida -1 mouth-piece, alike to contracted vein.97 “ 1 

Flues, Corrugated. (I Vm. Parker.) IOO ° —— = Working stress in 

lbs. per sq. inch. T representing thickness in 16 ths of an inch, and D diameter in ins. 

Steel, corrugations 1.5 ins. deep. Experiments upon a furnace 31.875 ins. in 
diam., 6.75 feet in length, and with 13 corrugations. 

Foundation Piles. When piles are driven to a solid foundation, they act 
as columns of support, and are designated Columns, and when they deriv" their 
supporting power from the friction of the soil alone, they are termed Piles. 

Authorities differ greatly as to the factor of safety for Piles, varying .1 to .01 of 
impact of ram. ( Weisbach. ) 

As columns, their safe load may be taken at from 750 to 900 lbs. per sq. inch. 
Authorities give a higher value (Rankine and Mahon. 1000); but it is to be borne 
in mind that when piles are driven to a solid resistance, they are frequently split, 
and consequently their resistance is much decreased. 

As a rule, the following coefficients for ordinary structures are submitted: 

When the piles are wholly free from vibration consequent upon external impulse, 
.35 to .4, and when the structures are heavy and exposed to irregular loading, as 
storehouses, etc., .15 to .2. 

Ordinarily, the bearing of a properly driven pile not less than 10 ins. in diam. may 
be taken at 10 tons. 

Friction, of Bottoms of Vessels. At a velocity of 7 knots per 
hour, a foul bottom requires 2.42 EP over that for a clean bottom. 

Friction of Planed. Brass Surfaces in muddy water is .4 pressure. 

Gas, Steam, and Hot-air Engines. Relative costs of gas, steam, 
and air engines per IP: Otto Gas engine, 8.75; Steam engine, 3.5; and Hot-air 
engine, 4. 

Heat. Available heat) _ __ * 643 * 535 __ _ 

expended per IIP per hour) — Total heat of combustion X Coefficient for fuel 
consumption of coal per IIP. 

Coal 14000X772 units = 10808000. Theoretical evaporative power =15 lbs. 

water. Efficiency of furnace =. 5; then 10 808 000 X • 5 = 5 404 000, and 43 1 5 j 5 

5 404 000 

= 3.04 lbs. per IIP per hour. 

Ice Boats, Speed of. Maj.-Gen. Z. B. Tower, U. S. A., assigns the speed of 
Ice boats at twice that of the wind, and the angle of sail, to attain greatest speed, 
to be less than 90 0 . 

japan Coal. Analysis of Bituminous. —Specific Gravity, 1.231. Carbon, 
77.59. Hydrogen, 5.28. Oxygen, 3.26. Nitrogen, 2.75. Sulphur, 1.65 Ash, 8.49. 
and loss, .98. 

Its evaporative effect = 4.16 lbs. water per lb. of coal. 










910 


MEMORANDA. 


Lee-waj\ A full modelled vessel, with an immersed section of i to 6 of her 
longitudinal section, and with an area of 36 sq. feet of sails to 1 of immersed sec¬ 
tion, will drift to leeward 1 mile in 6. A medium modelled vessel, with an im¬ 
mersed section of 1 to 8, and with like areas of sail and section, will drift 1 in 9. 

DigHt, Standard, of. Photometric , English.—Spermaceti candles, 6 per 
lb.; 120 grains per hour. Carcel burner = 9.5 candles. 

Locomotive .A-xles, Friction of. .016 of weight. Hence, if radius of 

wheel = .1, axle friction at periphery -f-10 = 3.73 at periphery. 

00 

MIercnrial Grange. To prevent freezing, apply or introduce Glycerine on 
top of column. 

Mietal Drodncts of XT. S., 1 SS 3 . Value, $222000000. 

Nlississippi Diver, Silt in. Near St. Charles the volume of silt 
borne per day in 1879 was 475457 cube yards, and on one day, July 3, it was 
4113 600. At times the volume equals 3 ozs. per cube foot of water. 

Motive Power. A sailing vessel having a length 6 times‘that of her 
breadth, requires, for a speed of 10 knots per hour, an impelling force of 48 lbs. per 
sq. foot of immersed section. 

Mowing Machine. Kirby's (Auburn, N. Y.)—670 lbs., 2 horses, 1 acre 
heavy clover in 46 min. 

Ordnance, Energy of. In a competitive test ot a 9-inch Woolwich 
gun, and a 5.75-inch Krupp, the energy per inch of circumference of bore was re¬ 
spectively 118 and 123 foot-tons; their penetration therefore by the wrought-iron 
standard being about the same, but their total energies were respectively 16400 
and 5800 foot-tons. 

At Mepper a shot of no lbs., with a velocity of 1749 feet per second, and a strik¬ 
ing energy of 2300 foot-tons, passed through a target composed of two plates of soft 
wrought iron 7 ins. thick, with 10 ins. of wood between them, and passed 800 yards 
beyond. 

Petroleum. One lb. crude oil heated 1 lb. water 315.75 0 — 28.21 lbs. water 
at 6o° converted to steam at 212°. Relative evaporative effects of Oil and Anthra¬ 
cite coal as 1 to 3.45. 

Dopnlation, Comparative Density of, and IN'nm'bei.' of 
Persons living in a House in different Cities. 

Chicago, 4; Baltimore and Naples, 4.5; Philadelphia, 6; London, Boston, and 
Cairo, 8; Marseilles, 9; Pekin, 10; Amsterdam, 11 ; New York, 13.5; Hamburg, 
17.07; Rome and Munich, 27; Paris, 29; Buda Pesth, 34.2; Madrid, 40; St. Peters¬ 
burg, 43.9; Vienna, 60.5; and in Berlin, 63. 

Dower of a Volcano. An eruption of that of Cotopaxi has projected 
a mass of rock of a volume of 100 cube yards a distance of 9 miles. 

Dower Deqnired to Draw a Vessel or Load up an In¬ 
clined Hydrostatic Dail or Slip Way. (Wm. Boyd, Eng.) 


W I — R; Cd¥-i-D = F; and P d' c —f W representing weight of vessel, or 
load and cradle , I inclination of ways, as length-r- rise, R resistance of vessel or load, 
¥ friction of cradle and rollers, and ffriction of plunger in stuffing-box, all in tons, 
C and c coefficients of friction of cradle and stuffing-box, d diameter of axle of rollers, 
d' product of circumference of plunger and depth of collar or stuffing, all in ins ., and 
¥ pressure per sq. inch on plunger, in lbs. 

rise 

Hence, W y c ~ g t ^ = I, and R + F +/=power in tons. 

Illustration. — Assume weight of a vessel and cradle 2000 tons, pressure on 
plunger 2500 lbs. per sq. inch, inclination of ways 1 in 20, diameters of axle of roll¬ 
ers and of rollers 3 and 10 ins., depth of collar 2 ins., and circumference of plunger 
50; what would be the power required? C = .2, and c = .6. 


Then 


2000 


, .2X3X2000 2500X2X50X.6 

: 100 tons; - - — i2o tons: - -— =1 

10 224.0 


and 100-f- 120-j-67 = 287 tons. 







MEMORANDA, 


91 I 


^Propeller Steamer, Ordinary Distritmtion of Power 
in a. Power developed by engine, 88 IIP; Power expended in its operation, 12. 

Per cent. 

Power expended by slip of propeller_ 14 

“ “ in propulsion. 71 


Per cent. 

Friction of load.7.5 

“ of propeller.7.5 


3 ?nmp, Centrifugal, has lifted water 28 to 29 feet, drawn it horizontally 800 
feet, and then lifted it 15 feet. Also drawn it 24 feet, and projected it 50 feet. 

Pail way Trains. Power and Resistance. — A railway train running at 

rate of 60 miles per hour = 88 feet per second, and velocity a body would acquire 
_ 2 

in falling from 88 feet = 88 - 4 - 8.02 = 120.3 f ee L Consequently, in addition to power 
expended in frictional and atmospheric resistance to train, as much power must be 
expended to put it in motion at this speed, as would lift it in mass to a height of 
121 feet in a second. 

If the train weighed 100 tons = 224000 lbs., then 224000 X 120.3 = 26747200 
foot-lbs., and if this result was obtained in a period of 5 minutes, it would require 
120.3 = 5 X 224000-4-33000 = 163.3 IP in addition to that required for frictional 
resistances. 

To raise the speed of a train from 40 (58.66 feet per second) to 45 (66 feet per sec¬ 
ond) miles per hour, the power required in addition to that of friction would be as 

_ 2 __ 2 

58.66 - 4 - 8.02 = 53.44 feet is to 66 - 4 - 8.02 = 67.57 feet = 67.57 — 53.44 = 14-13 feet. 

Assume a train of 100 tons, running at rate of 60 miles per hour, and total retard¬ 
ing power at. 1 its weight 100- 4 -10 = 10. Then 224 000 x 10X 120.3 = 26 947 200-4- 
22400= 1203 feet, which train would run before stopping. If, however, train was 
ascending a grade of 1 in 100, the retarding force = .11 (11 - 4 - 100) of weight = 
24640, distance in which train would come to rest would be 26947200=24640 = 
1093.6 feet. 

Pelative Nfon-condnctitoility of NEaterials. 


Material. 

Per cent. 

Hair felt. 

IOO 

Mineral wool, No. 2 

83.2 

“ “ and tar 

71 • 5 

Sawdust. 

68 


Material. 

Per cent. 

Material. 

Per cent. 

Mineral wool, No. 1 
Charcoal. 

6 7-5 

63.2 

55-3 

55 

Lime, slacked .... 
Asbestos . 

48 

36.3 

34-5 

13.6 

Pine wood. 

Coal ashes. 

Loam. 

Air space, 2 ins... 


Resistance to a Steam-vessel in Air and Water. In air 
10 per cent, of IIP, and in water, at a speed of 20 miles per hour, 90 per cent., or 8 
IIP per sq. foot of immersed amidship section. 

Saws, Circular. 30 ins. in diameter, are run at 2000 revolutions per minute 
= 3.57 miles. 

Spur Gear has been driven at a velocity of 1 mile per minute. 

Sugar IVIill Rollers. 5 feet by 28 ins., at 2.5 revolutions per minute, 
requires 20 IP, and 18 feet per minute is proper speed of such rolls. 

Surface Condensation, Experiments on. (B. G. Nichol.) 
Tube of Brass, .75 Inch External Diameter. No. 18 B W G. Surface — 1.0656 
sq. feet. Duration of Experiment , 20 Minutes. 

Horizontal. 


Steam. 

Vertical. 

Temperature. 


256° 

18.25 lbs. 

29- 9585 “ 

84-34 “ 

30 - 4375 “ 

Pressure per sq. inch per gauge... 

Condensation by tube surface_ 

“ per sq. ft. of “ per hour 
Condensed during experiment.... 

17.75^ lbs. 
18.5835 “ 
52.32 “ 

19.0625 “ 


253 0 

254 ° 

16.75 lbs. 

17.25 lbs. 

24.0835 “ 

43-0835 “ 

67.8 “ 

121.29 u 

24.5625 “ 

43-5625“ 


Steamers’ Engines, "Weights of. 
Fittings ready for Service per IIP. 

Mercantile steamer. 480 lbs. 

English Naval “ . 360 “ 


Engine, Boiler , Water, and all 


Light draught. 280 lbs. 

Torpedoes. 60 “ 


Ordinary Marine Boiler with Water.196 lbs. 

"Wind, Pressure of. Estimate of, upon Structures. — 30 lbs. per sq. foot. 
Per lineal foot of a locomotive train = 10 feet in height, 300 lbs. per sq. foot. 

A Tornado has developed a pressure of 93 lbs. per sq. foot. 










































912 


MEMORANDA. 


Via Suez Canal. Passages by Steamers. —1882, “ Stirling Castle Shang¬ 
hai to Gravesend, in 29 days 22 hours and 15 min., including 1 day 22 hours and 30 
min. in coaling and detentions. 

“ Glenaref Amoy to New York, N. Y., in 44 days and 12 hours, including deten¬ 
tion at Suez. From Gibraltar in n days. 

Zinc Foil in Steam-boilers. Zinc in an iron steam-boiler consti¬ 
tutes a voltaic element, which decomposes the water, liberating oxygen and hydro¬ 
gen. The oxygen combines with fatty acids and makes soap, which, coating the 
tubes, prevents the adhesion of the salts left by evaporation. The niealy deposit 
can then bo readily removed. 

Files. To Compute Extreme Load a Foundation Pile will Sustain. 

R2 ft 

p - ^ ^ ^ — L. R representing weight of ram, P weight of pile, and L extreme 

load, all in lbs.; h height of fall of ram, and s distance of depression of pile with last 
blows, both in feet. 


Illustration. —Assume a ram 1000 lbs. to fall 20 feet upon a pile of 400 lbs., 
what resistance will the earth bear, or what weight will the pile sustain when 
driven by the last blow, from a height of 20 feet, .5 inch? 

s = .5 of 12 ins. .0416. 


Then 


1000 2 X 26 
400 -f- 1000 X .0416 


20 000 000 
58.24 


== 343 4° 6 lbs. 


Perimeter. The limits or bounds of a figure, or sum of all its sides. 

Of a canal it is the length of the bottom and wet sides of its transverse section. 


Flood Wave. The flood wave of the Ohio River in March (1884) was 71 
feet 1 inch at Cincinnati, being higher than that of any previous record. 

Ice. Crushing Strength of, as determined by U. S. testing machine, ranged 
from 327 to 1000 lbs. per sq. inch. 





APPENDIX. 


9*3 


APPENDIX. 

River Steamboat. "Wood Side NWheels. 

Freight and. Passenger. 

“Bostona.”—Horizontal Lever Engines ( Non-condensing).—Length on deck, 
302 feet 10 ins.; beam, 43 feet 4 ins.; hold, 6 feet. Tons, 993.52. 

Immersed section of light draught of 26 ins., 83 sq.feet. Capacity for freight, 1200 
tons (2000 lbs.). 

Cylinders .—Two of 25 ins. in diam. by 8 feet stroke of piston. 

Boilers .—Four of steel, 47 ins. in diam. by 30 feet in length, 6 flues in each. 
Heating surface , 903 sq. feet. Grate surface , 98 sq. feet. 

Pressure of Steam, 154 lbs. per sq. inch, cut off at .625. 

Revolutions, -per minute. Speed, 10 miles per hour against current of upper 

Ohio, 3 to 5 miles. 

To Compute Meta-centre of Hull of a 'V'essel. 

Operation of Formula in Naval Architecture, page 660. 

Assume a sharp-modelled yacht, 45 feet in length, 13.5 feet beam, and 9.5 feet 
hold, with an immersed amidship section of 42 sq. feet, and a displacement of 900 
cube feet at a mean draught of water of 6 feet. 

2 p d x 

— / ■ — = Meta-centre. See pages 650, 659. 


Ordinates [dx) taken at intervals of 2.5 feet are as follows: 


y 3 = 

O = 

.O 

yS 3 = 

6.53 

= 287.496 

yl6 3 = 

3-25 

= 34.328 


.63 = 

.216 

y 98 — 

6.73 

= 300.763 

y x 7 3 = 

2.4 

= 13.824 

y 2 3 — 

I.3 3 = 

2.197 

y IC)3 — 

6.75 

— 3 ° 7-547 

y is3 = 

i -5 

— 3-375 

y 3 3 = 

2 3 = 

8 

yI * 3 = 

6-5 

= 287.496 

y x 9 = 

.8 

= .512 

y 4 3 =r 

00 

II 

21.952 

y 1 = 

6.25 

= 244.14 

y 2 ° 3 = 

O 

= .O 

ys 3 = 

3.63 = 

46.656 

y I3 ° = 

5-8 

—195.112 



2272.814 

y<> 3 = 

5 3 = 

125 

y^ 3 - 

5 

= 125 



2-5 

y 7 3 = 

5.83 = 

195.112 

y is3 = 

4.2 

= 74.088 



5682.035 


Summation of function of cubes of ordinates for value of fy^dx — 5682.035. 


, . 2 _ 5682.035 

And — of --— 


3 9 °° 3 

Note.— The other elements of this vessel are: 


of 6.31 = 4.21 feet. 


Area of load-line, 401.12 sq.feet; Displacement in weight, 27.974 tons ; do. at load- 
draught, .955 tons per inch; Depth of centre of gravity of displacement below load- 
line, 1.49 feet; Volume of displacement, to volume of immersed dimensions, 26.8 
per cent. 

To Compute Height of J"et in. a Conduit 3 ?ipe from a 
Constant Head. (Weisbach.) 


i+ { c+c 'i){V) 


v 2 V 

: — = h', and - — h' 

2 g z 


h, h', and h" representing heights 


due to velocity of efflux, loss of head and of ascent, l length of pipe or conduit, and d 
and d' diameters of pipe and jet, all in feet, v velocity of efflux in feet per second, C 
and C' coefficients of friction of inlet of pipe and outlet, and z a divisor determined 
by experiment with diameters of. 5 to 1.25 ins., ranging from 1.06 to 1.08. 

Illustration. —If conduit pipe for a fountain is 350 feet in length, and 2 ins. in 
diameter, to what height will a jet of .5 inch ascend under a head of 40 feet? 

Assume C and C' .8 and .5, h=z2$feet, d=z 2 ins. —.166, and .5 = .5-4-12 = .0416. 


Then 



25 _ 

350 ^ / .Q4*6 \4 
.1 66J \ .166/ 


4.9 feet. 


4 H 











9H 


APPENDIX. 


To Compute Head and. Discharge of Water in Pipes of 

Grreat Length. 

It becomes necessary first to determine the velocity of the flow, which is = 
a V V 

— - - — v — 1-273 —, independent of friction. V representing volume of water 

3.1416 d 2 d 2 

in cube feet, and d diameter of pipe in ins. 

V 2 g h 

When head, length, and diameter of pipe are given, 


= v. 


\j 1 + C + < 


Coefficients of friction C, for velocity of flow, range from .0234 to .0191 for veloci¬ 
ties from 3 to 13 feet per second, and c that for the pipe as a mean at .5. See Weis- 
bach’s Mechanics, Vol. i., page 431. 

Illustration.— What head must be given to a pipe 150 feet in length and 5 ins. 
in diameter, to discharge 25 cube feet of water per minute, and what velocity will 
it attain at that head? 02 4 andc= 5 

25 X 12 2 

Then 1.273 ^ ^ — 3 - — 1.273 X 2.4 = 3.055 feet velocity per second , and 


60 X 5 2 


(1 +-5 + -024 


150 X i2 \ 3.05s 2 
5 / 64.33 


= 1.5 -J- 8.64 X • 14 = 1.42 feet head. 


Vd 5 /( V 2 

Or, 4.72 — - = Y in cube feet per minute , and . 538 ?/ — - = d in ins: 

V 1-7- h V h 

Illustration.—A ssume elements of preceding case. 

'A' 25 -x = *5.(7 cube fid, and .538 


Then 4.72 


" 4 ‘ 72 


10.28 


V150 -I- x.42 
= • 53 8 X •^'69 607 =. 538 X 9.301 = 5 ins 


1.42 


To Compute Pall of a Canal or Open Conduit to Con¬ 
duct and Discharge a Given Volume of Water per 
Second. 

Coefficient of friction in such case is assumed by Du Buat and others at 
•007 565. 

C If x —- — h. h representing height of fall, l length of canal, and p net perime- 
A 2 g 

ter , all in feet; A area of section of canal in sq. feet, and v velocity of flow in feet 
per second. 

Illustration i. —What fall should be given to a canal with a section of 3 feet at 
bottom, 7 at top, and 3 in depth, and a length of 2600 feet, to conduct 40 cube feet 
of water per second ? _ 

_ n —l— o ^ o 

C — .0076, j9=:3 + (V3 2 + 22 x 2) = 10.21 feet, A = -- --= 15 sq. feet, and 

v — — =3 2.66 feet. 

15 

„ , 2600 X 10.21 2.66 s „ , , 

•Then .0076 - X >— 13-45 X .11 = 1.48 feet. 

15 64.33 

2.—What is volume of water conducted by a canal, with a section of 4 feet at 
bottom, 12 at top, and 5 in depth, with a fall of 3 feet, and a length of 5800 feet? 

X 2 gh = v. A=z 12 4 X 5 — 4 o sq. feet, and p = 4 -f (V5 2 -f 4 s X 2) — 


C ip 
16.8 feet. 


Then 


f 


40 


X 64.33 X 3 


j 40 

V 740.544 


X 193 = 3.23 feet, 


V .0076 x 5800 x 16.8 
40 X 3-23 feet velocity = 129.2 cube feet. 

For Dimensions of transverse profile of a canal, see Weisbach, page 492, vol. i. 


and 


























ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 915 

ORTHOGRAPHY OF TECHNICAL .WORDS AND TERMS. 

Orthography in ordinary use of following words and terms is so varied, 
that they are here given for the purpose of aiding in the establishment of 
a uniformity of expression. 

Abut. To meet, to adjoin to at the end, to border upon. Abut end of a log, etc., 
is that having the greatest diameter or side. 

But and Butt end, when applied in this manner, are corruptions. 

Adit. In Mining , the opening into a mine. 

Amidships. The middle or ceutre of a vessel, either fore and aft or athwartships. 
The amidship frame of a vessel is at 0 , and is termed dead flat. 

Arabesque. Applied to painted and carved or sculptured ornaments of imaginary 
foliage and animals, in which there are no perfect figures of either. Synonymous 
with Moresque. 

Arbor. The principal axis or spindle of a machine of revolution. 

Arris. A term in Mechanics, the line in which the two straight or curved sur¬ 
faces of a body, forming an exterior angle, meet each other. The edges of a body, 
as a brick, are arrises. 

Ashlar. In Masonry , stones roughly squared, or when faced. 

Athwart. Across, from side to side, transverse, across the line of a vessel’s 
course. 

Athwartships, reaching across a vessel, from side to side. 

Bagasse. Sugar-cane in its crushed state, as delivered from the rollers of a mill. 

Balk. In Carpentry , a piece of timber from 4 to 10 ins. square. 

Baluster. A small-column or pilaster; a collection of them, joined by a rail, forms 
a balustrade. 

Banister is a corruption of balustrade. 

Bark. A ship without a mizzen-topsail, and formerly a small ship. 

Bateau. A light boat, with great length proportionate to its beam, and wider at 
its centre than at its ends. 

Batten. In Carpentry , a piece of wood from 1 to 2.5 ins. thick, and from 1 to 7 
ins. in breadth. When less than 6 feet in length, it is termed a deal-end. 

Berme. In Fortifications and Engineering , a space of ground between a rampart 
and a moat or fosse, to arrest the ruins of a rampart. The level top of the embank¬ 
ment of a canal, opposite to and alike to the towpath. 

Bevel. A term for a plane having any other angle than 45 0 or 90 0 . 

Binnacle. The case in which the compass, or compasses (when two are used), is 
set on board of a vessel. 

Bit. The part of a bridle which is put into an animal’s mouth. In Carpentry, a 
boring instrument. 

Bitter End. The inboard end of a vessel’s cable abaft the bitts. 

Bitts. A vertical frame upon a deck of a vessel, around or upon which is secured 
cables, hawsers, sheets, etc. 

Bogie. Pivoted truck, to ease the running of an engine or car around a curve. 

Boomkin. A short spar projecting from the bow or quarter of a vessel, to extend 
the tack of a sail to windward. 

Bowlder. A stone rounded by natural attrition; a rounded mass of rock trans¬ 
ported from its original bed. 

Breast-summer. A lintel beam in the exterior wall of a building. 

Buhr-stone. A stone which is nearly pure silex, full of pores and cavities, and 
used for Mills. 

Bunting. Woolen texture of which colors and flags are made. 

Burden. A load. The quantity that a ship will carry. Hence burdensome. 

Cag. A small cask, differing from a barrel only in size. Commonly written Keg. 


916 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 

Caliber. An instrument with semi-circular legs, to measure diameters of spheres, 
or exterior and interior diameters of cylinders, bores, etc. 

A pair of Calibers is superfluous and improper. 

Calk. To stop seams and pay them with pitch, etc. To point an iron shoe so as 
to prevent its slipping. 

Cam. An irregular curved instrument, having its axis eccentric to the shaft 
upon winch it is lixed. 

Camber. To camber is to cut a beam or mold a structure archwise, as deck- 
beams of a vessel. 

Camboose. The stove or range in which the cooking in a vessel is effected. The 
cooking-room of a vessel; this term is usually confined to merchant vessels; in 
vessels of w r ar it is termed Galley. 

Camel. In Engineering , a decked vessel, having great stability, designed for use 
in the lifting of sunken vessels or structures. Also to transport loads of great 
weight or bulk. 

A Scow is open decked. 

Cantle. A fragment; a piece; the raised portion of the hind part of a saddle. 

Cantline. The space between the sides of two casks stow 7 ed aside of each other. 
When a cask is laid in the cantline of two others, it is said to be stowed bilge and 
cantline. 

Capstan. A vertical windlass. 

Caravel. A small vessel (of 25 or 30 tons’ burden) used upon the coast of France 
in herring fisheries. 

Carlings. Pieces of timber set fore and aft from the deck beams of a vessel, to 
receive the ends of the ledges in framing a deck. 

Carvel built .—A term applied to the manner of construction of small boats, to 
signify that the edges of their bottom planks are laid to each other like to the man¬ 
ner of planking vessels. Opposed to the term Clincher. 

Caster. A small phial or bottle for the table. Casters. Small wheels placed 
upon the legs of tables, etc., to allow them to be moved with facility. 

Catamaran. A small raft of logs, usually consisting of three, the centre one be¬ 
ing longer and wider than the others, and designed for use in an open roadstead 
and upon a sea-coast. 

Chamfer. A slope, groove, or small gutter cut in wood, metal, or stone. 

Chapelling. Wearing a ship around without bracing her fore yards. 

Chimney. The flue of a fireplace or furnace, constructed of masonry in houses 
and furnaces, and of metal, as in a steam boiler. See Pipe. 

Chinse. To chinse is to calk slightly with a knife or chisel. 

Chock. In Naval Architecture , small pieces of wood used to make good any de¬ 
ficiency in a piece of timber, frame, etc. See Furrings. 

Choice. To stop, to obstruct, to block up, to hinder, etc. 

Cleats. Pieces of wood or metal of various shapes, according to their uses, either 
to belay ropes upon, to resist or support weights or strains, as sheet , shoar, beam 
cleats, etc. 

Clincher built. A term applied to the construction of vessels’ bottoms, w r hen 
the lower edges of the planks overlay the next under them. 

Coak. A cylinder, cube, or triangle of hard wood let into the ends or faces of two 
pieces of timber to be secured together. The metallic eyes in a sheave through 
which the pin runs. In Naval Architecture , the oblong ridges banded on the masts 
of ships. 

Coamings. Raised borders around the edges of hatches. 

Coble. A small fishing-boat! 

Cocoon. The case wdiich certain insects make for a covering during the period 
of their metamorphosis to the pupa state. 


ORTHOGRAPHY OP TECHNICAL WORDS AND TERMS. 917 


Cog. In Mechanics , a short piece of wood or other material let into the faces of 
a body to impart motion to another. A term applied to a tooth in a wheel when it 
is made of a different material than that of the wheel. In Mining , an intrusion of 
matter into Assures of rocks, as when a mass of unstratified rocks appears to be in¬ 
jected into a rent in the stratified rocks. 

Cogging. In Carpentry , the cutting of a piece of timber so as to leave a part 
alike to a cog, and the notching of the upper piece so as to conform to and receive 
it. Alike to indenting or tabling. , 

Colter. The fore iron of a plough that cuts earth or sod. 

Compass. In Geometry , an instrument for describing circles, measuring figures, etc. 

A pair of Compasses is superfluous and improper. 

Connecting Rod. In Mechanics , the connection between a prime and secondary 
mover, as between the piston-rod of a steam-engine and the crank of a water-wheel 
or fly-wheel shaft. 

The term Pitman is local, and altogether inapplicable. 

Contrariwise. Conversely, opposite. Crossways is a corruption. 

Corridor. A gallery or passage in or around a building, connected with various 
departments, sometimes running within a quadrangle; it may be opened or enclosed. 
In Fortifications , a covert way. 

Cyma. A molding in a cornice. 

Damasquinerie. Inlaying in metal. 

Davit. A short boom fitted to hoist an anchor or boat. 

Deals. In Carpentry , the pieces of timber into which a log is cut or sawed up. 
Their usual thickness is 3 by 9 ins. and exceeding 6 feet in length. 

Improperly restricted to the wood of fir-trees. 

Dike. In Engineering , an embankment of greater length than breadth, imper¬ 
vious to water, and designed as a wall to a reservoir, a drain, or to resist the influx 
of a river or sea. 

Dingay (Nautical). A ship or vessel’s small boat. 

Dock. In Marine Architecture , an enclosure in a harbor or shore of a river, for 
the reception, repair, or security of vessels or timber. It may be wholly or only 
partially enclosed. See Pier. 

When applied to a single pier or jetty, it is a misapplication. 

Dowel. A pin of wood or metal inserted in the edge or face of two boards or 
pieces, so as to secure them together. 

This is very similar to coaking, but is used in a diminutive sense. An illustration of it is had in the 
manner a cooper secures two or more pieces in the head of a cask. 

Draught. A representation by delineation. The depth which a vessel or any 
floating body sinks into water. The act of drawing. A detachment of men from 
the main body, etc. 

Ordinarily written draft. 

Dutchman. In Mechanics , a piece of like material with the structure, let into a 
slack place, to cover slack or bad work. See Shim. 

Edgewise. An edge put into a particular direction. Hence endwise and sidewise 
have similar significations with reference to an end and a side. 

Edgeways is a corruption. 

Euphroe. A piece of wood by which the crowfoot of an awning is extended. 

Fault. In Mining , a break of strata, with displacement, which interrupts opera¬ 
tions. Also, fissures traversing the strata. 

Felloe , Felloes. The pieces of wood which form the rim of a wheel. 

Fetch. Length of a reservoir, pond, etc., along which the wind may blow towards 
the embankment or dam. 

Flange. A projection from an end or from the body of an instrument, or any 
part composing it, for the purpose of receiving, confining, or of securing it to a sup¬ 
port or to a second piece. 

Flier. In Carpentry , a straight line of steps in a stairway. 

Frap. To bind together with a rope, as to frap a fall, etc. 

4H* 


918 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 

Frieze. In Architecture , the part of the entablature of a column which is between 
the architrave and the cornice. 

Frustum. The part of a solid next the base, left by the removal of the top or 
segment. 

Frustrum, although used by some lexicographers, is erroneous. 

Furrings. Strips of timber or boards fastened to frames, joists, etc., in order to 
bring their faces to the required shape or level. 

Galeting. Putting galets into pointing-mortar or cement. 

Galets. Pieces of stone chipped off by the stroke of a chisel. See Spall. 

Galiot. A small galley built for speed, having one mast, and from 16 to 20 thwarts 
for rowers. A Dutch-constructed brigantine. 

Gate. In Mechanics , the hole through which molten metal is poured into a mold 
for casting. Geat aud Gett are corruptions. 

Gearing. A series of teeth or cogged wheels for transmitting motion. To gear a 
machine is to prepare to connect its parts as by an articulation. 

Gingle. To shake so as to produce a sharp, clattering noise, commonly Jingle. 

Girt. The circumference of a tree or piece of timber. Girth. The band or strap 
by which a saddle or burden is secured upon the back of an animal, by passing 
around his belly. In Printing , the bands of a press. 

Gnarled. Knotty. 

Grave. To clean a vessel’s bottom by burning. 

Graving. Burning off grass, shells, etc., from a ship’s bottom. Synonymous 
with Breaming. 

Grommet. A wreath or ring of rope. 

Gymbal Ring. A circular rynd for the connection of the upper mill-stone to the 
spindle by which the stone is suspended, so that it may vibrate upon all sides. 

Harpings. The fore part of the wales of a vessel which encompass her bows, 
and are fastened to the stem. Cat harpings, ropes which brace in the shrouds of 
the lower masts of a vessel. 

Hogging. A term applied to the hull of a vessel when her ends drop below her 
centre. See Sagging. 

Horsing. In Naval Architecture , calking with a large maul or beetle. 

Jam. To press, to crowd, to wedge in. In Nautical language , to squeeze tight. 

Jamb. A pier; the sides of an opening in a wall. 

Jib. The projecting beam of a crane from which the pulleys and weight are sus¬ 
pended. A sail in a vessel. 

Jibe. To shift a boom-sail from one tack to another; hence Jibing , the shifting 
of a boom. 

Jigging. Washing minerals in a sieve. 

Keelson. The timber within a vessel laid upon the middle of the floor timbers, 
and exactly over the keel. When located on the floors or at the sides, it is termed a 
sisters or a side keelson. 

Kerf. Slit made by cut of a saw. 

Kevel. Large wooden cleats to belay hawsers and ropes to, commonly Cavil. 

Lacquer. A spirituous solution of lac. To varnish with lacquer. 

Lagan. Articles sunk in the water with a buoy attached. 

Laitance. A pulpy, gelatinous fluid washed from the cement of concrete depos¬ 
ited in water. 

Lap-sided. A term expressive of the condition of a vessel or any body when it 
will not float or sit upright. 

Lay-to. To arrest headway of a vessel, without anchoring or securing her to a 
buoy, etc., as by counterbracing her yards, or stopping her engine. 

Leat. A trench to conduct water to or from a mill-wheel. 

Leech. In Nautical language, the perpendicular or slanting edge of a sail when 
not secured to a spar or stay. 


ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 9I9 

Luf The fullest part of the bow of a vessel. 

Mall. A large double-headed wooden hammer. 

Mantle. To expand, to spread. Mantelpiece. The shelf over a fireplace in front 
of a chimney. 

Marquetry. Checkered or inlaid work in wood. 

Matrass. A' chemical vessel with a body alike to an egg, and a tapering neck. 
Mattress. A quilted bed; a bed stuffed with hair, moss, etc., and quilted. 

Mitred. In Mechanics, cut to an angle of 45 0 , or two pieces joined so as to make 
a right angle. 

Mizzen-mast. The aftermost mast in a three-masted vessel. 

Mold. In Mechanics , a matrix in -which a casting is formed. A number of pieces 
of vellum or like substance, between which gold and silver are laid for the purpose 
of being beaten. Thin pieces of materials cut to curves or any required figure. In 
Naval Architecture , pieces of thin board cut to the lines of a vessel’s timbers, etc. 

Fine earth, such as constitutes soil. A substance which forms upon bodies in 
warm and confined damp air. 

This orthography is by analogy, as gold, sold, old, bold, cold, fold, etc. 

Molding. In Architecture , a projection beyond a wall, from a column, wainscot, etc 
Moresque. See Arabesque. 

Mortise. A hole cut in any material to receive the end or tenon of another piece. 
Muclc. A mass of dung in a moist state, or of dung and putrefied vegetable matter. 

Mullion. A vertical bar dividing the lights in a window ; the horizontal are 
termed transoms. 

Net. Clear of deductions, as net weight. 

Newel. An upright post, around wTiich winding stairs turn. 

Nigged. Stone hewed with a pick or pointed hammer instead of a chisel. 

Ogee. A molding with a concave and convex outline, like to an S. See Cyrna 
and Talon. 

Paillasse. Masonry raised upon a floor. A bed. 

Pargeting. In Architecture, rough plastering, alike to that upon chimneys. 
Parquetry. Inlaying of wood in figures. See Marquetry. 

Parral. The rope by -which a yard is secured to a mast at its centre. 

Pawl. The catch which stops, or holds, or falls on to a ratchet wheel. 

Peek. The upper or pointed corner of a sail extended by a gaff, or a yard set ob¬ 
liquely to a mast. To peek a yard is to point it perpendicularly to a mast. 

Pendant. A short rope over the head of a mast for the attachment of tackles 
thereto; a tackle, etc. 

Pennant. A small pointed flag. 

Pier. In Marine Architecture , a mole or jetty, projecting into a river or sea, to 
protect vessels from the sea, or for convenience of their lading. See Dock. 
Erroneously termed a Bock. 

Pile. In Engineering, spars pointed at one end and driven into soil to support a 
Superstructure or holdfast. Spile is a corruption. 

Pipe. In Mechanics , a metallic tube. The flue of a fireplace or furnace when 
constructed of metal; usually of a cylindrical form. 

The term or application of Stack (which refers solely to masonry) to a metallic pipe is a misappli¬ 
cation. 

Piragua. A small vessel with two masts and two boom-sails. 

Commonly termed Perry-augur. 

Pirogue. A canoe formed from a single log, propelled by paddles or by a sail, 
with the aid of an outrigger. 

Plastering. In Architecture, covering with plaster cement or mortar upon walls 
or laths. In England, termed laying, if in one or two coat wmrk; and pricking up, 
if in three-coat work. 

Plumber block. A bearing to receive and support the journal of a shaft. 

Polacre. Masts of one piece, without tops. 


920 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 


Poppets. In Naval Architecture , pieces of timber set perpendicular to a vessel’s 
bilge-ways, and extending to her bottom, to support her in launching. 

Porch. An arched vestibule at the entrance of a building. A vestibule supported 
by columns. A portico. 

Portico. A gallery near to the ground, the sides being open. A piazza encom¬ 
passed with arches supported by columns, where persons may walk; the roof may 
be flat or vaulted. 

Pozzuolana. A loose, porous, volcanic substance, composed of silicious, argilla¬ 
ceous, and calcareous earths and iron. 

Prize. In Mechanics , to raise with a lever. To pry and a pry are corruptions. 

Proa, Flying. A narrow canoe, the outer or lee side being nearly flat. A frame¬ 
work, projecting several feet to the windward side, supports a solid bearing, in the 
form of a canoe. Used in the Ladrone Islands. 

Purlin. In Carpentry, a piece of timber laid horizontal upon the rafters of a 
roof, to support the covering. 

Ramp. In Architecture , a flight of steps on a line tangential to the steps. A 
concave sweep connecting a higher and lower portion of a railing, wall, etc. A 
sloping line of a surface, as an inclined platform. 

Rarefaction. The act or process of distending bodies, by separating their parts 
and rendering them more rare or porous. It is opposed to Condensation. 

Rebate. In Mechanics, to pare down an edge of a board or a plate for the purpose 

of receiving another board or plate by lapping. To lap and unite edges of boards 
and plates. In Naval Architecture , the grooves in the side of the keel for receiving 
the garboard strake of plank. 

Commonly written Rabbet. 

Remou. Eddy water without progressive action, in bed of a river; a return of 
water against direction of flow of a river. 

Rendering. In Architecture , laying plaster or mortar upon mortar or walls. 
Rendered and Set refers to two coats or layers, and Rendered , Floated , and Set, to 
three coats or layers. 

Reniform. Kidney-shaped. 

Resin. The residuum of the distillation of turpentine. R os in is a corruption. 

Riband. In Naval Architecture, a long, narrow, flexible piece of timber. 

Rimer. A bit or boring tool for making a tapering hole. In Mechanics , to Rime 
is to bevel out a hole. Riming. The opening of the seams between the planks of a 
vessel for the purpose of calking them. 

Rotary. Turning upon an axis, as a wheel. 

Rynd. The metallic collar in the upper mill-stone by which it is connected to 
the spindle. 

Sagging. A term applied to the hull of a vessel when her centre drops below her 
ends. The converse of Hogging. 

Scallop. To mark or cut an edge into segments of circles. 

Scarcement. A set back in the face of a wall or in a bank of earth. A footing. 

Scarf. To join; to piece; to unite two pieces of timber at their ends by running 
the end of one over and upon the other, and bolting or securing them together. 

Scend. The settling of a vessel below the level of her keel. 

Selvagee. A strap made of rope-yarns, without being twisted or laid up, and re¬ 
tained in form by knotting it at intervals. 

Sennit. Braided cordage. 

Sewage. The matter borne off by a sewer. 

Sewed. In nautical language, the condition of a vessel aground; she is said to be 
sewed by as much as the difference in depth of water around her and her floating 
depth. 

Sewerage. The system of sewers. 

Shaky. Cracked or split, or as timber loosely put together. 

Shammy. Leather prepared from the skin of a chamois goat. 


ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 92 1 

Sheer. In Naval Architecture , the curve or bend of a ship’s deck or sides. To 
sheer , to slip or move aside. 

Sheers. Elevated spars connected at the upper ends, and used to elevate heavy 
bodies, as masts, etc. 

Shim. In Naval Architecture , a piece of wood or iron let into a slack place in a 
frame, plank, or plate to fill out to a fair surface or line. 

Shoal. A great multitude; a crowd; a multitude of fish. 

School is a corruption. 

Shoar. An oblique brace, the upper end resting against the substance to be sup¬ 
ported. 

Sholes. Pieces of plank under the heels of shoars, etc. 

Shoot. A passage-way on the side of a steep hill, down which wood, coal, etc., are 
thrown or slid. The artificial or natural contraction of a river. A young pig. 

Sidewise. See Edgewise. 

Signalled. Communicated by signals. 

Signalized, when applied to signals, is a misapplication of words. 

Sill. A piece of timber upon which a building rests; the horizontal piece of tim¬ 
ber or stone at the bottom of a framed case. 

Siphon. A curved tube or pipe designed to draw fluids out of vessels. 

Skeg. The extreme after-part of the keel of a vessel; the portion that supports 
the rudder-post. 

Slantwise. Oblique; not perpendicular. 

Sleek. To make smooth. Refuse; small coal. 

Sleeker. A spherical-shaped, curved, or plane-surfaced instrument with which to 
smooth surfaces. 

Slue. The turning of a substance upon an axis within its figure. 

Snying. A term applied to planks when their edges at their ends are curved or 
rounded upward, as a strake at the ends of a full-modelled vessel. 

Spall. A piece of stone, etc., chipped off by the stroke of a hammer or the force 
of a blow. Spalling , breaking up of ore into small pieces. 

Spandrel. In Architecture, the irregular triangular space between the outer lines 
Or extrados of an arch, a horizontal line drawn from its apex, and a vertical line 
from its springing. 

Sponson. An addition to the outer side of the hull of a steam vessel, commencing 
near the light water-line and running up to the wheel guards; applied for the pur¬ 
pose of shielding the deck-beams from the shock of a sea. 

Sponson-sided. The hull of a vessel is so termed when her frames have the out¬ 
line of a sponson, and the space afforded by the curvature is included in the hold. 

Sponding, Sponsing, etc., are corruptions. 

Squilgee. A wooden instrument, alike to a hoe, its edge faced with leather or 
vulcanized rubber, used to facilitate the drying of wet floors, or decks of a vessel. 

Stack. In Masonry , a number of chimneys or pipes standing together. The 
chimney of a blast furnace. 

The application of this word to the smoke-pipe of a steam-hoiler is wholly erroneous. 

Staqe. In Engineering , the interval or distance between two elevations, in shovel¬ 
ling, throwing, or lifting. 

Steering. The elevation of a vessel’s bowsprit, cathead, etc. 

Strake. A breadth of plank. 

Strut. An oblique brace to support a rafter. 

Style. The gnomon of a sun-dial. 

Sumy. In Mining , a pit or well into which water may be led from a mine or work. 

Surcingle. A belt, band, or girth, which passes over a saddle or blanket upon a 
horse’s back. 

Swage. To bear or force down. An instrument having a groove on its under 
side for the purpose of giving shape to any piece subjected to it when receiving a 
blow from a hammer. 


922 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 


Syphered. Overlapping the chamfered edge of one plank upon the chamfered 
edge of another in such a manner that the joint shall be a plane surface. 

Talus. In Architecture , the slope or batter of a wall, parapet, etc. In Geology , 
a sloping heap of rubble at foot of a cliff. 

Template. In Architecture, a wooden bearing to receive the end of a girder to 
distribute its weight. 

Templet. A mold cut to an exact section of any piece or structure. 

Tenon. The end of a piece of wood, cut into the form of a rectangular prism, de¬ 
signed to be set into a cavity of a like form in another piece, which is termed the 
mortise. 

Terring. The earth overlying a quarry. 

Tester. The top covering of a bedstead. 

Tholes. The pins in the gunwale of a boat which are used as rowlocks. 

Thwarts. The athwartship seats in a boat. 

Tide-rode. The situation of a vessel at anchor, when she rides in direction of the 
current instead of the wind. 

Tire. The metal hoop that binds the felloes of a wheel. 

Tompion. The stopper of a piece of ordnance. The iron bottom to which grape- 
shot are secured. 

Treenails. Wooden pins employed to secure the planking of a vessel to the 
frames. 

Trepan. In Mining , the instrument used in the comminution of rock in earth¬ 
boring at great depths. 

Trestle. The frame of a table; a movable form of support. In Mast-making, two 
pieces of timber set horizontally upon opposite sides of a mast-head. 

Trice. In Seamanship, to haul or tie up by means of a rope or tricing-line. 

Tue-iron or Tuyere. The nozzle of a bellows or blast-pipe in a forge or smelting- 
furnace. 

Vice. In Mechanics , a press to hold fast anything to he worked upon. 

Voyal. In Seamanship, a purchase applied to the weighing of an anchor, leading 
to a capstan. 

Wagon. An open or partially enclosed four-wheeled vehicle, adapted for the 
transportation of persons, goods, etc. 

Wear. In nautical language, to put a vessel upon a contrary tack by turning 
her around stem to the wind. 

Weir. A dam across a river or stream to arrest the water; a fence of twigs or 
stakes in a stream to divert the run of fish. 

Whipple-tree. The bar to which the traces of harness are fastened. 

Wind-rode. The situation of a vessel at anchor, when she rides in direction of 
the wind instead of the current. 

Windrow. A row or line of hay, etc., raked together. 

Withe. An instrument fitted to the end of a boom or mast, with a ring, through 
which a boom is rigged out or mast set up. 

Woold. To wind; particularly to bind a rope around a spar, etc. 


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